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Sample records for penalty discontinuous galerkin

  1. An optimal order interior penalty discontinuous Galerkin discretization of the compressible Navier Stokes equations

    NASA Astrophysics Data System (ADS)

    Hartmann, Ralf; Houston, Paul

    2008-11-01

    In this article we propose a new symmetric version of the interior penalty discontinuous Galerkin finite element method for the numerical approximation of the compressible Navier-Stokes equations. Here, particular emphasis is devoted to the construction of an optimal numerical method for the evaluation of certain target functionals of practical interest, such as the lift and drag coefficients of a body immersed in a viscous fluid. With this in mind, the key ingredients in the construction of the method include: (i) an adjoint consistent imposition of the boundary conditions; (ii) an adjoint consistent reformulation of the underlying target functional of practical interest; (iii) design of appropriate interior penalty stabilization terms. Numerical experiments presented within this article clearly indicate the optimality of the proposed method when the error is measured in terms of both the L2-norm, as well as for certain target functionals. Computational comparisons with other discontinuous Galerkin schemes proposed in the literature, including the second scheme of Bassi and Rebay, cf. [F. Bassi, S. Rebay, GMRES discontinuous Galerkin solution of the compressible Navier-Stokes equations, in: B. Cockburn, G. Karniadakis, C.-W. Shu (Eds.), Discontinuous Galerkin Methods, Lecture Notes in Comput. Sci. Engrg., vol. 11, Springer, Berlin, 2000, pp. 197-208; F. Bassi, S. Rebay, Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier-Stokes equations, Int. J. Numer. Methods Fluids 40 (2002) 197-207], the standard SIPG method outlined in [R. Hartmann, P. Houston, Symmetric interior penalty DG methods for the compressible Navier-Stokes equations. I: Method formulation, Int. J. Numer. Anal. Model. 3(1) (2006) 1-20], and an NIPG variant of the new scheme will be undertaken.

  2. A discontinuous Galerkin method with a modified penalty flux for the propagation and scattering of acousto-elastic waves

    NASA Astrophysics Data System (ADS)

    Ye, Ruichao; de Hoop, Maarten V.; Petrovitch, Christopher L.; Pyrak-Nolte, Laura J.; Wilcox, Lucas C.

    2016-05-01

    We develop an approach for simulating acousto-elastic wave phenomena, including scattering from fluid-solid boundaries, where the solid is allowed to be anisotropic, with the discontinuous Galerkin method. We use a coupled first-order elastic strain-velocity, acoustic velocity-pressure formulation, and append penalty terms based on interior boundary continuity conditions to the numerical (central) flux so that the consistency condition holds for the discretized discontinuous Galerkin weak formulation. We incorporate the fluid-solid boundaries through these penalty terms and obtain a stable algorithm. Our approach avoids the diagonalization into polarized wave constituents such as in the approach based on solving elementwise Riemann problems.

  3. FAST TRACK PAPER: The interior penalty discontinuous Galerkin method for elastic wave propagation: grid dispersion

    NASA Astrophysics Data System (ADS)

    De Basabe, Jonás D.; Sen, Mrinal K.; Wheeler, Mary F.

    2008-10-01

    Recently, there has been an increased interest in applying the discontinuous Galerkin method (DGM) to wave propagation. In this work, we investigate the applicability of the interior penalty DGM to elastic wave propagation by analysing it's grid dispersion properties, with particular attention to the effect that different basis functions have on the numerical dispersion. We consider different types of basis functions that naturally yield a diagonal mass matrix. This is relevant to seismology because a diagonal mass matrix is tantamount to an explicit and efficient time marching scheme. We find that the Legendre basis functions that are traditionally used in the DGM introduce numerical dispersion and anisotropy. Furthermore, we find that using Lagrange basis functions along with the Gauss nodes has attractive advantages for numerical wave propagation.

  4. An efficient Monte Carlo interior penalty discontinuous Galerkin method for elastic wave scattering in random media

    NASA Astrophysics Data System (ADS)

    Feng, X.; Lorton, C.

    2017-03-01

    This paper develops and analyzes an efficient Monte Carlo interior penalty discontinuous Galerkin (MCIP-DG) method for elastic wave scattering in random media. The method is constructed based on a multi-modes expansion of the solution of the governing random partial differential equations. It is proved that the mode functions satisfy a three-term recurrence system of partial differential equations (PDEs) which are nearly deterministic in the sense that the randomness only appears in the right-hand side source terms, not in the coefficients of the PDEs. Moreover, the same differential operator applies to all mode functions. A proven unconditionally stable and optimally convergent IP-DG method is used to discretize the deterministic PDE operator, an efficient numerical algorithm is proposed based on combining the Monte Carlo method and the IP-DG method with the $LU$ direct linear solver. It is shown that the algorithm converges optimally with respect to both the mesh size $h$ and the sampling number $M$, and practically its total computational complexity is only amount to solving very few deterministic elastic Helmholtz equations using the $LU$ direct linear solver. Numerically experiments are also presented to demonstrate the performance and key features of the proposed MCIP-DG method.

  5. Hybridized Multiscale Discontinuous Galerkin Methods for Multiphysics

    DTIC Science & Technology

    2015-09-14

    4302. Respondents should be aware that notwithstanding any other provision of law , no person shall be subject to any penalty for failing to comply with...ABSTRACT We have continued the development of the Hybridized Discontinuous Galerkin (HDG) method for the solution of systems of conservation laws . The...2015 Title: Hybridizable Discontinuous Galerkin (HDG) Methods for Systems of Conservation Laws In recent years, discontinuous Galerkin (DG) methods

  6. The Reverse Time Migration technique coupled with Interior Penalty Discontinuous Galerkin method.

    NASA Astrophysics Data System (ADS)

    Baldassari, C.; Barucq, H.; Calandra, H.; Denel, B.; Diaz, J.

    2009-04-01

    Seismic imaging is based on the seismic reflection method which produces an image of the subsurface from reflected waves recordings by using a tomography process and seismic migration is the industrial standard to improve the quality of the images. The migration process consists in replacing the recorded wavefields at their actual place by using various mathematical and numerical methods but each of them follows the same schedule, according to the pioneering idea of Claerbout: numerical propagation of the source function (propagation) and of the recorded wavefields (retropropagation) and next, construction of the image by applying an imaging condition. The retropropagation step can be realized accouting for the time reversibility of the wave equation and the resulting algorithm is currently called Reverse Time Migration (RTM). To be efficient, especially in three dimensional domain, the RTM requires the solution of the full wave equation by fast numerical methods. Finite element methods are considered as the best discretization method for solving the wave equation, even if they lead to the solution of huge systems with several millions of degrees of freedom, since they use meshes adapted to the domain topography and the boundary conditions are naturally taken into account in the variational formulation. Among the different finite element families, the spectral element one (SEM) is very interesting because it leads to a diagonal mass matrix which dramatically reduces the cost of the numerical computation. Moreover this method is very accurate since it allows the use of high order finite elements. However, SEM uses meshes of the domain made of quadrangles in 2D or hexaedra in 3D which are difficult to compute and not always suitable for complex topographies. Recently, Grote et al. applied the IPDG (Interior Penalty Discontinuous Galerkin) method to the wave equation. This approach is very interesting since it relies on meshes with triangles in 2D or tetrahedra in 3D

  7. An MPI/GPU parallelization of an interior penalty discontinuous Galerkin time domain method for Maxwell's equations

    NASA Astrophysics Data System (ADS)

    Dosopoulos, Stylianos; Gardiner, Judith D.; Lee, Jin-Fa

    2011-06-01

    In this paper we discuss our approach to the MPI/GPU implementation of an Interior Penalty Discontinuous Galerkin Time domain (IPDGTD) method to solve the time dependent Maxwell's equations. In our approach, we exploit the inherent DGTD parallelism and describe a combined MPI/GPU and local time stepping implementation. This combination is aimed at increasing efficiency and reducing computational time, especially for multiscale applications. The CUDA programming model was used, together with non-blocking MPI calls to overlap communications across the network. A 10× speedup compared to CPU clusters is observed for double precision arithmetic. Finally, for p = 1 basis functions, a good scalability with parallelization efficiency of 85% for up to 40 GPUs and 80% for up to 160 CPU cores was achieved on the Ohio Supercomputer Center's Glenn cluster.

  8. A multiscale discontinuous Galerkin method.

    SciTech Connect

    Bochev, Pavel Blagoveston; Scovazzi, Guglielmo; Hughes, Thomas J. R.

    2005-04-01

    We propose a new class of Discontinuous Galerkin (DG) methods based on variational multiscale ideas. Our approach begins with an additive decomposition of the discontinuous finite element space into continuous (coarse) and discontinuous (fine) components. Variational multiscale analysis is used to define an interscale transfer operator that associates coarse and fine scale functions. Composition of this operator with a donor DG method yields a new formulation that combines the advantages of DG methods with the attractive and more efficient computational structure of a continuous Galerkin method. The new class of DG methods is illustrated for a scalar advection-diffusion problem.

  9. Discontinuous Galerkin finite element solution for poromechanics

    NASA Astrophysics Data System (ADS)

    Liu, Ruijie

    This dissertation focuses on applying discontinuous Galerkin (DG) methods to poromechanics problems. A few challenges have been presented in traditional and popular continuous Galerkin (CG) finite element methods for solving complex coupled thermal, flow and solid mechanics. For example, nonphysical pore pressure oscillations often occur in CG solutions for poroelasticity problems with low permeability. A robust and practical numerical scheme for removing or alleviating the oscillation is not available. In modeling thermoporoelastoplasticity, CG methods require the use of very small time steps to obtain a convergent solution. The temperature profile predicted by CG methods in the fine mesh zones is often seriously polluted by large errors produced in coarse mesh zones in the case where the convection dominates the thermal process. The nonphysical oscillations in pore pressure and temperature solutions induced by CG methods at very early time stages seriously corrupt the solutions at longer time. We propose DG methods to handle these challenges because they are physics driven, provide local conservation of mass and momentum, have high stability and robustness, are locking-free, and because of their meshing and implementation capabilities. We first apply a family of DG methods, including Oden-Babuska-Baumann (OBB), Nonsymmetric Interior Penalty Galerkin (NIPG), Symmetric Interior Penalty Galerkin (SIPG) and Incomplete Interior Penalty Galerkin (IIPG), to 3D linear elasticity problems. This family of DG methods is tested and evaluated by using a cantilever beam problem with nearly incompressible materials. It is shown that DG methods are simple, robust and locking-free in dealing with nearly incompressible materials. Based on the success of DG methods in elasticity, we extend the DG theory into plasticity problems. A DG formulation has been implemented for solving 3D poroelasticity problems with low permeability. Numerical examples solved by DG methods demonstrate

  10. Discontinuous Galerkin Methods: Theory, Computation and Applications

    SciTech Connect

    Cockburn, B.; Karniadakis, G. E.; Shu, C-W

    2000-12-31

    This volume contains a survey article for Discontinuous Galerkin Methods (DGM) by the editors as well as 16 papers by invited speakers and 32 papers by contributed speakers of the First International Symposium on Discontinuous Galerkin Methods. It covers theory, applications, and implementation aspects of DGM.

  11. Modeling Storm Surges Using Discontinuous Galerkin Methods

    DTIC Science & Technology

    2016-06-01

    Abbreviations AMR Adaptive Mesh Refinement BE Backward Euler CG Continuous Galerkin Method CFL Courant -Friedrichs-Lewy Condition DG Discontinuous Galerkin Method...large time- steps whereas the explicit has to satisfy the Courant -Friedrichs-Lewy (CFL) condition. The CFL condition is a restriction on the maximum...time in- 17 tegrators are restricted by their time step. To determine the time step for explicit time integrators, we use the Courant -Friedrichs-Lewy

  12. Nodal discontinuous Galerkin methods on graphics processors

    NASA Astrophysics Data System (ADS)

    Klöckner, A.; Warburton, T.; Bridge, J.; Hesthaven, J. S.

    2009-11-01

    Discontinuous Galerkin (DG) methods for the numerical solution of partial differential equations have enjoyed considerable success because they are both flexible and robust: They allow arbitrary unstructured geometries and easy control of accuracy without compromising simulation stability. Lately, another property of DG has been growing in importance: The majority of a DG operator is applied in an element-local way, with weak penalty-based element-to-element coupling. The resulting locality in memory access is one of the factors that enables DG to run on off-the-shelf, massively parallel graphics processors (GPUs). In addition, DG's high-order nature lets it require fewer data points per represented wavelength and hence fewer memory accesses, in exchange for higher arithmetic intensity. Both of these factors work significantly in favor of a GPU implementation of DG. Using a single US$400 Nvidia GTX 280 GPU, we accelerate a solver for Maxwell's equations on a general 3D unstructured grid by a factor of around 50 relative to a serial computation on a current-generation CPU. In many cases, our algorithms exhibit full use of the device's available memory bandwidth. Example computations achieve and surpass 200 gigaflops/s of net application-level floating point work. In this article, we describe and derive the techniques used to reach this level of performance. In addition, we present comprehensive data on the accuracy and runtime behavior of the method.

  13. Parallel Implementation of the Discontinuous Galerkin Method

    NASA Technical Reports Server (NTRS)

    Baggag, Abdalkader; Atkins, Harold; Keyes, David

    1999-01-01

    This paper describes a parallel implementation of the discontinuous Galerkin method. Discontinuous Galerkin is a spatially compact method that retains its accuracy and robustness on non-smooth unstructured grids and is well suited for time dependent simulations. Several parallelization approaches are studied and evaluated. The most natural and symmetric of the approaches has been implemented in all object-oriented code used to simulate aeroacoustic scattering. The parallel implementation is MPI-based and has been tested on various parallel platforms such as the SGI Origin, IBM SP2, and clusters of SGI and Sun workstations. The scalability results presented for the SGI Origin show slightly superlinear speedup on a fixed-size problem due to cache effects.

  14. Adaptive Discontinuous Galerkin Methods in Multiwavelets Bases

    SciTech Connect

    Archibald, Richard K; Fann, George I; Shelton Jr, William Allison

    2011-01-01

    We use a multiwavelet basis with the Discontinuous Galerkin (DG) method to produce a multi-scale DG method. We apply this Multiwavelet DG method to convection and convection-diffusion problems in multiple dimensions. Merging the DG method with multiwavelets allows the adaptivity in the DG method to be resolved through manipulation of multiwavelet coefficients rather than grid manipulation. Additionally, the Multiwavelet DG method is tested on non-linear equations in one dimension and on the cubed sphere.

  15. Unstructured discontinuous Galerkin for seismic inversion.

    SciTech Connect

    van Bloemen Waanders, Bart Gustaaf; Ober, Curtis Curry; Collis, Samuel Scott

    2010-04-01

    This abstract explores the potential advantages of discontinuous Galerkin (DG) methods for the time-domain inversion of media parameters within the earth's interior. In particular, DG methods enable local polynomial refinement to better capture localized geological features within an area of interest while also allowing the use of unstructured meshes that can accurately capture discontinuous material interfaces. This abstract describes our initial findings when using DG methods combined with Runge-Kutta time integration and adjoint-based optimization algorithms for full-waveform inversion. Our initial results suggest that DG methods allow great flexibility in matching the media characteristics (faults, ocean bottom and salt structures) while also providing higher fidelity representations in target regions. Time-domain inversion using discontinuous Galerkin on unstructured meshes and with local polynomial refinement is shown to better capture localized geological features and accurately capture discontinuous-material interfaces. These approaches provide the ability to surgically refine representations in order to improve predicted models for specific geological features. Our future work will entail automated extensions to directly incorporate local refinement and adaptive unstructured meshes within the inversion process.

  16. General spline filters for discontinuous Galerkin solutions.

    PubMed

    Peters, Jörg

    2015-09-01

    The discontinuous Galerkin (dG) method outputs a sequence of polynomial pieces. Post-processing the sequence by Smoothness-Increasing Accuracy-Conserving (SIAC) convolution not only increases the smoothness of the sequence but can also improve its accuracy and yield superconvergence. SIAC convolution is considered optimal if the SIAC kernels, in the form of a linear combination of B-splines of degree d, reproduce polynomials of degree 2d. This paper derives simple formulas for computing the optimal SIAC spline coefficients for the general case including non-uniform knots.

  17. Spacetime Discontinuous Galerkin FEM: Spectral Response

    NASA Astrophysics Data System (ADS)

    Abedi, R.; Omidi, O.; Clarke, P. L.

    2014-11-01

    Materials in nature demonstrate certain spectral shapes in terms of their material properties. Since successful experimental demonstrations in 2000, metamaterials have provided a means to engineer materials with desired spectral shapes for their material properties. Computational tools are employed in two different aspects for metamaterial modeling: 1. Mircoscale unit cell analysis to derive and possibly optimize material's spectral response; 2. macroscale to analyze their interaction with conventional material. We compare two different approaches of Time-Domain (TD) and Frequency Domain (FD) methods for metamaterial applications. Finally, we discuss advantages of the TD method of Spacetime Discontinuous Galerkin finite element method (FEM) for spectral analysis of metamaterials.

  18. Discontinuous Galerkin for Stiff Hyperbolic Systems

    SciTech Connect

    Lowrie, R.B.; Morel, J.E.

    1999-06-27

    A Discontinuous Galerkin (DG) method is applied to hyperbolic systems that contain stiff relaxation terms. We demonstrate that when the relaxation time is under-resolved, DG is accurate in the sense that the method accurately represents the system's Chapman-Enskog (or ''diffusion'') approximation. Moreover, we demonstrate that a high-resolution, finite-volume method using the same time-integration method as DG is very inaccurate in the diffusion limit. Results for DG are presented for the hyperbolic heat equation, the Broadwell model of gas kinetics, and coupled radiation-hydrodynamics.

  19. Discontinuous Galerkin methods for extended hydrodynamics

    NASA Astrophysics Data System (ADS)

    Suzuki, Yoshifumi

    This dissertation presents a step towards high-order methods for continuum-transition flows. In order to achieve maximum accuracy and efficiency for numerical methods on a distorted mesh, it is desirable that both governing equations and corresponding numerical methods are in some sense compact. We argue our preference for a physical model described solely by first-order partial differential equations called hyperbolic-relaxation equations, and, among various numerical methods, for the discontinuous Galerkin method. Hyperbolic-relaxation equations can be generated as moments of the Boltzmann equation and can describe continuum-transition flows. Two challenging properties of hyperbolic-relaxation equations are the presence of a stiff source term, which drives the system towards equilibrium, and the accompanying change of eigenstructure. The first issue can be solved by an implicit treatment of the source term. To cope with the second difficulty, we develop a space-time discontinuous Galerkin method, based on Huynh's "upwind moment scheme." It is called the DG(1)--Hancock method. The DG(1)--Hancock method for one- and two-dimensional meshes is described, and Fourier analyses for both linear advection and linear hyperbolic-relaxation equations are conducted. The analyses show that the DG(1)--Hancock method is not only accurate but efficient in terms of turnaround time in comparison to other semi- and fully discrete finite-volume and discontinuous Galerkin methods. Numerical tests confirm the analyses, and also show the properties are preserved for nonlinear equations; the efficiency is superior by an order of magnitude. Subsequently, discontinuous Galerkin and finite-volume spatial discretizations are applied to more practical equations, in particular, to the set of 10-moment equations, which are gas dynamics equations that include a full pressure/temperature tensor among the flow variables. Results for flow around a micro-airfoil are compared to experimental data and

  20. General spline filters for discontinuous Galerkin solutions

    PubMed Central

    Peters, Jörg

    2015-01-01

    The discontinuous Galerkin (dG) method outputs a sequence of polynomial pieces. Post-processing the sequence by Smoothness-Increasing Accuracy-Conserving (SIAC) convolution not only increases the smoothness of the sequence but can also improve its accuracy and yield superconvergence. SIAC convolution is considered optimal if the SIAC kernels, in the form of a linear combination of B-splines of degree d, reproduce polynomials of degree 2d. This paper derives simple formulas for computing the optimal SIAC spline coefficients for the general case including non-uniform knots. PMID:26594090

  1. Adaptive Discontinuous Evolution Galerkin Method for Dry Atmospheric Flow

    DTIC Science & Technology

    2013-04-02

    standard one-dimensional approximate Riemann solver used for the flux integration demonstrate better stability, accuracy as well as reliability of the...discontinuous evolution Galerkin method for dry atmospheric convection. Comparisons with the standard one-dimensional approximate Riemann solver used...instead of a standard one- dimensional approximate Riemann solver, the flux integration within the discontinuous Galerkin method is now realized by

  2. A multiscale discontinuous Galerkin method with the computational structure of a continuous Galerkin method.

    SciTech Connect

    Buffa, Annalisa; Bochev, Pavel Blagoveston; Scovazzi, Guglielmo; Hughes, Thomas J. R.

    2005-03-01

    Proliferation of degrees-of-freedom has plagued discontinuous Galerkin methodology from its inception over 30 years ago. This paper develops a new computational formulation that combines the advantages of discontinuous Galerkin methods with the data structure of their continuous Galerkin counterparts. The new method uses local, element-wise problems to project a continuous finite element space into a given discontinuous space, and then applies a discontinuous Galerkin formulation. The projection leads to parameterization of the discontinuous degrees-of-freedom by their continuous counterparts and has a variational multiscale interpretation. This significantly reduces the computational burden and, at the same time, little or no degradation of the solution occurs. In fact, the new method produces improved solutions compared with the traditional discontinuous Galerkin method in some situations.

  3. A multiscale discontinuous galerkin method with the computational structure of a continuous galerkin method.

    SciTech Connect

    Sangalli, Giancarlo; Buffa, Annalisa; Bochev, Pavel Blagoveston; Scovazzi, Guglielmo; Hughes, Thomas J. R.

    2005-07-01

    Proliferation of degrees-of-freedom has plagued discontinuous Galerkin methodology from its inception over 30 years ago. This paper develops a new computational formulation that combines the advantages of discontinuous Galerkin methods with the data structure of their continuous Galerkin counterparts. The new method uses local, element-wise problems to project a continuous finite element space into a given discontinuous space, and then applies a discontinuous Galerkin formulation. The projection leads to parameterization of the discontinuous degrees-of-freedom by their continuous counterparts and has a variational multiscale interpretation. This significantly reduces the computational burden and, at the same time, little or no degradation of the solution occurs. In fact, the new method produces improved solutions compared with the traditional discontinuous Galerkin method in some situations.

  4. Discontinuous Galerkin Methods for Turbulence Simulation

    NASA Technical Reports Server (NTRS)

    Collis, S. Scott

    2002-01-01

    A discontinuous Galerkin (DG) method is formulated, implemented, and tested for simulation of compressible turbulent flows. The method is applied to turbulent channel flow at low Reynolds number, where it is found to successfully predict low-order statistics with fewer degrees of freedom than traditional numerical methods. This reduction is achieved by utilizing local hp-refinement such that the computational grid is refined simultaneously in all three spatial coordinates with decreasing distance from the wall. Another advantage of DG is that Dirichlet boundary conditions can be enforced weakly through integrals of the numerical fluxes. Both for a model advection-diffusion problem and for turbulent channel flow, weak enforcement of wall boundaries is found to improve results at low resolution. Such weak boundary conditions may play a pivotal role in wall modeling for large-eddy simulation.

  5. Discontinuous Galerkin computation of the Maxwell eigenvalues on simplicial meshes

    NASA Astrophysics Data System (ADS)

    Buffa, Annalisa; Houston, Paul; Perugia, Ilaria

    2007-07-01

    This paper is concerned with the discontinuous Galerkin approximation of the Maxwell eigenproblem. After reviewing the theory developed in [A. Buffa, I. Perugia, Discontinuous Galerkin approximation of the Maxwell eigenproblem, Technical Report 24-PV, IMATI-CNR, Pavia, Italy, 2005 ], we present a set of numerical experiments which both validate the theory, and provide further insight regarding the practical performance of discontinuous Galerkin methods, particularly in the case when non-conforming meshes, characterized by the presence of hanging nodes, are employed.

  6. Local discontinuous Galerkin approximations to Richards’ equation

    NASA Astrophysics Data System (ADS)

    Li, H.; Farthing, M. W.; Dawson, C. N.; Miller, C. T.

    2007-03-01

    We consider the numerical approximation to Richards' equation because of its hydrological significance and intrinsic merit as a nonlinear parabolic model that admits sharp fronts in space and time that pose a special challenge to conventional numerical methods. We combine a robust and established variable order, variable step-size backward difference method for time integration with an evolving spatial discretization approach based upon the local discontinuous Galerkin (LDG) method. We formulate the approximation using a method of lines approach to uncouple the time integration from the spatial discretization. The spatial discretization is formulated as a set of four differential algebraic equations, which includes a mass conservation constraint. We demonstrate how this system of equations can be reduced to the solution of a single coupled unknown in space and time and a series of local constraint equations. We examine a variety of approximations at discontinuous element boundaries, permeability approximations, and numerical quadrature schemes. We demonstrate an optimal rate of convergence for smooth problems, and compare accuracy and efficiency for a wide variety of approaches applied to a set of common test problems. We obtain robust and efficient results that improve upon existing methods, and we recommend a future path that should yield significant additional improvements.

  7. On cell entropy inequality for discontinuous Galerkin methods

    NASA Technical Reports Server (NTRS)

    Jiang, Guangshan; Shu, Chi-Wang

    1993-01-01

    We prove a cell entropy inequality for a class of high order discontinuous Galerkin finite element methods approximating conservation laws, which implies convergence for the one dimensional scalar convex case.

  8. A thermodynamically consistent discontinuous Galerkin formulation for interface separation

    SciTech Connect

    Versino, Daniele; Mourad, Hashem M.; Dávila, Carlos G.; Addessio, Francis L.

    2015-07-31

    Our paper describes the formulation of an interface damage model, based on the discontinuous Galerkin (DG) method, for the simulation of failure and crack propagation in laminated structures. The DG formulation avoids common difficulties associated with cohesive elements. Specifically, it does not introduce any artificial interfacial compliance and, in explicit dynamic analysis, it leads to a stable time increment size which is unaffected by the presence of stiff massless interfaces. This proposed method is implemented in a finite element setting. Convergence and accuracy are demonstrated in Mode I and mixed-mode delamination in both static and dynamic analyses. Significantly, numerical results obtained using the proposed interface model are found to be independent of the value of the penalty factor that characterizes the DG formulation. By contrast, numerical results obtained using a classical cohesive method are found to be dependent on the cohesive penalty stiffnesses. The proposed approach is shown to yield more accurate predictions pertaining to crack propagation under mixed-mode fracture because of the advantage. Furthermore, in explicit dynamic analysis, the stable time increment size calculated with the proposed method is found to be an order of magnitude larger than the maximum allowable value for classical cohesive elements.

  9. A thermodynamically consistent discontinuous Galerkin formulation for interface separation

    DOE PAGES

    Versino, Daniele; Mourad, Hashem M.; Dávila, Carlos G.; ...

    2015-07-31

    Our paper describes the formulation of an interface damage model, based on the discontinuous Galerkin (DG) method, for the simulation of failure and crack propagation in laminated structures. The DG formulation avoids common difficulties associated with cohesive elements. Specifically, it does not introduce any artificial interfacial compliance and, in explicit dynamic analysis, it leads to a stable time increment size which is unaffected by the presence of stiff massless interfaces. This proposed method is implemented in a finite element setting. Convergence and accuracy are demonstrated in Mode I and mixed-mode delamination in both static and dynamic analyses. Significantly, numerical resultsmore » obtained using the proposed interface model are found to be independent of the value of the penalty factor that characterizes the DG formulation. By contrast, numerical results obtained using a classical cohesive method are found to be dependent on the cohesive penalty stiffnesses. The proposed approach is shown to yield more accurate predictions pertaining to crack propagation under mixed-mode fracture because of the advantage. Furthermore, in explicit dynamic analysis, the stable time increment size calculated with the proposed method is found to be an order of magnitude larger than the maximum allowable value for classical cohesive elements.« less

  10. Discontinuous Galerkin Methods for Neutrino Radiation Transport

    NASA Astrophysics Data System (ADS)

    Endeve, Eirik; Hauck, Cory; Xing, Yulong; Mezzacappa, Anthony

    2015-04-01

    We are developing new computational methods for simulation of neutrino transport in core-collapse supernovae, which is challenging since neutrinos evolve from being diffusive in the proto-neutron star to nearly free streaming in the critical neutrino heating region. To this end, we consider conservative formulations of the Boltzmann equation, and aim to develop robust, high-order accurate methods. Runge-Kutta discontinuous Galerkin (DG) methods, offer several attractive properties, including (i) high-order accuracy on a compact stencil and (ii) correct asymptotic behavior in the diffusion limit. We have recently developed a new DG method for the advection part for the transport solve, which is high-order accurate and strictly preserves the physical bounds of the distribution function; i.e., f ∈ [ 0 , 1 ] . We summarize the main ingredients of our bound-preserving DG method and discuss ongoing work to include neutrino-matter interactions in the scheme. Research sponsored in part by Oak Ridge National Laboratory, managed by UT-Battelle, LLC for the U. S. Department of Energy

  11. Adaptive Discontinuous Galerkin Approximation to Richards' Equation

    NASA Astrophysics Data System (ADS)

    Li, H.; Farthing, M. W.; Miller, C. T.

    2006-12-01

    Due to the occurrence of large gradients in fluid pressure as a function of space and time resulting from nonlinearities in closure relations, numerical solutions to Richards' equations are notoriously difficult for certain media properties and auxiliary conditions that occur routinely in describing physical systems of interest. These difficulties have motivated a substantial amount of work aimed at improving numerical approximations to this physically important and mathematically rich model. In this work, we build upon recent advances in temporal and spatial discretization methods by developing spatially and temporally adaptive solution approaches based upon the local discontinuous Galerkin method in space and a higher order backward difference method in time. Spatial step-size adaption, h adaption, approaches are evaluated and a so-called hp-adaption strategy is considered as well, which adjusts both the step size and the order of the approximation. Solution algorithms are advanced and performance is evaluated. The spatially and temporally adaptive approaches are shown to be robust and offer significant increases in computational efficiency compared to similar state-of-the-art methods that adapt in time alone. In addition, we extend the proposed methods to two dimensions and provide preliminary numerical results.

  12. IMEX time marching for discontinuous Galerkin methods

    NASA Astrophysics Data System (ADS)

    Shu, Chi-Wang

    2017-07-01

    In this talk we give a short summary of our recent work [9, 10, 11, 7], jointly with H. Wang, Q. Zhang, S. Wang and Y. Liu, on the development, analysis and application of implicit-explicit (IMEX) Runge-Kutta or multi-step time marching methods for discontinuous Galerkin (DG) methods approximating convection diffusion equations. For such DG methods, explicit time marching is expensive since the time step is restricted by the square of the spatial mesh size, while fully implicit methods would require the solution of a non-symmetric, non-positive definite and possibly nonlinear system in each time step. The high order accurate IMEX Runge-Kutta or multi-step time marching would treat the diffusion term implicitly (which is often linear, resulting in a linear positive-definite solver) and the convection term (often nonlinear) explicitly, hence it can greatly improve computational efficiency. We prove that certain IMEX time discretizations, up to third order accuracy, coupled with local DG (LDG) method for the diffusion term treated implicitly, and regular DG method for the convection term treated explicitly, are unconditionally stable (the time step is upper bounded only by a constant depending on the diffusion coefficient but not on the spatial mesh size) and optimally convergent. The results also hold for drift-diffusion model in semiconductor device simulations, where a convection diffusion equation is coupled with an electrical potential equation. Numerical experiments confirm the good performance of such schemes.

  13. Discontinuous Galerkin schemes for nonstationary convection-diffusion problems

    NASA Astrophysics Data System (ADS)

    Dautov, R. Z.; Fedotov, E. M.

    2016-11-01

    In this paper we analyse an approximation of linear nonstationary convection- diffusion problem based on combination of discontinues Galerkin method for time discretisation in conjunction with hybridized discontinues Galerkin for spatial approximation. Such discrete schemes can be used for the solution of equations degenerating in the leading part and are formulated in term of solution of the problem, its gradient, the diffusional flux, and the restriction of the solution to the boundaries of elements. Conditions responsible for the solvability, stability and accuracy of the schemes are presented.

  14. Discontinuous Galerkin finite element methods for gradient plasticity.

    SciTech Connect

    Garikipati, Krishna.; Ostien, Jakob T.

    2010-10-01

    In this report we apply discontinuous Galerkin finite element methods to the equations of an incompatibility based formulation of gradient plasticity. The presentation is motivated with a brief overview of the description of dislocations within a crystal lattice. A tensor representing a measure of the incompatibility with the lattice is used in the formulation of a gradient plasticity model. This model is cast in a variational formulation, and discontinuous Galerkin machinery is employed to implement the formulation into a finite element code. Finally numerical examples of the model are shown.

  15. A Discontinuous Galerkin Chimera Overset Solver

    NASA Astrophysics Data System (ADS)

    Galbraith, Marshall Christopher

    This work summarizes the development of an accurate, efficient, and flexible Computational Fluid Dynamics computer code that is an improvement relative to the state of the art. The improved accuracy and efficiency is obtained by using a high-order discontinuous Galerkin (DG) discretization scheme. In order to maximize the computational efficiency, quadrature-free integration and numerical integration optimized as matrix-vector multiplications is employed and implemented through a pre-processor (PyDG). Using the PyDG pre-processor, a C++ polynomial library has been developed that uses overloaded operators to design an efficient Domain Specific Language (DSL) that allows expressions involving polynomials to be written as if they are scalars. The DSL, which makes the syntax of computer code legible and intuitive, promotes maintainability of the software and simplifies the development of additional capabilities. The flexibility of the code is achieved by combining the DG scheme with the Chimera overset method. The Chimera overset method produces solutions on a set of overlapping grids that communicate through an exchange of data on grid boundaries (known as artificial boundaries). Finite volume and finite difference discretizations use fringe points, which are layers of points on the artificial boundaries, to maintain the interior stencil on artificial boundaries. The fringe points receive solution values interpolated from overset grids. Proper interpolation requires fringe points to be contained in overset grids. Insufficient overlap must be corrected by modifying the grid system. The Chimera scheme can also exclude regions of grids that lie outside the computational domain; a process commonly known as hole cutting. The Chimera overset method has traditionally enabled the use of high-order finite difference and finite volume approaches such as WENO and compact differencing schemes, which require structured meshes, for modeling fluid flow associated with complex

  16. Discontinuous Galerkin for Hyperbolic Systems with Stiff Relaxation

    SciTech Connect

    Lowrie, R.B.; Morel, J.E.

    1999-05-24

    A Discontinuous Galerkin method is applied to hyperbolic systems that contain stiff relaxation terms. We demonstrate that when the relaxation time is unresolved, the method is accurate in the sense that it accurately represents the system's Chapman-Enskog approximation. Results are presented for the hyperbolic heat equation and coupled radiation-hydrodynamics.

  17. Error Analysis for Discontinuous Galerkin Method for Parabolic Problems

    NASA Technical Reports Server (NTRS)

    Kaneko, Hideaki

    2004-01-01

    In the proposal, the following three objectives are stated: (1) A p-version of the discontinuous Galerkin method for a one dimensional parabolic problem will be established. It should be recalled that the h-version in space was used for the discontinuous Galerkin method. An a priori error estimate as well as a posteriori estimate of this p-finite element discontinuous Galerkin method will be given. (2) The parameter alpha that describes the behavior double vertical line u(sub t)(t) double vertical line 2 was computed exactly. This was made feasible because of the explicitly specified initial condition. For practical heat transfer problems, the initial condition may have to be approximated. Also, if the parabolic problem is proposed on a multi-dimensional region, the parameter alpha, for most cases, would be difficult to compute exactly even in the case that the initial condition is known exactly. The second objective of this proposed research is to establish a method to estimate this parameter. This will be done by computing two discontinuous Galerkin approximate solutions at two different time steps starting from the initial time and use them to derive alpha. (3) The third objective is to consider the heat transfer problem over a two dimensional thin plate. The technique developed by Vogelius and Babuska will be used to establish a discontinuous Galerkin method in which the p-element will be used for through thickness approximation. This h-p finite element approach, that results in a dimensional reduction method, was used for elliptic problems, but the application appears new for the parabolic problem. The dimension reduction method will be discussed together with the time discretization method.

  18. An Application of the Quadrature-Free Discontinuous Galerkin Method

    NASA Technical Reports Server (NTRS)

    Lockard, David P.; Atkins, Harold L.

    2000-01-01

    The process of generating a block-structured mesh with the smoothness required for high-accuracy schemes is still a time-consuming process often measured in weeks or months. Unstructured grids about complex geometries are more easily generated, and for this reason, methods using unstructured grids have gained favor for aerodynamic analyses. The discontinuous Galerkin (DG) method is a compact finite-element projection method that provides a practical framework for the development of a high-order method using unstructured grids. Higher-order accuracy is obtained by representing the solution as a high-degree polynomial whose time evolution is governed by a local Galerkin projection. The traditional implementation of the discontinuous Galerkin uses quadrature for the evaluation of the integral projections and is prohibitively expensive. Atkins and Shu introduced the quadrature-free formulation in which the integrals are evaluated a-priori and exactly for a similarity element. The approach has been demonstrated to possess the accuracy required for acoustics even in cases where the grid is not smooth. Other issues such as boundary conditions and the treatment of non-linear fluxes have also been studied in earlier work This paper describes the application of the quadrature-free discontinuous Galerkin method to a two-dimensional shear layer problem. First, a brief description of the method is given. Next, the problem is described and the solution is presented. Finally, the resources required to perform the calculations are given.

  19. An Application of the Quadrature-Free Discontinuous Galerkin Method

    NASA Technical Reports Server (NTRS)

    Lockard, David P.; Atkins, Harold L.

    2000-01-01

    The process of generating a block-structured mesh with the smoothness required for high-accuracy schemes is still a time-consuming process often measured in weeks or months. Unstructured grids about complex geometries are more easily generated, and for this reason, methods using unstructured grids have gained favor for aerodynamic analyses. The discontinuous Galerkin (DG) method is a compact finite-element projection method that provides a practical framework for the development of a high-order method using unstructured grids. Higher-order accuracy is obtained by representing the solution as a high-degree polynomial whose time evolution is governed by a local Galerkin projection. The traditional implementation of the discontinuous Galerkin uses quadrature for the evaluation of the integral projections and is prohibitively expensive. Atkins and Shu introduced the quadrature-free formulation in which the integrals are evaluated a-priori and exactly for a similarity element. The approach has been demonstrated to possess the accuracy required for acoustics even in cases where the grid is not smooth. Other issues such as boundary conditions and the treatment of non-linear fluxes have also been studied in earlier work This paper describes the application of the quadrature-free discontinuous Galerkin method to a two-dimensional shear layer problem. First, a brief description of the method is given. Next, the problem is described and the solution is presented. Finally, the resources required to perform the calculations are given.

  20. Computational aeroacoustics applications based on a discontinuous Galerkin method

    NASA Astrophysics Data System (ADS)

    Delorme, Philippe; Mazet, Pierre; Peyret, Christophe; Ventribout, Yoan

    2005-09-01

    CAA simulation requires the calculation of the propagation of acoustic waves with low numerical dissipation and dispersion error, and to take into account complex geometries. To give, at the same time, an answer to both challenges, a Discontinuous Galerkin Method is developed for Computational AeroAcoustics. Euler's linearized equations are solved with the Discontinuous Galerkin Method using flux splitting technics. Boundary conditions are established for rigid wall, non-reflective boundary and imposed values. A first validation, for induct propagation is realized. Then, applications illustrate: the Chu and Kovasznay's decomposition of perturbation inside uniform flow in term of independent acoustic and rotational modes, Kelvin-Helmholtz instability and acoustic diffraction by an air wing. To cite this article: Ph. Delorme et al., C. R. Mecanique 333 (2005).

  1. A Streaming Language Implementation of the Discontinuous Galerkin Method

    NASA Technical Reports Server (NTRS)

    Barth, Timothy; Knight, Timothy

    2005-01-01

    We present a Brook streaming language implementation of the 3-D discontinuous Galerkin method for compressible fluid flow on tetrahedral meshes. Efficient implementation of the discontinuous Galerkin method using the streaming model of computation introduces several algorithmic design challenges. Using a cycle-accurate simulator, performance characteristics have been obtained for the Stanford Merrimac stream processor. The current Merrimac design achieves 128 Gflops per chip and the desktop board is populated with 16 chips yielding a peak performance of 2 Teraflops. Total parts cost for the desktop board is less than $20K. Current cycle-accurate simulations for discretizations of the 3-D compressible flow equations yield approximately 40-50% of the peak performance of the Merrimac streaming processor chip. Ongoing work includes the assessment of the performance of the same algorithm on the 2 Teraflop desktop board with a target goal of achieving 1 Teraflop performance.

  2. Development of Discontinuous Galerkin Method for the Linearized Euler Equations

    DTIC Science & Technology

    2003-02-01

    ESktbkX-(9) i=1 k=O Since the LEE are linear, Fj(Uh) is expanded in a natural way as can be seen from Eq.(7). Furthermore, Atkins and Lockard [5...Discontinuous Galerkin method for Hyperbolic Equations, AIAA Journal, Vol. 36, pp. 775-782, 1998. [5] H.L. Atkins and D.P. Lockard , A High-Order Method using

  3. Discontinuous Galerkin methods for Hamiltonian ODEs and PDEs

    NASA Astrophysics Data System (ADS)

    Tang, Wensheng; Sun, Yajuan; Cai, Wenjun

    2017-02-01

    In this article, we present a unified framework of discontinuous Galerkin (DG) discretizations for Hamiltonian ODEs and PDEs. We show that with appropriate numerical fluxes the numerical algorithms deduced from DG discretizations can be combined with the symplectic methods in time to derive the multi-symplectic PRK schemes. The resulting numerical discretizations are applied to the linear and nonlinear Schrödinger equations. Some conservative properties of the numerical schemes are investigated and confirmed in the numerical experiments.

  4. Fluorescence lifetime optical tomography with Discontinuous Galerkin discretisation scheme.

    PubMed

    Soloviev, Vadim Y; D'Andrea, Cosimo; Mohan, P Surya; Valentini, Gianluca; Cubeddu, Rinaldo; Arridge, Simon R

    2010-09-20

    We develop discontinuous Galerkin framework for solving direct and inverse problems in fluorescence diffusion optical tomography in turbid media. We show the advantages and the disadvantages of this method by comparing it with previously developed framework based on the finite volume discretization. The reconstruction algorithm was used with time-gated experimental dataset acquired by imaging a highly scattering cylindrical phantom concealing small fluorescent tubes. Optical parameters, quantum yield and lifetime were simultaneously reconstructed. Reconstruction results are presented and discussed.

  5. Spectral element discontinuous Galerkin simulations for wake potential calculations : NEKCEM.

    SciTech Connect

    Min, M.; Fischer, P. F.; Chae, Y.-C.

    2008-01-01

    In this paper we present high-order spectral element discontinuous Galerkin simulations for wake field and wake potential calculations. Numerical discretizations are based on body-conforming hexagonal meshes on Gauss-Lobatto-Legendre grids. We demonstrate wake potential profiles for cylindrically symmetric cavity structures in 3D, including the cases for linear and quadratic transitions between two cross sections. Wake potential calculations are carried out on 2D surfaces for various bunch sizes.

  6. Application of the Discontinuous Galerkin Method to Acoustic Scatter Problems

    NASA Technical Reports Server (NTRS)

    Atkins, H. L.

    1997-01-01

    The application of the quadrature-free form of the discontinuous Galerkin method to two problems from Category 1 of the Second Computational Aeroacoustics Workshop on Benchmark problems is presented. The method and boundary conditions relevant to this work are described followed by two test problems, both of which involve the scattering of an acoustic wave off a cylinder. The numerical test performed to evaluate mesh-resolution requirements and boundary-condition effectiveness are also described.

  7. Application of the Discontinuous Galerkin Method to Acoustic Scatter Problems

    NASA Technical Reports Server (NTRS)

    Atkins, H. L.

    1997-01-01

    The application of the quadrature-free form of the discontinuous Galerkin method to two problems from Category 1 of the Second Computational Aeroacoustics Workshop on Benchmark problems is presented. The method and boundary conditions relevant to this work are described followed by two test problems, both of which involve the scattering of an acoustic wave off a cylinder. The numerical test performed to evaluate mesh-resolution requirements and boundary-condition effectiveness are also described.

  8. Discontinuous Galerkin Finite Element Method for Parabolic Problems

    NASA Technical Reports Server (NTRS)

    Kaneko, Hideaki; Bey, Kim S.; Hou, Gene J. W.

    2004-01-01

    In this paper, we develop a time and its corresponding spatial discretization scheme, based upon the assumption of a certain weak singularity of parallel ut(t) parallel Lz(omega) = parallel ut parallel2, for the discontinuous Galerkin finite element method for one-dimensional parabolic problems. Optimal convergence rates in both time and spatial variables are obtained. A discussion of automatic time-step control method is also included.

  9. Polymorphic nodal elements and their application in discontinuous Galerkin methods

    NASA Astrophysics Data System (ADS)

    Gassner, Gregor J.; Lörcher, Frieder; Munz, Claus-Dieter; Hesthaven, Jan S.

    2009-03-01

    In this work, we discuss two different but related aspects of the development of efficient discontinuous Galerkin methods on hybrid element grids for the computational modeling of gas dynamics in complex geometries or with adapted grids. In the first part, a recursive construction of different nodal sets for hp finite elements is presented. They share the property that the nodes along the sides of the two-dimensional elements and along the edges of the three-dimensional elements are the Legendre-Gauss-Lobatto points. The different nodal elements are evaluated by computing the Lebesgue constants of the corresponding Vandermonde matrix. In the second part, these nodal elements are applied within the modal discontinuous Galerkin framework. We still use a modal based formulation, but introduce a nodal based integration technique to reduce computational cost in the spirit of pseudospectral methods. We illustrate the performance of the scheme on several large scale applications and discuss its use in a recently developed space-time expansion discontinuous Galerkin scheme.

  10. Entropy-bounded discontinuous Galerkin scheme for Euler equations

    NASA Astrophysics Data System (ADS)

    Lv, Yu; Ihme, Matthias

    2015-08-01

    An entropy-bounded Discontinuous Galerkin (EBDG) scheme is proposed in which the solution is regularized by constraining the entropy. The resulting scheme is able to stabilize the solution in the vicinity of discontinuities and retains the optimal accuracy for smooth solutions. The properties of the limiting operator according to the entropy-minimum principle are proofed, and an optimal CFL-criterion is derived. We provide a rigorous description for locally imposing entropy constraints to capture multiple discontinuities. Significant advantages of the EBDG-scheme are the general applicability to arbitrary high-order elements and its simple implementation for multi-dimensional configurations. Numerical tests confirm the properties of the scheme, and particular focus is attributed to the robustness in treating discontinuities on arbitrary meshes.

  11. A discontinuous Galerkin method for unsteady two-dimensional convective flows

    NASA Astrophysics Data System (ADS)

    Aristotelous, A. C.; Papanicolaou, N. C.

    2016-10-01

    We develop a High-Order Symmetric Interior Penalty (SIP) Discontinuous Galerkin (DG) Finite Element Method (FEM) to investigate two-dimensional in space natural convective flows in a vertical cavity. The physical problem is modeled by a coupled nonlinear system of partial differential equations and admits various solutions including stable and unstable modes in the form of traveling and/or standing waves, depending on the governing parameters. These flows are characterized by steep boundary and internal layers which evolve with time and can be well-resolved by high-order methods that also are adept to adaptive meshing. The standard no-slip boundary conditions which apply on the lateral walls, and the periodic conditions prescribed on the upper and lower boundaries, present additional challenges. The numerical scheme proposed herein is shown to successfully address these issues and furthermore, large Prandtl number values can be handled naturally. Discontinuous source terms and coefficients are an innate feature of multiphase flows involving heterogeneous fluids and will be a topic of subsequent work. Spatially adaptive Discontinuous Galerkin Finite Elements are especially suited to such problems.

  12. Discontinuous Galerkin Methods and Local Time Stepping for Wave Propagation

    SciTech Connect

    Grote, M. J.; Mitkova, T.

    2010-09-30

    Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical simulation of time dependent wave phenomena. To overcome that stability restriction, local time-stepping methods are developed, which allow arbitrarily small time steps precisely where small elements in the mesh are located. When combined with a discontinuous Galerkin finite element discretization in space, which inherently leads to a diagonal mass matrix, the resulting numerical schemes are fully explicit. Starting from the classical Adams-Bashforth multi-step methods, local time stepping schemes of arbitrarily high accuracy are derived. Numerical experiments validate the theory and illustrate the usefulness of the proposed time integration schemes.

  13. Local Analysis of Shock Capturing Using Discontinuous Galerkin Methodology

    NASA Technical Reports Server (NTRS)

    Atkins, H. L.

    1997-01-01

    The compact form of the discontinuous Galerkin method allows for a detailed local analysis of the method in the neighborhood of the shock for a non-linear model problem. Insight gained from the analysis leads to new flux formulas that are stable and that preserve the compactness of the method. Although developed for a model equation, the flux formulas are applicable to systems such as the Euler equations. This article presents the analysis for methods with a degree up to 5. The analysis is accompanied by supporting numerical experiments using Burgers' equation and the Euler equations.

  14. GPU-accelerated discontinuous Galerkin methods on hybrid meshes

    NASA Astrophysics Data System (ADS)

    Chan, Jesse; Wang, Zheng; Modave, Axel; Remacle, Jean-Francois; Warburton, T.

    2016-08-01

    We present a time-explicit discontinuous Galerkin (DG) solver for the time-domain acoustic wave equation on hybrid meshes containing vertex-mapped hexahedral, wedge, pyramidal and tetrahedral elements. Discretely energy-stable formulations are presented for both Gauss-Legendre and Gauss-Legendre-Lobatto (Spectral Element) nodal bases for the hexahedron. Stable timestep restrictions for hybrid meshes are derived by bounding the spectral radius of the DG operator using order-dependent constants in trace and Markov inequalities. Computational efficiency is achieved under a combination of element-specific kernels (including new quadrature-free operators for the pyramid), multi-rate timestepping, and acceleration using Graphics Processing Units.

  15. Parallel adaptive discontinuous Galerkin approximation for thin layer avalanche modeling

    NASA Astrophysics Data System (ADS)

    Patra, A. K.; Nichita, C. C.; Bauer, A. C.; Pitman, E. B.; Bursik, M.; Sheridan, M. F.

    2006-08-01

    This paper describes the development of highly accurate adaptive discontinuous Galerkin schemes for the solution of the equations arising from a thin layer type model of debris flows. Such flows have wide applicability in the analysis of avalanches induced by many natural calamities, e.g. volcanoes, earthquakes, etc. These schemes are coupled with special parallel solution methodologies to produce a simulation tool capable of very high-order numerical accuracy. The methodology successfully replicates cold rock avalanches at Mount Rainier, Washington and hot volcanic particulate flows at Colima Volcano, Mexico.

  16. Properties of Discontinuous Galerkin Algorithms and Implications for Edge Gyrokinetics

    NASA Astrophysics Data System (ADS)

    Hammett, G. W.; Hakim, A.; Shi, E. L.; Abel, I. G.; Stoltzfus-Dueck, T.

    2015-11-01

    The continuum gyrokinetic code Gkeyll uses Discontinuous Galerkin (DG) algorithms, which have a lot of flexibility in the choice of basis functions and inner product norm that can be useful in designing algorithms for particular problems. Rather than use regular polynomial basis functions, we consider here Maxwellian-weighted basis functions (which have similarities to Gaussian radial basis functions). The standard Galerkin approach loses particle and energy conservation, but this can be restored with a particular weight for the inner product (this is equivalent to a Petrov-Galerkin method). This allows a full- F code to have some benefits similar to the Gaussian quadrature used in gyrokinetic δf codes to integrate Gaussians times some polynomials exactly. In tests of Gkeyll for electromagnetic fluctuations, we found it is important to use consistent basis functions where the potential is in a higher-order continuity subspace of the space for the vector potential A| |. A regular projection method to this subspace is a non-local operation, while we show a self-adjoint averaging operator that can preserve locality and energy conservation. This does not introduce damping, but like gyro-averaging involves only the reactive part of the dynamics. Supported by the Max-Planck/Princeton Center for Plasma Physics, the SciDAC Center for the Study of Plasma Microturbulence, and DOE Contract DE-AC02-09CH11466.

  17. A discontinuous Galerkin method for studying elasticity and variable viscosity Stokes problems

    NASA Astrophysics Data System (ADS)

    Schnepp, Sascha M.; Charrier, Dominic; May, Dave

    2015-04-01

    Traditionally in the geodynamics community, staggered grid finite difference schemes and mixed Finite Elements (FE) have been utilised to discretise the variable viscosity (VV) Stokes problem. These methods have been demonstrated to be sufficiently robust and accurate for a wide range of variable viscosity problems involving both smooth viscosity structures possessing large spatial variations, and for discontinuous viscosity structures. One caveat of the aforementioned discretisations is that they tend to have inf-sup constants which are highly dependent on the cell aspect ratio. Whilst high order mixed FE approaches utilising spaces defined via Qk - Qk-2, k ≥ 3, alleviate this shortcoming, such elements are seldomly used as they are computationally expensive, the definition of multi-level preconditioners is complex, and spectral accuracy is often not obtained. Discontinuous Galerkin (DG) methods offer the advantage that spaces can be constructed which have both low order in velocity and pressure and inf-sup constants which are not sensitive to the element aspect ratio. To date, DG discretisations have not been extensively used within geodynamic applications associated with VV Stokes formulations. Here we rigorously evaluate the applicability of two Interior Penalty Discontinuous Galerkin methods, namely the Nonsymmetric and Symmetric Interior Penalty Galerkin methods (NIPG and SIPG) for compressible elasticity and incompressible, variable viscosity Stokes problems. Through a suite of numerical experiments, our evaluation considers the stability, order of accuracy and robustness of the NIPG and SIPG discretisations for cases with both smooth and discontinuous coefficients. Using manufactured solutions, we confirm that both DG formulations are stable and result in convergent solutions for displacement based elasticity formulations, even in the limit of Poisson ratio approaching 0.5. With regards to incompressible flow simulations, using the analytic solution Sol

  18. Discontinuous Galerkin flood model formulation: Luxury or necessity?

    NASA Astrophysics Data System (ADS)

    Kesserwani, Georges; Wang, Yueling

    2014-08-01

    The finite volume Godunov-type flood model formulation is the most comprehensive amongst those currently employed for flood risk modeling. The local Discontinuous Galerkin method constitutes a more complex, rigorous, and extended local Godunov-type formulation. However, the practical merit associated with such an increase in the level of complexity of the formulation is yet to be decided. This work makes the case for a second-order Runge-Kutta Discontinuous Galerkin (RKDG2) formulation and contrasts it with the equivalently accurate finite volume (MUSCL) formulation, both of which solve the Shallow Water Equations (SWE) in two space dimensions. The numerical complexity of both formulations are presented and their capabilities are explored for wide-ranging diagnostic and real-scale tests, incorporating all challenging features relevant to flood inundation modeling. Our findings reveal that the extra complexity associated with the RKDG2 model pays off by providing higher-quality solution behavior on very coarse meshes and improved velocity predictions. The practical implication of this is that improved accuracy for flood modeling simulations will result when terrain data are limited or of a low resolution.

  19. A Discontinuous Galerkin Discretization of the Eikonal Equation on Curved Piecewise Isoparametric Triangulated Manifolds

    NASA Technical Reports Server (NTRS)

    Barth, TIm

    2002-01-01

    This viewgraph presentation provides information on optimizing the travel distance between two points on a curved surface. The presentation addresses the single source shortest path problem, fast algorithms for estimating the eikonal equation, fast schemes and barrier theorems, and the discontinuous Galerkin method, including hyperbolic causality, finite element method, scalars, and marching the discontinuous Galerkin Eikonal approximation.

  20. Simplified Discontinuous Galerkin Methods for Systems of Conservation Laws with Convex Extension

    NASA Technical Reports Server (NTRS)

    Barth, Timothy J.

    1999-01-01

    Simplified forms of the space-time discontinuous Galerkin (DG) and discontinuous Galerkin least-squares (DGLS) finite element method are developed and analyzed. The new formulations exploit simplifying properties of entropy endowed conservation law systems while retaining the favorable energy properties associated with symmetric variable formulations.

  1. Adaptive multiresolution semi-Lagrangian discontinuous Galerkin methods for the Vlasov equations

    NASA Astrophysics Data System (ADS)

    Besse, N.; Deriaz, E.; Madaule, É.

    2017-03-01

    We develop adaptive numerical schemes for the Vlasov equation by combining discontinuous Galerkin discretisation, multiresolution analysis and semi-Lagrangian time integration. We implement a tree based structure in order to achieve adaptivity. Both multi-wavelets and discontinuous Galerkin rely on a local polynomial basis. The schemes are tested and validated using Vlasov-Poisson equations for plasma physics and astrophysics.

  2. A reconstructed discontinuous Galerkin method for magnetohydrodynamics on arbitrary grids

    NASA Astrophysics Data System (ADS)

    Karami Halashi, Behrouz; Luo, Hong

    2016-12-01

    A reconstructed discontinuous Galerkin (rDG) method, designed not only to enhance the accuracy of DG methods but also to ensure the nonlinear stability of the rDG method, is developed for solving the Magnetohydrodynamics (MHD) equations on arbitrary grids. In this rDG(P1P2) method, a quadratic polynomial solution (P2) is first obtained using a Hermite Weighted Essentially Non-oscillatory (WENO) reconstruction from the underlying linear polynomial (P1) discontinuous Galerkin solution to ensure linear stability of the rDG method and to improves efficiency of the underlying DG method. By taking advantage of handily available and yet invaluable information, namely the first derivatives in the DG formulation, the stencils used in reconstruction involve only Von Neumann neighborhood (adjacent face-neighboring cells) and thus are compact. The first derivatives of the quadratic polynomial solution are then reconstructed using a WENO reconstruction in order to eliminate spurious oscillations in the vicinity of strong discontinuities, thus ensuring the nonlinear stability of the rDG method. The HLLD Riemann solver introduced in the literature for one-dimensional MHD problems is adopted in normal direction to compute numerical fluxes. The divergence free constraint is satisfied using the Locally Divergence Free (LDF) approach. The developed rDG method is used to compute a variety of 2D and 3D MHD problems on arbitrary grids to demonstrate its accuracy, robustness, and non-oscillatory property. Our numerical experiments indicate that the rDG(P1P2) method is able to capture shock waves sharply essentially without any spurious oscillations, and achieve the designed third-order of accuracy: one order accuracy higher than the underlying DG method.

  3. Discontinuous Galerkin method for predicting heat transfer in hypersonic environments

    NASA Astrophysics Data System (ADS)

    Ching, Eric; Lv, Yu; Ihme, Matthias

    2016-11-01

    This study is concerned with predicting surface heat transfer in hypersonic flows using high-order discontinuous Galerkin methods. A robust and accurate shock capturing method designed for steady calculations that uses smooth artificial viscosity for shock stabilization is developed. To eliminate parametric dependence, an optimization method is formulated that results in the least amount of artificial viscosity necessary to sufficiently suppress nonlinear instabilities and achieve steady-state convergence. Performance is evaluated in two canonical hypersonic tests, namely a flow over a circular half-cylinder and flow over a double cone. Results show this methodology to be significantly less sensitive than conventional finite-volume techniques to mesh topology and inviscid flux function. The method is benchmarked against state-of-the-art finite-volume solvers to quantify computational cost and accuracy. Financial support from a Stanford Graduate Fellowship and the NASA Early Career Faculty program are gratefully acknowledged.

  4. Scalable parallel Newton-Krylov solvers for discontinuous Galerkin discretizations

    SciTech Connect

    Persson, P.-O.

    2008-12-31

    We present techniques for implicit solution of discontinuous Galerkin discretizations of the Navier-Stokes equations on parallel computers. While a block-Jacobi method is simple and straight-forward to parallelize, its convergence properties are poor except for simple problems. Therefore, we consider Newton-GMRES methods preconditioned with block-incomplete LU factorizations, with optimized element orderings based on a minimum discarded fill (MDF) approach. We discuss the difficulties with the parallelization of these methods, but also show that with a simple domain decomposition approach, most of the advantages of the block-ILU over the block-Jacobi preconditioner are still retained. The convergence is further improved by incorporating the matrix connectivities into the mesh partitioning process, which aims at minimizing the errors introduced from separating the partitions. We demonstrate the performance of the schemes for realistic two- and three-dimensional flow problems.

  5. Discontinuous Galerkin method based on non-polynomial approximation spaces

    SciTech Connect

    Yuan Ling . E-mail: lyuan@dam.brown.edu; Shu Chiwang . E-mail: shu@dam.brown.edu

    2006-10-10

    In this paper, we develop discontinuous Galerkin (DG) methods based on non-polynomial approximation spaces for numerically solving time dependent hyperbolic and parabolic and steady state hyperbolic and elliptic partial differential equations (PDEs). The algorithm is based on approximation spaces consisting of non-polynomial elementary functions such as exponential functions, trigonometric functions, etc., with the objective of obtaining better approximations for specific types of PDEs and initial and boundary conditions. It is shown that L {sup 2} stability and error estimates can be obtained when the approximation space is suitably selected. It is also shown with numerical examples that a careful selection of the approximation space to fit individual PDE and initial and boundary conditions often provides more accurate results than the DG methods based on the polynomial approximation spaces of the same order of accuracy.

  6. Discontinuous Galerkin time-domain computations of metallic nanostructures.

    PubMed

    Stannigel, Kai; König, Michael; Niegemann, Jens; Busch, Kurt

    2009-08-17

    We apply the three-dimensional Discontinuous-Galerkin Time-Domain method to the investigation of the optical properties of bar- and V-shaped metallic nanostructures on dielectric substrates. A flexible finite element-like mesh together with an expansion into high-order basis functions allows for an accurate resolution of complex geometries and strong field gradients. In turn, this provides accurate results on the optical response of realistic structures. We study in detail the influence of particle size and shape on resonance frequencies as well as on scattering and absorption efficiencies. Beyond a critical size which determines the onset of the quasi-static limit we find significant deviations from the quasi-static theory. Furthermore, we investigate the influence of the excitation by comparing normal illumination and attenuated total internal reflection setups. Finally, we examine the possibility of coherently controlling the local field enhancement of V-structures via chirped pulses.

  7. Discontinuous Galerkin Methods for NonLinear Differential Systems

    NASA Technical Reports Server (NTRS)

    Barth, Timothy; Mansour, Nagi (Technical Monitor)

    2001-01-01

    This talk considers simplified finite element discretization techniques for first-order systems of conservation laws equipped with a convex (entropy) extension. Using newly developed techniques in entropy symmetrization theory, simplified forms of the discontinuous Galerkin (DG) finite element method have been developed and analyzed. The use of symmetrization variables yields numerical schemes which inherit global entropy stability properties of the PDE (partial differential equation) system. Central to the development of the simplified DG methods is the Eigenvalue Scaling Theorem which characterizes right symmetrizers of an arbitrary first-order hyperbolic system in terms of scaled eigenvectors of the corresponding flux Jacobian matrices. A constructive proof is provided for the Eigenvalue Scaling Theorem with detailed consideration given to the Euler equations of gas dynamics and extended conservation law systems derivable as moments of the Boltzmann equation. Using results from kinetic Boltzmann moment closure theory, we then derive and prove energy stability for several approximate DG fluxes which have practical and theoretical merit.

  8. GPU Accelerated Discontinuous Galerkin Methods for Shallow Water Equations

    NASA Astrophysics Data System (ADS)

    Gandham, Rajesh; Medina, David; Warburton, Timothy

    2015-07-01

    We discuss the development, verification, and performance of a GPU accelerated discontinuous Galerkin method for the solutions of two dimensional nonlinear shallow water equations. The shallow water equations are hyperbolic partial differential equations and are widely used in the simulation of tsunami wave propagations. Our algorithms are tailored to take advantage of the single instruction multiple data (SIMD) architecture of graphic processing units. The time integration is accelerated by local time stepping based on a multi-rate Adams-Bashforth scheme. A total variational bounded limiter is adopted for nonlinear stability of the numerical scheme. This limiter is coupled with a mass and momentum conserving positivity preserving limiter for the special treatment of a dry or partially wet element in the triangulation. Accuracy, robustness and performance are demonstrated with the aid of test cases. We compare the performance of the kernels expressed in a portable threading language OCCA, when cross compiled with OpenCL, CUDA, and OpenMP at runtime.

  9. Benchmarking a Discontinuous Galerkin Vlasov-Poisson solver in Gkeyll

    NASA Astrophysics Data System (ADS)

    Juno, James; Hakim, Ammar; Hammett, Greg

    2013-10-01

    Gkeyll is a discontinuous Galerkin (DG) code under development to solve a variety of kinetic equations in plasmas, with the aim of solving 5D gyrokinetic equations for edge turbulence. For bench-marking the code, we solved the Vlasov-Poisson system of equations, modeling the two-stream instability, as well as current research in nonlinear wave modes of plasmas, such as Bernstein, Green, and Kruskal (BGK) and kinetic electrostatic electron nonlinear (KEEN) wave modes. Results from Gkeyll compared well with linear and nonlinear analysis of the two-stream instability, and matched results of other computational techniques being used to examine BGK and KEEN wave modes. On bounded domains, we explored sheath plasma physics, and modeled the Pierce Diode connected to an external circuit. These sheath studies will contribute to an understanding of boundary conditions for gyrokinetic equations in the edge of tokamaks. Supported by the National Undergraduate Fellowship in Plasma Physics and Fusion Energy Sciences.

  10. A unified discontinuous Galerkin framework for time integration.

    PubMed

    Zhao, Shan; Wei, G W

    2014-05-15

    We introduce a new discontinuous Galerkin approach for time integration. On the basis of the method of weighted residual, numerical quadratures are employed in the finite element time discretization to account for general nonlinear ordinary differential equations. Many different conditions, including explicit, implicit, and symplectic conditions, are enforced for the test functions in the variational analysis to obtain desirable features of the resulting time-stepping scheme. The proposed discontinuous Galerkin approach provides a unified framework to derive various time-stepping schemes, such as low-order one-step methods, Runge-Kutta methods, and multistep methods. On the basis of the proposed framework, several explicit Runge-Kutta methods of different orders are constructed. The derivation of symplectic Runge-Kutta methods has also been realized. The proposed framework allows the optimization of new schemes in terms of several characteristics, such as accuracy, sparseness, and stability. The accuracy optimization is performed on the basis of an analytical form of the error estimation function for a linear test initial value problem. Schemes with higher formal order of accuracy are found to provide more accurate solutions. We have also explored the optimization potential of sparseness, which is related to the general compressive sensing in signal/imaging processing. Two critical dimensions of the stability region, that is, maximal intervals along the imaginary and negative real axes, are employed as the criteria for stability optimization. This gives the largest Courant-Friedrichs-Lewy time steps in solving hyperbolic and parabolic partial differential equations, respectively. Numerical experiments are conducted to validate the optimized time-stepping schemes.

  11. A unified discontinuous Galerkin framework for time integration

    PubMed Central

    Zhao, Shan; Wei, G. W.

    2013-01-01

    We introduce a new discontinuous Galerkin approach for time integration. On the basis of the method of weighted residual, numerical quadratures are employed in the finite element time discretization to account for general nonlinear ordinary differential equations. Many different conditions, including explicit, implicit, and symplectic conditions, are enforced for the test functions in the variational analysis to obtain desirable features of the resulting time-stepping scheme. The proposed discontinuous Galerkin approach provides a unified framework to derive various time-stepping schemes, such as low-order one-step methods, Runge–Kutta methods, and multistep methods. On the basis of the proposed framework, several explicit Runge–Kutta methods of different orders are constructed. The derivation of symplectic Runge–Kutta methods has also been realized. The proposed framework allows the optimization of new schemes in terms of several characteristics, such as accuracy, sparseness, and stability. The accuracy optimization is performed on the basis of an analytical form of the error estimation function for a linear test initial value problem. Schemes with higher formal order of accuracy are found to provide more accurate solutions. We have also explored the optimization potential of sparseness, which is related to the general compressive sensing in signal/imaging processing. Two critical dimensions of the stability region, that is, maximal intervals along the imaginary and negative real axes, are employed as the criteria for stability optimization. This gives the largest Courant–Friedrichs–Lewy time steps in solving hyperbolic and parabolic partial differential equations, respectively. Numerical experiments are conducted to validate the optimized time-stepping schemes. PMID:25382889

  12. A discontinuous Galerkin method to solve chromatographic models.

    PubMed

    Javeed, Shumaila; Qamar, Shamsul; Seidel-Morgenstern, Andreas; Warnecke, Gerald

    2011-10-07

    This article proposes a discontinuous Galerkin method for solving model equations describing isothermal non-reactive and reactive chromatography. The models contain a system of convection-diffusion-reaction partial differential equations with dominated convective terms. The suggested method has capability to capture sharp discontinuities and narrow peaks of the elution profiles. The accuracy of the method can be improved by introducing additional nodes in the same solution element and, hence, avoids the expansion of mesh stencils normally encountered in the high order finite volume schemes. Thus, the method can be uniformly applied up to boundary cells without loosing accuracy. The method is robust and well suited for large-scale time-dependent simulations of chromatographic processes where accuracy is highly demanding. Several test problems of isothermal non-reactive and reactive chromatographic processes are presented. The results of the current method are validated against flux-limiting finite volume schemes. The numerical results verify the efficiency and accuracy of the investigated method. The proposed scheme gives more resolved solutions than the high resolution finite volume schemes.

  13. Continued Development of the Discontinuous Galerkin Method for Computational Aeroacoustic Applications

    NASA Technical Reports Server (NTRS)

    Atkins, H. L.

    1997-01-01

    The formulation and the implementation of boundary conditions within the context of the quadrature-free form of the discontinuous Galerkin method are presented for several types of boundary conditions for the Euler equations. An important feature of the discontinuous Galerkin method is that the interior point algorithm is well behaved in the neighborhood of the boundary and requires no modifications. This feature leads to a simple and accurate treatment for wall boundary conditions and simple inflow and outflow boundary conditions. Curved walls are accurately treated with only minor changes to the implementation described in earlier work. The 'perfectly matched layer' approach to nonreflecting boundary conditions is easily applied to the discontinuous Galerkin. The compactness of the discontinuous Galerkin method makes it better suited for buffer-zone-type methods than high-order finite-difference methods. Results are presented for wall, characteristic inflow and outflow, and nonreflecting boundary conditions.

  14. A Discontinuous Galerkin Finite Element Method for Hamilton-Jacobi Equations

    NASA Technical Reports Server (NTRS)

    Hu, Changqing; Shu, Chi-Wang

    1998-01-01

    In this paper, we present a discontinuous Galerkin finite element method for solving the nonlinear Hamilton-Jacobi equations. This method is based on the Runge-Kutta discontinuous Galerkin finite element method for solving conservation laws. The method has the flexibility of treating complicated geometry by using arbitrary triangulation, can achieve high order accuracy with a local, compact stencil, and are suited for efficient parallel implementation. One and two dimensional numerical examples are given to illustrate the capability of the method.

  15. A CLASS OF RECONSTRUCTED DISCONTINUOUS GALERKIN METHODS IN COMPUTATIONAL FLUID DYNAMICS

    SciTech Connect

    Hong Luo; Yidong Xia; Robert Nourgaliev

    2011-05-01

    A class of reconstructed discontinuous Galerkin (DG) methods is presented to solve compressible flow problems on arbitrary grids. The idea is to combine the efficiency of the reconstruction methods in finite volume methods and the accuracy of the DG methods to obtain a better numerical algorithm in computational fluid dynamics. The beauty of the resulting reconstructed discontinuous Galerkin (RDG) methods is that they provide a unified formulation for both finite volume and DG methods, and contain both classical finite volume and standard DG methods as two special cases of the RDG methods, and thus allow for a direct efficiency comparison. Both Green-Gauss and least-squares reconstruction methods and a least-squares recovery method are presented to obtain a quadratic polynomial representation of the underlying linear discontinuous Galerkin solution on each cell via a so-called in-cell reconstruction process. The devised in-cell reconstruction is aimed to augment the accuracy of the discontinuous Galerkin method by increasing the order of the underlying polynomial solution. These three reconstructed discontinuous Galerkin methods are used to compute a variety of compressible flow problems on arbitrary meshes to assess their accuracy. The numerical experiments demonstrate that all three reconstructed discontinuous Galerkin methods can significantly improve the accuracy of the underlying second-order DG method, although the least-squares reconstructed DG method provides the best performance in terms of both accuracy, efficiency, and robustness.

  16. High order discontinuous Galerkin discretizations with discontinuity resolution within the cell

    NASA Astrophysics Data System (ADS)

    Ekaterinaris, John; Panourgias, Konstantinos

    2016-11-01

    The nonlinear filter of Yee et al. and used for low dissipative well-balanced high order accurate finite-difference schemes is adapted to the finite element context of discontinuous Galerkin (DG) discretizations. The performance of the proposed nonlinear filter for DG discretizations is demonstrated for different orders of expansions for one- and multi-dimensional problems with exact solutions. It is shown that for higher order discretizations discontinuity resolution within the cell is achieved and the design order of accuracy is preserved. The filter is applied for inviscid and viscous flow test problems including strong shocks interactions to demonstrate that the proposed dissipative mechanism for DG discretizations yields superior results compared to the results obtained with the TVB limiter and high-order hierarchical limiting. The proposed approach is suitable for p-adaptivity in order to locally enhance resolution of three-dimensional flow simulations.

  17. Application of wall-models to discontinuous Galerkin LES

    NASA Astrophysics Data System (ADS)

    Frère, Ariane; Carton de Wiart, Corentin; Hillewaert, Koen; Chatelain, Philippe; Winckelmans, Grégoire

    2017-08-01

    Wall-resolved Large-Eddy Simulations (LES) are still limited to moderate Reynolds number flows due to the high computational cost required to capture the inner part of the boundary layer. Wall-modeled LES (WMLES) provide more affordable LES by modeling the near-wall layer. Wall function-based WMLES solve LES equations up to the wall, where the coarse mesh resolution essentially renders the calculation under-resolved. This makes the accuracy of WMLES very sensitive to the behavior of the numerical method. Therefore, best practice rules regarding the use and implementation of WMLES cannot be directly transferred from one methodology to another regardless of the type of discretization approach. Whilst numerous studies present guidelines on the use of WMLES, there is a lack of knowledge for discontinuous finite-element-like high-order methods. Incidentally, these methods are increasingly used on the account of their high accuracy on unstructured meshes and their strong computational efficiency. The present paper proposes best practice guidelines for the use of WMLES in these methods. The study is based on sensitivity analyses of turbulent channel flow simulations by means of a Discontinuous Galerkin approach. It appears that good results can be obtained without the use of a spatial or temporal averaging. The study confirms the importance of the wall function input data location and suggests to take it at the bottom of the second off-wall element. These data being available through the ghost element, the suggested method prevents the loss of computational scalability experienced in unstructured WMLES. The study also highlights the influence of the polynomial degree used in the wall-adjacent element. It should preferably be of even degree as using polynomials of degree two in the first off-wall element provides, surprisingly, better results than using polynomials of degree three.

  18. An Investigation of Wave Propagations in Discontinuous Galerkin Method

    NASA Technical Reports Server (NTRS)

    Hu, Fang Q.

    2004-01-01

    Analysis of the discontinuous Galerkin method has been carried out for one- and two-dimensional system of hyperbolic equations. Analytical, as well as numerical, properties of wave propagation in a DGM scheme are derived and verified with direct numerical simulations. In addition to a systematic examination of the dissipation and dispersion errors, behaviours of a DG scheme at an interface of two different grid topologies are also studied. Under the same framework, a quantitative discrete analysis of various artificial boundary conditions is also conducted. Progress has been made in numerical boundary condition treatment that is closely related to the application of DGM in aeroacoustics problems. Finally, Fourier analysis of DGM for the Convective diffusion equation has also be studied in connection with the application of DG schemes for the Navier-Stokes equations. This research has resulted in five(5) publications, plus one additional manuscript in preparation, four(4) conference presentations, and three(3) departmental seminars, as summarized in part II. Abstracts of papers are given in part 111 of this report.

  19. A Discontinuous Galerkin Framework for Electronic Structure Calculations

    NASA Astrophysics Data System (ADS)

    Baczewski, Andrew; Shanker, Balasubramaniam; Mahanti, Subhendra; Levine, Benjamin

    2012-02-01

    It is generally accepted that a good basis set for any calculation should possess a number of salient features, including systematic improvability, adaptive resolution of multiscale features, and fidelity in capturing the pertinent physics. Considering the progenitors of most modern electronic structure basis sets to be Gaussian-type orbitals or planewaves, descendants of these methods have inherited features that address either systematic improvability (planewaves) or adaptive resolution (Gaussians) separately, and use a variety of tricks to differentiate the core and valence physics. Discontinuous Galerkin methods provide a framework for defining adaptive local basis sets, that may be based on these canonical basis sets, that can be mixed and matched to simultaneously achieve all of these goals. Our group is presently developing a new electronic structure code to enable Density Functional and Hartree-Fock calculations within this framework, particularly in the context of all-electron formulations wherein the accurate resolution of both core and valence states is necessary. Numerous implementation details will be addressed, including the incorporation of hardware- and software-based acceleration, such as GPU-based parallelism, and fast electrostatics solvers.

  20. Regional wave propagation using the discontinuous Galerkin method

    NASA Astrophysics Data System (ADS)

    Wenk, S.; Pelties, C.; Igel, H.; Käser, M.

    2012-08-01

    We present an application of the discontinuous Galerkin (DG) method to regional wave propagation. The method makes use of unstructured tetrahedral meshes, combined with a time integration scheme solving the arbitrary high-order derivative (ADER) Riemann problem. The ADER-DG method is high-order accurate in space and time, beneficial for reliable simulations of high-frequency wavefields over long propagation distances. Due to the ease with which tetrahedral grids can be adapted to complex geometries, undulating topography of the Earth's surface and interior interfaces can be readily implemented in the computational domain. The ADER-DG method is benchmarked for the accurate radiation of elastic waves excited by an explosive and a shear dislocation source. We compare real data measurements with synthetics of the 2009 L'Aquila event (central Italy). We take advantage of the geometrical flexibility of the approach to generate a European model composed of the 3-D EPcrust model, combined with the depth-dependent ak135 velocity model in the upper-mantle. The results confirm the applicability of the ADER-DG method for regional scale earthquake simulations, which provides an alternative to existing methodologies.

  1. Regional wave propagation using the discontinuous Galerkin method

    NASA Astrophysics Data System (ADS)

    Wenk, S.; Pelties, C.; Igel, H.; Käser, M.

    2013-01-01

    We present an application of the discontinuous Galerkin (DG) method to regional wave propagation. The method makes use of unstructured tetrahedral meshes, combined with a time integration scheme solving the arbitrary high-order derivative (ADER) Riemann problem. This ADER-DG method is high-order accurate in space and time, beneficial for reliable simulations of high-frequency wavefields over long propagation distances. Due to the ease with which tetrahedral grids can be adapted to complex geometries, undulating topography of the Earth's surface and interior interfaces can be readily implemented in the computational domain. The ADER-DG method is benchmarked for the accurate radiation of elastic waves excited by an explosive and a shear dislocation source. We compare real data measurements with synthetics of the 2009 L'Aquila event (central Italy). We take advantage of the geometrical flexibility of the approach to generate a European model composed of the 3-D EPcrust model, combined with the depth-dependent ak135 velocity model in the upper mantle. The results confirm the applicability of the ADER-DG method for regional scale earthquake simulations, which provides an alternative to existing methodologies.

  2. Studies of Coherent Synchrotron Radiation with the Discontinuous Galerkin Method

    NASA Astrophysics Data System (ADS)

    Bizzozero, David A.

    In this thesis, we present methods for integrating Maxwell's equations in Frenet-Serret coordinates in several settings using discontinuous Galerkin (DG) finite element method codes in 1D, 2D, and 3D. We apply these routines to the study of coherent synchrotron radiation, an important topic in accelerator physics. We build upon the published computational work of T. Agoh and D. Zhou in solving Maxwell's equations in the frequency-domain using a paraxial approximation which reduces Maxwell's equations to a Schrodinger-like system. We also evolve Maxwell's equations in the time-domain using a Fourier series decomposition with 2D DG motivated by an experiment performed at the Canadian Light Source. A comparison between theory and experiment has been published (Phys. Rev. Lett. 114, 204801 (2015)). Lastly, we devise a novel approach to integrating Maxwell's equations with 3D DG using a Galilean transformation and demonstrate proof-of-concept. In the above studies, we examine the accuracy, efficiency, and convergence of DG.

  3. An Investigation of Wave Propagations in Discontinuous Galerkin Method

    NASA Technical Reports Server (NTRS)

    Hu, Fang Q.

    2004-01-01

    Analysis of the discontinuous Galerkin method has been carried out for one- and two-dimensional system of hyperbolic equations. Analytical, as well as numerical, properties of wave propagation in a DGM scheme are derived and verified with direct numerical simulations. In addition to a systematic examination of the dissipation and dispersion errors, behaviours of a DG scheme at an interface of two different grid topologies are also studied. Under the same framework, a quantitative discrete analysis of various artificial boundary conditions is also conducted. Progress has been made in numerical boundary condition treatment that is closely related to the application of DGM in aeroacoustics problems. Finally, Fourier analysis of DGM for the Convective diffusion equation has also be studied in connection with the application of DG schemes for the Navier-Stokes equations. This research has resulted in five(5) publications, plus one additional manuscript in preparation, four(4) conference presentations, and three(3) departmental seminars, as summarized in part II. Abstracts of papers are given in part 111 of this report.

  4. A Quadrature Free Discontinuous Galerkin Conservative Level Set Scheme

    NASA Astrophysics Data System (ADS)

    Czajkowski, Mark; Desjardins, Olivier

    2010-11-01

    In an effort to improve the scalability and accuracy of the Accurate Conservative Level Set (ACLS) scheme [Desjardins et al., J COMPUT PHYS 227 (2008)], a scheme based on the quadrature free discontinuous Galerkin (DG) methodology has been developed. ACLS relies on a hyperbolic tangent level set function that is transported and reinitialized using conservative schemes in order to alleviate mass conservation issues known to plague level set methods. DG allows for an arbitrarily high order representation of the interface by using a basis of high order polynomials while only using data from the faces of neighboring cells. The small stencil allows DG to have excellent parallel scalability. The diffusion term present in the conservative reinitialization equation is handled using local DG method [Cockburn et al., SIAM J NUMER ANAL 39, (2001)] while the normals are computed from a limited form of the level set function in order to avoid spurious oscillations. The resulting scheme is shown to be both robust, accurate, and highly scalable, making it a method of choice for large-scale simulations of multiphase flows with complex interfacial topology.

  5. Adaptive local discontinuous Galerkin approximation to Richards’ equation

    NASA Astrophysics Data System (ADS)

    Li, H.; Farthing, M. W.; Miller, C. T.

    2007-09-01

    We propose a spatially and temporally adaptive solution to Richards' equation based upon a local discontinuous Galerkin approximation in space and a high-order, backward difference method in time. We cast our approach in terms of a general, decoupled adaption algorithm based upon operators. We define non-unique instances of all operators to result in an adaption method from within the general class of methods that is defined. We formally decouple the spatial adaption from the temporal adaption using a method of lines approach and limit the temporal truncation error so that the total error is dominated by the spatial component. We use a multiple grid approach to guide adaption and support the data structures. Spatial adaption decisions are based upon error and regularity indicators, which are economical to compute. The resultant methods are compared for two test problems. The results show that the proposed adaption methods are superior to methods that adapt only in time and that in cases in which the problem has sufficient smoothness, adapting the order of the elements in addition to the grid spacing can further improve the efficiency of this robust solution approach.

  6. A Reconstructed Discontinuous Galerkin Method for the Compressible Navier-Stokes Equations on Arbitrary Grids

    SciTech Connect

    Hong Luo; Luqing Luo; Robert Nourgaliev; Vincent A. Mousseau

    2010-09-01

    A reconstruction-based discontinuous Galerkin (RDG) method is presented for the solution of the compressible Navier–Stokes equations on arbitrary grids. The RDG method, originally developed for the compressible Euler equations, is extended to discretize viscous and heat fluxes in the Navier–Stokes equations using a so-called inter-cell reconstruction, where a smooth solution is locally reconstructed using a least-squares method from the underlying discontinuous DG solution. Similar to the recovery-based DG (rDG) methods, this reconstructed DG method eliminates the introduction of ad hoc penalty or coupling terms commonly found in traditional DG methods. Unlike rDG methods, this RDG method does not need to judiciously choose a proper form of a recovered polynomial, thus is simple, flexible, and robust, and can be used on arbitrary grids. The developed RDG method is used to compute a variety of flow problems on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results indicate that this RDG method is able to deliver the same accuracy as the well-known Bassi–Rebay II scheme, at a half of its computing costs for the discretization of the viscous fluxes in the Navier–Stokes equations, clearly demonstrating its superior performance over the existing DG methods for solving the compressible Navier–Stokes equations.

  7. A Reconstructed Discontinuous Galerkin Method for the Compressible Flows on Unstructured Tetrahedral Grids

    SciTech Connect

    Hong Luo; Yidong Xia; Robert Nourgaliev; Chunpei Cai

    2011-06-01

    A reconstruction-based discontinuous Galerkin (RDG) method is presented for the solution of the compressible Navier-Stokes equations on unstructured tetrahedral grids. The RDG method, originally developed for the compressible Euler equations, is extended to discretize viscous and heat fluxes in the Navier-Stokes equations using a so-called inter-cell reconstruction, where a smooth solution is locally reconstructed using a least-squares method from the underlying discontinuous DG solution. Similar to the recovery-based DG (rDG) methods, this reconstructed DG method eliminates the introduction of ad hoc penalty or coupling terms commonly found in traditional DG methods. Unlike rDG methods, this RDG method does not need to judiciously choose a proper form of a recovered polynomial, thus is simple, flexible, and robust, and can be used on unstructured grids. The preliminary results indicate that this RDG method is stable on unstructured tetrahedral grids, and provides a viable and attractive alternative for the discretization of the viscous and heat fluxes in the Navier-Stokes equations.

  8. A Reconstructed Discontinuous Galerkin Method for the Compressible Navier-Stokes Equations on Arbitrary Grids

    SciTech Connect

    Hong Luo; Luqing Luo; Robert Nourgaliev; Vincent A. Mousseau

    2010-01-01

    A reconstruction-based discontinuous Galerkin (RDG) method is presented for the solution of the compressible Navier-Stokes equations on arbitrary grids. The RDG method, originally developed for the compressible Euler equations, is extended to discretize viscous and heat fluxes in the Navier-Stokes equations using a so-called inter-cell reconstruction, where a smooth solution is locally reconstructed using a least-squares method from the underlying discontinuous DG solution. Similar to the recovery-based DG (rDG) methods, this reconstructed DG method eliminates the introduction of ad hoc penalty or coupling terms commonly found in traditional DG methods. Unlike rDG methods, this RDG method does not need to judiciously choose a proper form of a recovered polynomial, thus is simple, flexible, and robust, and can be used on arbitrary grids. The developed RDG method is used to compute a variety of flow problems on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results indicate that this RDG method is able to deliver the same accuracy as the well-known Bassi-Rebay II scheme, at a half of its computing costs for the discretization of the viscous fluxes in the Navier-Stokes equations, clearly demonstrating its superior performance over the existing DG methods for solving the compressible Navier-Stokes equations.

  9. Thermoelastic Simulations Based on Discontinuous Galerkin Methods: Formulation and Application in Gas Turbines

    NASA Astrophysics Data System (ADS)

    Hao, Zengrong; Gu, Chunwei; Song, Yin

    2016-06-01

    This study extends the discontinuous Galerkin (DG) methods to simulations of thermoelasticity. A thermoelastic formulation of interior penalty DG (IP-DG) method is presented and aspects of the numerical implementation are discussed in matrix form. The content related to thermal expansion effects is illustrated explicitly in the discretized equation system. The feasibility of the method for general thermoelastic simulations is validated through typical test cases, including tackling stress discontinuities caused by jumps of thermal expansive properties and controlling accompanied non-physical oscillations through adjusting the magnitude of IP term. The developed simulation platform upon the method is applied to the engineering analysis of thermoelastic performance for a turbine vane and a series of vanes with various types of simplified thermal barrier coating (TBC) systems. This analysis demonstrates that while TBC properties on heat conduction are generally the major consideration for protecting the alloy base vanes, the mechanical properties may have more significant effects on protections of coatings themselves. Changing characteristics of normal tractions on TBC/base interface, closely related to the occurrence of coating failures, over diverse components distributions along TBC thickness of the functional graded materials are summarized and analysed, illustrating the opposite tendencies in situations with different thermal-stress-free temperatures for coatings.

  10. Plane wave discontinuous Galerkin methods for acoustic scattering

    NASA Astrophysics Data System (ADS)

    Kapita, Shelvean

    We apply the Plane Wave Discontinuous Galerkin (PWDG) method to study the direct scattering of acoustic waves from impenetrable obstacles. In the first part of the thesis we consider the full exterior scattering problem with smooth boundaries. This problem is modeled by the Helmholtz equation in the unbounded domain exterior to the scatterer. To compute the scattered field, an artificial boundary is introduced to reduce the infinite domain to a finite computational domain. We then apply Dirichlet-to-Neumann (DtN) and Neumann-to-Dirichlet (NtD) boundary conditions on a circular artificial boundary. By using asymptotic properties of Hankel functions, we are able to prove wavenumber explicit L2-norm error estimates for the DtN-PWDG method on quasi-uniform meshes. Numerical experiments indicate that the accuracy of the PWDG method for the scattering problem is improved by the use of DtN and NtD boundary conditions. The second part of the thesis concerns acoustic scattering from domains with corners. In such domains, quasi-uniform meshes are not efficient so we derive error indicators to drive the selective refinement of the mesh in an adaptive algorithm. We prove a posteriori L2-norm error estimates for the Helmholtz equation with impedance boundary conditions on the artificial boundary. Numerical results demonstrate the efficiency of the proposed indicators. This adaptive strategy is compatible with the DtN and NtD truncation of the infinite domain problem and the combination would significantly improve the accuracy and reliability of PWDG simulations.

  11. Localized Artificial Viscosity Stabilization of Discontinuous Galerkin Methods for Nonhydrostatic Mesoscale Atmospheric Modeling

    DTIC Science & Technology

    2014-01-01

    Pedinotti and M. Savini, "A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows," in 2nd European...Conference on Turbomachinery Fluid Dynamics and Thermodynamics, Antwerpen, Belgium, 1997. [27] R. Hartmann, "Adaptive discontinuous Galerkin methods

  12. Quadrature-Free Implementation of the Discontinuous Galerkin Method for Hyperbolic Equations

    NASA Technical Reports Server (NTRS)

    Atkins, H. L.; Shu, Chi-Wang

    1996-01-01

    A discontinuous Galerkin formulation that avoids the use of discrete quadrature formulas is described and applied to linear and nonlinear test problems in one and two space dimensions. This approach requires less computational time and storage than conventional implementations but preserves the compactness and robustness inherent to the discontinuous Galerkin method. Test problems include both linear and nonlinear one-dimensional scalar advection of botH smooth and discontinuous initial value problems, two-dimensional scalar advection of smooth initial value problems discretized by using unstructured grids with varying degrees of smoothness and regularity, and two-dimensional linear Euler solutions on unstructured grids.

  13. Quadrature-Free Implementation of the Discontinuous Galerkin Method for Hyperbolic Equations

    NASA Technical Reports Server (NTRS)

    Atkins, H. L.; Shu, Chi-Wang

    1996-01-01

    A discontinuous Galerkin formulation that avoids the use of discrete quadrature formulas is described and applied to linear and nonlinear test problems in one and two space dimensions. This approach requires less computational time and storage than conventional implementations but preserves the compactness and robustness inherent to the discontinuous Galerkin method. Test problems include both linear and nonlinear one-dimensional scalar advection of botH smooth and discontinuous initial value problems, two-dimensional scalar advection of smooth initial value problems discretized by using unstructured grids with varying degrees of smoothness and regularity, and two-dimensional linear Euler solutions on unstructured grids.

  14. A nonlinear filter for high order discontinuous Galerkin discretizations with discontinuity resolution within the cell

    NASA Astrophysics Data System (ADS)

    Panourgias, Konstantinos T.; Ekaterinaris, John A.

    2016-12-01

    The nonlinear filter introduced by Yee et al. (1999) [27] and extensively used in the development of low dissipative well-balanced high order accurate finite-difference schemes is adapted to the finite element context of discontinuous Galerkin (DG) discretizations. The filter operator is constructed in the canonical computational domain for the standard cubical element where it is applied to the computed conservative variables in a direction per direction basis. Filtering becomes possible for all element types in unstructured meshes using collapsed coordinate transformations. The performance of the proposed nonlinear filter for DG discretizations is demonstrated and evaluated for different orders of expansions for one-dimensional and multidimensional problems with exact solutions. It is shown that for higher order discretizations discontinuity resolution within the cell is achieved and the design order of accuracy is preserved. The filter is applied for a number of standard inviscid flow test problems including strong shocks interactions to demonstrate that the proposed dissipative mechanism for DG discretizations yields superior results compared to the results obtained with the total variation bounded (TVB) limiter and high-order hierarchical limiting. The proposed approach is suitable for p-adaptivity in order to locally enhance resolution of three-dimensional flow simulations that include discontinuities and complex flow features.

  15. Accelerated GPU simulation of compressible flow by the discontinuous evolution Galerkin method

    NASA Astrophysics Data System (ADS)

    Block, B. J.; Lukáčová-Medvid'ová, M.; Virnau, P.; Yelash, L.

    2012-08-01

    The aim of the present paper is to report on our recent results for GPU accelerated simulations of compressible flows. For numerical simulation the adaptive discontinuous Galerkin method with the multidimensional bicharacteristic based evolution Galerkin operator has been used. For time discretization we have applied the explicit third order Runge-Kutta method. Evaluation of the genuinely multidimensional evolution operator has been accelerated using the GPU implementation. We have obtained a speedup up to 30 (in comparison to a single CPU core) for the calculation of the evolution Galerkin operator on a typical discretization mesh consisting of 16384 mesh cells.

  16. Discontinuous Galerkin method for multicomponent chemically reacting flows and combustion

    NASA Astrophysics Data System (ADS)

    Lv, Yu; Ihme, Matthias

    2014-08-01

    This paper presents the development of a discontinuous Galerkin (DG) method for application to chemically reacting flows in subsonic and supersonic regimes under the consideration of variable thermo-viscous-diffusive transport properties, detailed and stiff reaction chemistry, and shock capturing. A hybrid-flux formulation is developed for treatment of the convective fluxes, combining a conservative Riemann-solver and an extended double-flux scheme. A computationally efficient splitting scheme is proposed, in which advection and diffusion operators are solved in the weak form, and the chemically stiff substep is advanced in the strong form using a time-implicit scheme. The discretization of the viscous-diffusive transport terms follows the second form of Bassi and Rebay, and the WENO-based limiter due to Zhong and Shu is extended to multicomponent systems. Boundary conditions are developed for subsonic and supersonic flow conditions, and the algorithm is coupled to thermochemical libraries to account for detailed reaction chemistry and complex transport. The resulting DG method is applied to a series of test cases of increasing physico-chemical complexity. Beginning with one- and two-dimensional multispecies advection and shock-fluid interaction problems, computational efficiency, convergence, and conservation properties are demonstrated. This study is followed by considering a series of detonation and supersonic combustion problems to investigate the convergence-rate and the shock-capturing capability in the presence of one- and multistep reaction chemistry. The DG algorithm is then applied to diffusion-controlled deflagration problems. By examining convergence properties for polynomial order and spatial resolution, and comparing these with second-order finite-volume solutions, it is shown that optimal convergence is achieved and that polynomial refinement provides advantages in better resolving the localized flame structure and complex flow-field features

  17. Discontinuous Galerkin methods for modeling Hurricane storm surge

    NASA Astrophysics Data System (ADS)

    Dawson, Clint; Kubatko, Ethan J.; Westerink, Joannes J.; Trahan, Corey; Mirabito, Christopher; Michoski, Craig; Panda, Nishant

    2011-09-01

    Storm surge due to hurricanes and tropical storms can result in significant loss of life, property damage, and long-term damage to coastal ecosystems and landscapes. Computer modeling of storm surge can be used for two primary purposes: forecasting of surge as storms approach land for emergency planning and evacuation of coastal populations, and hindcasting of storms for determining risk, development of mitigation strategies, coastal restoration and sustainability. Storm surge is modeled using the shallow water equations, coupled with wind forcing and in some events, models of wave energy. In this paper, we will describe a depth-averaged (2D) model of circulation in spherical coordinates. Tides, riverine forcing, atmospheric pressure, bottom friction, the Coriolis effect and wind stress are all important for characterizing the inundation due to surge. The problem is inherently multi-scale, both in space and time. To model these problems accurately requires significant investments in acquiring high-fidelity input (bathymetry, bottom friction characteristics, land cover data, river flow rates, levees, raised roads and railways, etc.), accurate discretization of the computational domain using unstructured finite element meshes, and numerical methods capable of capturing highly advective flows, wetting and drying, and multi-scale features of the solution. The discontinuous Galerkin (DG) method appears to allow for many of the features necessary to accurately capture storm surge physics. The DG method was developed for modeling shocks and advection-dominated flows on unstructured finite element meshes. It easily allows for adaptivity in both mesh ( h) and polynomial order ( p) for capturing multi-scale spatial events. Mass conservative wetting and drying algorithms can be formulated within the DG method. In this paper, we will describe the application of the DG method to hurricane storm surge. We discuss the general formulation, and new features which have been added to

  18. A combined discontinuous Galerkin method for the dipolar Bose-Einstein condensation

    NASA Astrophysics Data System (ADS)

    Li, Xiang-Gui; Zhu, Jiang; Zhang, Rong-Pei; Cao, Shengshan

    2014-10-01

    In this work, a combined discontinuous Galerkin (DG) method, which is a hybridized mixed discontinuous Galerkin (HMDG) method combined with the direct discontinuous Galerkin (DDG) method, is proposed to compute ground states and dynamics of dipolar Bose-Einstein condensates (BECs) described by a multi-dimensional Gross-Pitaevskii equation (GPE) coupled with a first-order velocity system. Due to the adaption of the first-order velocity system instead of dipolar interactions, the proposed combined DG method avoids to evaluate integrals with high singularity. Additionally, this method keeps the conservation of the particle number. The Krylov semi-implicit method is applied to the time discretization. Finally, numerical examples are presented to demonstrate the accuracy and capability of the proposed method.

  19. The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems

    NASA Technical Reports Server (NTRS)

    Cockburn, Bernardo; Shu, Chi-Wang

    1997-01-01

    In this paper, we study the Local Discontinuous Galerkin methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge-Kutta Discontinuous Galerkin methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, their high-order formal accuracy, and their easy handling of complicated geometries, for convection dominated problems. It is proven that for scalar equations, the Local Discontinuous Galerkin methods are L(sup 2)-stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are k-th order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown.

  20. A High Order Discontinuous Galerkin Method for 2D Incompressible Flows

    NASA Technical Reports Server (NTRS)

    Liu, Jia-Guo; Shu, Chi-Wang

    1999-01-01

    In this paper we introduce a high order discontinuous Galerkin method for two dimensional incompressible flow in vorticity streamfunction formulation. The momentum equation is treated explicitly, utilizing the efficiency of the discontinuous Galerkin method The streamfunction is obtained by a standard Poisson solver using continuous finite elements. There is a natural matching between these two finite element spaces, since the normal component of the velocity field is continuous across element boundaries. This allows for a correct upwinding gluing in the discontinuous Galerkin framework, while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability The method is suitable for inviscid or high Reynolds number flows. Optimal error estimates are proven and verified by numerical experiments.

  1. A high-order discontinuous Galerkin method for 2D incompressible flows

    SciTech Connect

    Liu, J.G.; Shu, C.W.

    2000-05-20

    In this paper the authors introduce a high-order discontinuous Galerkin method for two-dimensional incompressible flow in the vorticity stream-function formulation. The momentum equation is treated explicitly, utilizing the efficiency of the discontinuous Galerkin method. The stream function is obtained by a standard Poisson solver using continuous finite elements. There is a natural matching between these two finite element spaces, since the normal component of the velocity field is continuous across element boundaries. This allows for a correct upwinding gluing in the discontinuous Galerkin framework, while still maintaining total energy conservation with no numerical dissipation and total entropy stability. The method is efficient for inviscid or high Reynolds number flows. Optimal error estimates are proved and verified by numerical experiments.

  2. Super-convergence of Discontinuous Galerkin Method Applied to the Navier-Stokes Equations

    NASA Technical Reports Server (NTRS)

    Atkins, Harold L.

    2009-01-01

    The practical benefits of the hyper-accuracy properties of the discontinuous Galerkin method are examined. In particular, we demonstrate that some flow attributes exhibit super-convergence even in the absence of any post-processing technique. Theoretical analysis suggest that flow features that are dominated by global propagation speeds and decay or growth rates should be super-convergent. Several discrete forms of the discontinuous Galerkin method are applied to the simulation of unsteady viscous flow over a two-dimensional cylinder. Convergence of the period of the naturally occurring oscillation is examined and shown to converge at 2p+1, where p is the polynomial degree of the discontinuous Galerkin basis. Comparisons are made between the different discretizations and with theoretical analysis.

  3. Multiwavelet Discontinuous Galerkin Accelerated ELP Method for the Shallow Water Equations on the Cubed Sphere

    SciTech Connect

    White III, James B; Archibald, Richard K; Evans, Katherine J; Drake, John

    2011-01-01

    In this paper we present a new approach to increase the time-step size for an explicit discontinuous Galerkin numerical method. The attributes of this approach are demonstrated on standard tests for the shallow-water equations on the sphere. The addition of multiwavelets to discontinuous Galerkin method, which has the benefit of being scalable, flexible, and conservative, provides a hierarchical scale structure that can be exploited to improve computational efficiency in both the spatial and temporal dimensions. This paper explains how combining a multiwavelet discontinuous Galerkin method with exact linear part time-evolution schemes, which can remain stable for implicit-sized time steps, can help increase the time-step size for shallow water equations on the sphere.

  4. Combination of the discontinuous Galerkin method with finite differences for simulation of seismic wave propagation

    SciTech Connect

    Lisitsa, Vadim; Tcheverda, Vladimir; Botter, Charlotte

    2016-04-15

    We present an algorithm for the numerical simulation of seismic wave propagation in models with a complex near surface part and free surface topography. The approach is based on the combination of finite differences with the discontinuous Galerkin method. The discontinuous Galerkin method can be used on polyhedral meshes; thus, it is easy to handle the complex surfaces in the models. However, this approach is computationally intense in comparison with finite differences. Finite differences are computationally efficient, but in general, they require rectangular grids, leading to the stair-step approximation of the interfaces, which causes strong diffraction of the wavefield. In this research we present a hybrid algorithm where the discontinuous Galerkin method is used in a relatively small upper part of the model and finite differences are applied to the main part of the model.

  5. Differential formulation of discontinuous Galerkin and related methods for compressible Euler and Navier-Stokes equations

    NASA Astrophysics Data System (ADS)

    Gao, Haiyang

    A new approach to high-order accuracy for the numerical solution of conservation laws introduced by Huynh and extended to simplexes by the current work is renamed CPR (correction procedure or collocation penalty via reconstruction). The CPR approach employs the differential form of the equation and accounts for the jumps in flux values at the cell boundaries by a correction procedure. In addition to being simple and economical, it unifies several existing methods including discontinuous Galerkin (DG), staggered grid, spectral volume (SV), and spectral difference (SD). The approach is then extended to diffusion equation and Navier-Stokes equations. In the discretization of the diffusion terms, the BR2 (Bassi and Rebay), interior penalty, compact DG (CDG), and I-continuous approaches are used. The first three of these approaches, originally derived using the integral formulation, were recast here in the CPR framework, whereas the I-continuous scheme, originally derived for a quadrilateral mesh, was extended to a triangular mesh. The current work also includes a study of high-order curve boundaries representations. A new boundary representation based on the Bezier curve is then developed and analyzed, which is shown to have several advantages for complicated geometries. To further enhance the efficiency, the capability of h/p mesh adaptation is developed for the CPR solver. The adaptation is driven by an efficient multi-p a posteriori error estimator. P-adaptation is applied to smooth regions of the flow field while h-adaptation targets the non-smooth regions, identified by accuracy-preserving TVD marker. Several numerical tests are presented to demonstrate the capability of the technique.

  6. Stretched-coordinate PMLs for Maxwell's equations in the discontinuous Galerkin time-domain method.

    PubMed

    König, Michael; Prohm, Christopher; Busch, Kurt; Niegemann, Jens

    2011-02-28

    The discontinuous Galerkin time-domain method (DGTD) is an emerging technique for the numerical simulation of time-dependent electromagnetic phenomena. For many applications it is necessary to model the infinite space which surrounds scatterers and sources. As a result, absorbing boundaries which mimic its properties play a key role in making DGTD a versatile tool for various kinds of systems. Popular techniques include the Silver-Mϋller boundary condition and uniaxial perfectly matched layers (UPMLs). We provide novel instructions for the implementation of stretched-coordinate perfectly matched layers in a discontinuous Galerkin framework and compare the performance of the three absorbers for a three-dimensional test system.

  7. ON THE ROLE OF INVOLUTIONS IN THE DISCONTINUOUS GALERKIN DISCRETIZATION OF MAXWELL AND MAGNETOHYDRODYNAMIC SYSTEMS

    NASA Technical Reports Server (NTRS)

    Barth, Timothy

    2005-01-01

    The role of involutions in energy stability of the discontinuous Galerkin (DG) discretization of Maxwell and magnetohydrodynamic (MHD) systems is examined. Important differences are identified in the symmetrization of the Maxwell and MHD systems that impact the construction of energy stable discretizations using the DG method. Specifically, general sufficient conditions to be imposed on the DG numerical flux and approximation space are given so that energy stability is retained These sufficient conditions reveal the favorable energy consequence of imposing continuity in the normal component of the magnetic induction field at interelement boundaries for MHD discretizations. Counterintuitively, this condition is not required for stability of Maxwell discretizations using the discontinuous Galerkin method.

  8. Fourth order exponential time differencing method with local discontinuous Galerkin approximation for coupled nonlinear Schrodinger equations

    DOE PAGES

    Liang, Xiao; Khaliq, Abdul Q. M.; Xing, Yulong

    2015-01-23

    In this paper, we study a local discontinuous Galerkin method combined with fourth order exponential time differencing Runge-Kutta time discretization and a fourth order conservative method for solving the nonlinear Schrödinger equations. Based on different choices of numerical fluxes, we propose both energy-conserving and energy-dissipative local discontinuous Galerkin methods, and have proven the error estimates for the semi-discrete methods applied to linear Schrödinger equation. The numerical methods are proven to be highly efficient and stable for long-range soliton computations. Finally, extensive numerical examples are provided to illustrate the accuracy, efficiency and reliability of the proposed methods.

  9. Runge-Kutta and Lax-Wendroff discontinuous Galerkin methods for linear conservation laws

    NASA Astrophysics Data System (ADS)

    Shu, Chi-Wang

    2017-07-01

    In this talk we give a short summary of our recent work [5], jointly with Z. Sun, on establishing the equivalency of the Runge-Kutta discontinuous Galerkin (RKDG) methods and a class of Lax-Wendroff discontinuous Galerkin (LWDG) methods for solving linear conservation laws, as well as on stability analysis and error estimates for the LWDG methods for solving one- and two-dimensional linear conservation laws, regardless of whether they are equivalent to the RKDG methods or not. Our stability analysis includes multidimensional problems with divergence-free coefficients, and our error estimates include those for both the solution u and its first order time derivative ut.

  10. On the simulation of industrial gas dynamic applications with the discontinuous Galerkin spectral element method

    NASA Astrophysics Data System (ADS)

    Hempert, F.; Hoffmann, M.; Iben, U.; Munz, C.-D.

    2016-06-01

    In the present investigation, we demonstrate the capabilities of the discontinuous Galerkin spectral element method for high order accuracy computation of gas dynamics. The internal flow field of a natural gas injector for bivalent combustion engines is investigated under its operating conditions. The simulations of the flow field and the aeroacoustic noise emissions were in a good agreement with the experimental data. We tested several shock-capturing techniques for the discontinuous Galerkin scheme. Based on the validated framework, we analyzed the development of the supersonic jets during different opening procedures of a compressed natural gas injector. The results suggest that a more gradual injector opening decreases the noise emission.

  11. A Parallel Reconstructed Discontinuous Galerkin Method for the Compressible Flows on Aritrary Grids

    SciTech Connect

    Hong Luo; Amjad Ali; Robert Nourgaliev; Vincent A. Mousseau

    2010-01-01

    A reconstruction-based discontinuous Galerkin method is presented for the solution of the compressible Navier-Stokes equations on arbitrary grids. In this method, an in-cell reconstruction is used to obtain a higher-order polynomial representation of the underlying discontinuous Galerkin polynomial solution and an inter-cell reconstruction is used to obtain a continuous polynomial solution on the union of two neighboring, interface-sharing cells. The in-cell reconstruction is designed to enhance the accuracy of the discontinuous Galerkin method by increasing the order of the underlying polynomial solution. The inter-cell reconstruction is devised to remove an interface discontinuity of the solution and its derivatives and thus to provide a simple, accurate, consistent, and robust approximation to the viscous and heat fluxes in the Navier-Stokes equations. A parallel strategy is also devised for the resulting reconstruction discontinuous Galerkin method, which is based on domain partitioning and Single Program Multiple Data (SPMD) parallel programming model. The RDG method is used to compute a variety of compressible flow problems on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results demonstrate that this RDG method is third-order accurate at a cost slightly higher than its underlying second-order DG method, at the same time providing a better performance than the third order DG method, in terms of both computing costs and storage requirements.

  12. A hybrid reconstructed discontinuous Galerkin and continuous Galerkin finite element method for incompressible flows on unstructured grids

    NASA Astrophysics Data System (ADS)

    Pandare, Aditya K.; Luo, Hong

    2016-10-01

    A hybrid reconstructed discontinuous Galerkin and continuous Galerkin method based on an incremental pressure projection formulation, termed rDG (PnPm) + CG (Pn) in this paper, is developed for solving the unsteady incompressible Navier-Stokes equations on unstructured grids. In this method, a reconstructed discontinuous Galerkin method (rDG (PnPm)) is used to discretize the velocity and a standard continuous Galerkin method (CG (Pn)) is used to approximate the pressure. The rDG (PnPm) + CG (Pn) method is designed to increase the accuracy of the hybrid DG (Pn) + CG (Pn) method and yet still satisfy Ladyženskaja-Babuška-Brezzi (LBB) condition, thus avoiding the pressure checkerboard instability. An upwind method is used to discretize the nonlinear convective fluxes in the momentum equations in order to suppress spurious oscillations in the velocity field. A number of incompressible flow problems for a variety of flow conditions are computed to numerically assess the spatial order of convergence of the rDG (PnPm) + CG (Pn) method. The numerical experiments indicate that both rDG (P0P1) + CG (P1) and rDG (P1P2) + CG (P1) methods can attain the designed 2nd order and 3rd order accuracy in space for the velocity respectively. Moreover, the 3rd order rDG (P1P2) + CG (P1) method significantly outperforms its 2nd order rDG (P0P1) + CG (P1) and rDG (P1P1) + CG (P1) counterparts: being able to not only increase the accuracy of the velocity by one order but also improve the accuracy of the pressure.

  13. Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods

    NASA Astrophysics Data System (ADS)

    Ainsworth, Mark

    2004-07-01

    The dispersive and dissipative properties of hp version discontinuous Galerkin finite element approximation are studied in three different limits. For the small wave-number limit hk→0, we show the discontinuous Galerkin gives a higher order of accuracy than the standard Galerkin procedure, thereby confirming the conjectures of Hu and Atkins [J. Comput. Phys. 182 (2) (2002) 516]. If the mesh is fixed and the order p is increased, it is shown that the dissipation and dispersion errors decay at a super-exponential rate when the order p is much larger than hk. Finally, if the order is chosen so that 2 p+1≈ κhk for some fixed constant κ>1, then it is shown that an exponential rate of decay is obtained.

  14. Mass conservative BDF-discontinuous Galerkin/explicit finite volume schemes for coupling subsurface and overland flows

    NASA Astrophysics Data System (ADS)

    Sochala, P.; Ern, A.; Piperno, S.

    2009-05-01

    Robust and accurate schemes are designed to simulate the coupling between subsurface and overland flows. The coupling conditions at the interface enforce the continuity of both the normal flux and the pressure. Richards' equation governing the subsurface flow is discretized using a Backward Differentiation Formula and a symmetric interior penalty Discontinuous Galerkin method. The kinematic wave equation governing the overland flow is discretized using a Godunov scheme. Both schemes individually are mass conservative and can be used within single-step or multi-step coupling algorithms that ensure overall mass conservation owing to a specific design of the interface fluxes in the multi-step case. Numerical results are presented to illustrate the performances of the proposed algorithms.

  15. Smoothness-Increasing Accuracy-Conserving (SIAC) Filters for Post-Processing Unstructured Discontinuous Galerkin Fields

    DTIC Science & Technology

    2015-08-27

    Robert M. Kirby School of Computing University of Utah Abstract Although discontinuous and continuous Galerkin methods have advantages mathematically ...of this proposal is to develop smoothness-increasing accuracy-conserving filters that respect the mathematical properties of the data while providing...tetrahedral meshes. In Particular, we propose to contribute both mathematically and algorithmically to the class of smoothness-increasing and accuracy

  16. A Local Discontinuous Galerkin Method for the Complex Modified KdV Equation

    SciTech Connect

    Li Wenting; Jiang Kun

    2010-09-30

    In this paper, we develop a local discontinuous Galerkin(LDG) method for solving complex modified KdV(CMKdV) equation. The LDG method has the flexibility for arbitrary h and p adaptivity. We prove the L{sup 2} stability for general solutions.

  17. A Runge-Kutta discontinuous Galerkin approach to solve reactive flows: The hyperbolic operator

    SciTech Connect

    Billet, G.; Ryan, J.

    2011-02-20

    A Runge-Kutta discontinuous Galerkin method to solve the hyperbolic part of reactive Navier-Stokes equations written in conservation form is presented. Complex thermodynamics laws are taken into account. Particular care has been taken to solve the stiff gaseous interfaces correctly with no restrictive hypothesis. 1D and 2D test cases are presented.

  18. A combined discontinuous Galerkin and finite volume scheme for multi-dimensional VPFP system

    SciTech Connect

    Asadzadeh, M.; Bartoszek, K.

    2011-05-20

    We construct a numerical scheme for the multi-dimensional Vlasov-Poisson-Fokker-Planck system based on a combined finite volume (FV) method for the Poisson equation in spatial domain and the streamline diffusion (SD) and discontinuous Galerkin (DG) finite element in time, phase-space variables for the Vlasov-Fokker-Planck equation.

  19. Spatial and spectral superconvergence of discontinuous Galerkin method for hyperbolic problems

    NASA Astrophysics Data System (ADS)

    Marchandise, Emilie; Chevaugeon, Nicolas; Remacle, Jean-Francois

    2008-06-01

    In this paper, we analyze the spatial and spectral superconvergence properties of one-dimensional hyperbolic conservation law by a discontinuous Galerkin (DG) method. The analyses combine classical mathematical arguments with MATLAB experiments. Some properties of the DG schemes are discovered using discrete Fourier analyses: superconvergence of the numerical wave numbers, Radau structure of the X spatial error.

  20. Discretizing the Gent-McWilliams velocity and isopycnal diffusion with a discontinuous Galerkin finite element method

    NASA Astrophysics Data System (ADS)

    Pestiaux, A.; Kärnä, T.; Melchior, S.; Lambrechts, J.; Remacle, J. F.; Deleersnijder, E.; Fichefet, T.

    2012-04-01

    The discretization of the Gent-McWilliams velocity and isopycnal diffusion with a discontinuous Galerkin finite element method is presented. Both processes are implemented in an ocean model thanks to a tensor related to the mesoscale eddies. The antisymmetric part of this tensor is computed from the Gent-McWilliams velocity and is subsequently included in the tracer advection equation. This velocity can be constructed to be divergence-free. The symmetric part that describes the diapycnal and isopycnal diffusions requires a special treatment. A stable and physically sound isopycnal tracer diffusion scheme is needed. Here, an interior penalty method is chosen that enables to build stable diffusion terms. However, due to the strong anisotropy of the diffusion, the common-usual penalty factor by Ern et al. (2008) is not sufficient. A novel method for computing the penalty term of Ern is then proposed for diffusion equations when both the diffusivity and the mesh are strongly anisotropic. Two test cases are resorted to validate the methodology and two more realistic applications illustrate the diapycnal and isopycnal diffusions, as well as the Gent-McWilliams velocity.

  1. Efficient high-order discontinuous Galerkin schemes with first-order hyperbolic advection-diffusion system approach

    NASA Astrophysics Data System (ADS)

    Mazaheri, Alireza; Nishikawa, Hiroaki

    2016-09-01

    We propose arbitrary high-order discontinuous Galerkin (DG) schemes that are designed based on a first-order hyperbolic advection-diffusion formulation of the target governing equations. We present, in details, the efficient construction of the proposed high-order schemes (called DG-H), and show that these schemes have the same number of global degrees-of-freedom as comparable conventional high-order DG schemes, produce the same or higher order of accuracy solutions and solution gradients, are exact for exact polynomial functions, and do not need a second-derivative diffusion operator. We demonstrate that the constructed high-order schemes give excellent quality solution and solution gradients on irregular triangular elements. We also construct a Weighted Essentially Non-Oscillatory (WENO) limiter for the proposed DG-H schemes and apply it to discontinuous problems. We also make some accuracy comparisons with conventional DG and interior penalty schemes. A relative qualitative cost analysis is also reported, which indicates that the high-order schemes produce orders of magnitude more accurate results than the low-order schemes for a given CPU time. Furthermore, we show that the proposed DG-H schemes are nearly as efficient as the DG and Interior-Penalty (IP) schemes as these schemes produce results that are relatively at the same error level for approximately a similar CPU time.

  2. Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives

    NASA Technical Reports Server (NTRS)

    Yan, Jue; Shu, Chi-Wang; Bushnell, Dennis M. (Technical Monitor)

    2002-01-01

    In this paper we review the existing and develop new continuous Galerkin methods for solving time dependent partial differential equations with higher order derivatives in one and multiple space dimensions. We review local discontinuous Galerkin methods for convection diffusion equations involving second derivatives and for KdV type equations involving third derivatives. We then develop new local discontinuous Galerkin methods for the time dependent bi-harmonic type equations involving fourth derivatives, and partial differential equations involving fifth derivatives. For these new methods we present correct interface numerical fluxes and prove L(exp 2) stability for general nonlinear problems. Preliminary numerical examples are shown to illustrate these methods. Finally, we present new results on a post-processing technique, originally designed for methods with good negative-order error estimates, on the local discontinuous Galerkin methods applied to equations with higher derivatives. Numerical experiments show that this technique works as well for the new higher derivative cases, in effectively doubling the rate of convergence with negligible additional computational cost, for linear as well as some nonlinear problems, with a local uniform mesh.

  3. Fractional Burgers equation with nonlinear non-locality: Spectral vanishing viscosity and local discontinuous Galerkin methods

    NASA Astrophysics Data System (ADS)

    Mao, Zhiping; Karniadakis, George Em

    2017-05-01

    We consider the viscous Burgers equation with a fractional nonlinear term as a model involving non-local nonlinearities in conservation laws, which, surprisingly, has an analytical solution obtained by a fractional extension of the Hopf-Cole transformation. We use this model and its inviscid limit to develop stable spectral and discontinuous Galerkin spectral element methods by employing the concept of spectral vanishing viscosity (SVV). For the global spectral method, SVV is very effective and the computational cost is O (N2), which is essentially the same as for the standard Burgers equation. We also develop a local discontinuous Galerkin (LDG) spectral element method to improve the accuracy around discontinuities, and we again stabilize the LDG method with the SVV operator. Finally, we solve numerically the inviscid fractional Burgers equation both with the spectral and the spectral element LDG methods. We study systematically the stability and convergence of both methods and determine the effectiveness of each method for different parameters.

  4. Robust multigrid for high-order discontinuous Galerkin methods: A fast Poisson solver suitable for high-aspect ratio Cartesian grids

    NASA Astrophysics Data System (ADS)

    Stiller, Jörg

    2016-12-01

    We present a polynomial multigrid method for nodal interior penalty and local discontinuous Galerkin formulations of the Poisson equation on Cartesian grids. For smoothing we propose two classes of overlapping Schwarz methods. The first class comprises element-centered and the second face-centered methods. Within both classes we identify methods that achieve superior convergence rates, prove robust with respect to the mesh spacing and the polynomial order, at least up to P = 32. Consequent structure exploitation yields a computational complexity of O (PN), where N is the number of unknowns. Further we demonstrate the suitability of the face-centered method for element aspect ratios up to 32.

  5. Discontinuous Galerkin finite element methods for radiative transfer in spherical symmetry

    NASA Astrophysics Data System (ADS)

    Kitzmann, D.; Bolte, J.; Patzer, A. B. C.

    2016-11-01

    The discontinuous Galerkin finite element method (DG-FEM) is successfully applied to treat a broad variety of transport problems numerically. In this work, we use the full capacity of the DG-FEM to solve the radiative transfer equation in spherical symmetry. We present a discontinuous Galerkin method to directly solve the spherically symmetric radiative transfer equation as a two-dimensional problem. The transport equation in spherical atmospheres is more complicated than in the plane-parallel case owing to the appearance of an additional derivative with respect to the polar angle. The DG-FEM formalism allows for the exact integration of arbitrarily complex scattering phase functions, independent of the angular mesh resolution. We show that the discontinuous Galerkin method is able to describe accurately the radiative transfer in extended atmospheres and to capture discontinuities or complex scattering behaviour which might be present in the solution of certain radiative transfer tasks and can, therefore, cause severe numerical problems for other radiative transfer solution methods.

  6. Wake fields in 9-cell TESLA accelerating structures : Spectral Element Discontinuous Galerkin (SEDG) simulations.

    SciTech Connect

    Min, M.; Fischer, P. F.; Chae, Y.-C.

    2007-01-01

    Using our recently developed high-order accurate Maxwell solver, NEKCEM, we carried out longitudinal wakefield calculations for a 9-cell TESLA cavity structure in 3D. Indirect method is used for wake potential calculations. Computational results with NEKCEM are compared with those of GdfidL. NEKCEM uses a spectral element discontinuous Galerkin (SEDG) method based on a domain decomposition approach using spectral-element discretizations on Gauss-Lobatto-Legendre grids with body-conforming hexahedral meshes. The numerical scheme is designed to ensure high-order spectral accuracy, using the discontinuous Galerkin form with boundary conditions weakly enforced through a flux term between elements. Concerns related to implementation on wake potential calculations are discussed, and wake potential calculations with indirect method by NEKCEM compared with the results of the finite difference time-domain code GdfidL.

  7. A space-time discontinuous Galerkin method for the incompressible Navier-Stokes equations

    NASA Astrophysics Data System (ADS)

    Rhebergen, Sander; Cockburn, Bernardo; van der Vegt, Jaap J. W.

    2013-01-01

    We introduce a space-time discontinuous Galerkin (DG) finite element method for the incompressible Navier-Stokes equations. Our formulation can be made arbitrarily high-order accurate in both space and time and can be directly applied to deforming domains. Different stabilizing approaches are discussed which ensure stability of the method. A numerical study is performed to compare the effect of the stabilizing approaches, to show the method's robustness on deforming domains and to investigate the behavior of the convergence rates of the solution. Recently we introduced a space-time hybridizable DG (HDG) method for incompressible flows [S. Rhebergen, B. Cockburn, A space-time hybridizable discontinuous Galerkin method for incompressible flows on deforming domains, J. Comput. Phys. 231 (2012) 4185-4204]. We will compare numerical results of the space-time DG and space-time HDG methods. This constitutes the first comparison between DG and HDG methods.

  8. Tensor-product preconditioners for higher-order space-time discontinuous Galerkin methods

    NASA Astrophysics Data System (ADS)

    Diosady, Laslo T.; Murman, Scott M.

    2017-02-01

    A space-time discontinuous-Galerkin spectral-element discretization is presented for direct numerical simulation of the compressible Navier-Stokes equations. An efficient solution technique based on a matrix-free Newton-Krylov method is developed in order to overcome the stiffness associated with high solution order. The use of tensor-product basis functions is key to maintaining efficiency at high-order. Efficient preconditioning methods are presented which can take advantage of the tensor-product formulation. A diagonalized Alternating-Direction-Implicit (ADI) scheme is extended to the space-time discontinuous Galerkin discretization. A new preconditioner for the compressible Euler/Navier-Stokes equations based on the fast-diagonalization method is also presented. Numerical results demonstrate the effectiveness of these preconditioners for the direct numerical simulation of subsonic turbulent flows.

  9. Efficient Implementations of the Quadrature-Free Discontinuous Galerkin Method

    NASA Technical Reports Server (NTRS)

    Lockard, David P.; Atkins, Harold L.

    1999-01-01

    The efficiency of the quadrature-free form of the dis- continuous Galerkin method in two dimensions, and briefly in three dimensions, is examined. Most of the work for constant-coefficient, linear problems involves the volume and edge integrations, and the transformation of information from the volume to the edges. These operations can be viewed as matrix-vector multiplications. Many of the matrices are sparse as a result of symmetry, and blocking and specialized multiplication routines are used to account for the sparsity. By optimizing these operations, a 35% reduction in total CPU time is achieved. For nonlinear problems, the calculation of the flux becomes dominant because of the cost associated with polynomial products and inversion. This component of the work can be reduced by up to 75% when the products are approximated by truncating terms. Because the cost is high for nonlinear problems on general elements, it is suggested that simplified physics and the most efficient element types be used over most of the domain.

  10. Discontinuous Galerkin finite element method applied to the 1-D spherical neutron transport equation

    SciTech Connect

    Machorro, Eric . E-mail: machorro@amath.washington.edu

    2007-04-10

    Discontinuous Galerkin finite element methods are used to estimate solutions to the non-scattering 1-D spherical neutron transport equation. Various trial and test spaces are compared in the context of a few sample problems whose exact solution is known. Certain trial spaces avoid unphysical behaviors that seem to plague other methods. Comparisons with diamond differencing and simple corner-balancing are presented to highlight these improvements.

  11. A toolbox for a class of discontinuous Petrov-Galerkin methods using trilinos.

    SciTech Connect

    Roberts, Nathaniel David; Bochev, Pavel Blagoveston; Demkowicz, Leszek D.; Ridzal, Denis

    2011-09-01

    The class of discontinuous Petrov-Galerkin finite element methods (DPG) proposed by L. Demkowicz and J. Gopalakrishnan guarantees the optimality of the solution in an energy norm and produces a symmetric positive definite stiffness matrix, among other desirable properties. In this paper, we describe a toolbox, implemented atop Sandia's Trilinos library, for rapid development of solvers for DPG methods. We use this toolbox to develop solvers for the Poisson and Stokes problems.

  12. Discontinuous Galerkin method for piecewise regular solutions to the nonlinear age-structured population model.

    PubMed

    Krzyzanowski, Piotr; Wrzosek, Dariusz; Wit, Dominik

    2006-10-01

    A discontinuous Galerkin approximation of the nonlinear Lotka-McKendrick equation is considered in the frequent case when the solution is only piecewise regular. An O(h(r+1/2)) error estimate for rth order polynomial finite elements is proved, as well as piecewise H(1)-regularity of the exact solution which guarantees the error estimate for r=0. Certain implementational details which improve the robustness of the method are also addressed.

  13. A Leap-Frog Discontinuous Galerkin Method for the Time-Domain Maxwell's Equations in Metamaterials

    SciTech Connect

    Li, J., Waters, J. W., Machorro, E. A.

    2012-06-01

    Numerical simulation of metamaterials play a very important role in the design of invisibility cloak, and sub-wavelength imaging. In this paper, we propose a leap-frog discontinuous Galerkin method to solve the time-dependent Maxwell’s equations in metamaterials. Conditional stability and error estimates are proved for the scheme. The proposed algorithm is implemented and numerical results supporting the analysis are provided.

  14. On discontinuous Galerkin for time integration in option pricing problems with adaptive finite differences in space

    NASA Astrophysics Data System (ADS)

    von Sydow, Lina

    2013-10-01

    The discontinuous Galerkin method for time integration of the Black-Scholes partial differential equation for option pricing problems is studied and compared with more standard time-integrators. In space an adaptive finite difference discretization is employed. The results show that the dG method are in most cases at least comparable to standard time-integrators and in some cases superior to them. Together with adaptive spatial grids the suggested pricing method shows great qualities.

  15. THE USE OF CLASSICAL LAX-FRIEDRICHS RIEMANN SOLVERS WITH DISCONTINUOUS GALERKIN METHODS

    SciTech Connect

    W. J. RIDER; R. B. LOWRIE

    2001-03-01

    While conducting a von Neumann stability analysis of discontinuous Galerkin methods we found that the standard Lax-Friedrichs (LxF) Riemann solver is unstable for all time-step sizes. A simple modification of the Riemann solver's dissipation returns the method to stability. Furthermore, the method has a smaller truncation error than the corresponding method with an upwind flux for the RK2-DG(1) method. These results are confirmed upon testing.

  16. Analysis of an Hp-Non-conforming Discontinuous Galerkin Spectral Element Method for Wave

    DTIC Science & Technology

    2011-04-01

    on hexahedral meshes using either exact integration with Legendre-Gauss quadrature or inexact integration with Legendre-Gauss- Lobatto quadrature. A...inexact integration with Legendre-Gauss- Lobatto quadrature. A mortar-based non-conforming approximation is developed to treat both h and p non...Legendre-Gauss- Lobatto ; consistency, stability, convergence. AMS subject classifications. 65N35, 65N12, 65N15 1. Introduction. The discontinuous Galerkin

  17. A Posteriori Error Estimation for Discontinuous Galerkin Approximations of Hyperbolic Systems

    NASA Technical Reports Server (NTRS)

    Larson, Mats G.; Barth, Timothy J.

    1999-01-01

    This article considers a posteriori error estimation of specified functionals for first-order systems of conservation laws discretized using the discontinuous Galerkin (DG) finite element method. Using duality techniques, we derive exact error representation formulas for both linear and nonlinear functionals given an associated bilinear or nonlinear variational form. Weighted residual approximations of the exact error representation formula are then proposed and numerically evaluated for Ringleb flow, an exact solution of the 2-D Euler equations.

  18. A Comparative Study of Different Reconstruction Schemes for a Reconstructed Discontinuous Galerkin Method on Arbitrary Grids

    SciTech Connect

    Hong Luo; Hanping Xiao; Robert Nourgaliev; Chunpei Cai

    2011-06-01

    A comparative study of different reconstruction schemes for a reconstruction-based discontinuous Galerkin, termed RDG(P1P2) method is performed for compressible flow problems on arbitrary grids. The RDG method is designed to enhance the accuracy of the discontinuous Galerkin method by increasing the order of the underlying polynomial solution via a reconstruction scheme commonly used in the finite volume method. Both Green-Gauss and least-squares reconstruction methods and a least-squares recovery method are implemented to obtain a quadratic polynomial representation of the underlying discontinuous Galerkin linear polynomial solution on each cell. These three reconstruction/recovery methods are compared for a variety of compressible flow problems on arbitrary meshes to access their accuracy and robustness. The numerical results demonstrate that all three reconstruction methods can significantly improve the accuracy of the underlying second-order DG method, although the least-squares reconstruction method provides the best performance in terms of both accuracy and robustness.

  19. SpECTRE: A task-based discontinuous Galerkin code for relativistic astrophysics

    NASA Astrophysics Data System (ADS)

    Kidder, Lawrence E.; Field, Scott E.; Foucart, Francois; Schnetter, Erik; Teukolsky, Saul A.; Bohn, Andy; Deppe, Nils; Diener, Peter; Hébert, François; Lippuner, Jonas; Miller, Jonah; Ott, Christian D.; Scheel, Mark A.; Vincent, Trevor

    2017-04-01

    We introduce a new relativistic astrophysics code, SpECTRE, that combines a discontinuous Galerkin method with a task-based parallelism model. SpECTRE's goal is to achieve more accurate solutions for challenging relativistic astrophysics problems such as core-collapse supernovae and binary neutron star mergers. The robustness of the discontinuous Galerkin method allows for the use of high-resolution shock capturing methods in regions where (relativistic) shocks are found, while exploiting high-order accuracy in smooth regions. A task-based parallelism model allows efficient use of the largest supercomputers for problems with a heterogeneous workload over disparate spatial and temporal scales. We argue that the locality and algorithmic structure of discontinuous Galerkin methods will exhibit good scalability within a task-based parallelism framework. We demonstrate the code on a wide variety of challenging benchmark problems in (non)-relativistic (magneto)-hydrodynamics. We demonstrate the code's scalability including its strong scaling on the NCSA Blue Waters supercomputer up to the machine's full capacity of 22 , 380 nodes using 671 , 400 threads.

  20. A Reconstructed Discontinuous Galerkin Method for the Euler Equations on Arbitrary Grids

    SciTech Connect

    Hong Luo; Luqing Luo; Robert Nourgaliev

    2012-11-01

    A reconstruction-based discontinuous Galerkin (RDG(P1P2)) method, a variant of P1P2 method, is presented for the solution of the compressible Euler equations on arbitrary grids. In this method, an in-cell reconstruction, designed to enhance the accuracy of the discontinuous Galerkin method, is used to obtain a quadratic polynomial solution (P2) from the underlying linear polynomial (P1) discontinuous Galerkin solution using a least-squares method. The stencils used in the reconstruction involve only the von Neumann neighborhood (face-neighboring cells) and are compact and consistent with the underlying DG method. The developed RDG method is used to compute a variety of flow problems on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results indicate that this RDG(P1P2) method is third-order accurate, and outperforms the third-order DG method (DG(P2)) in terms of both computing costs and storage requirements.

  1. A spectral-element discontinuous Galerkin lattice Boltzmann method for nearly incompressible flows

    NASA Astrophysics Data System (ADS)

    Min, Misun; Lee, Taehun

    2011-01-01

    We present a spectral-element discontinuous Galerkin lattice Boltzmann method for solving nearly incompressible flows. Decoupling the collision step from the streaming step offers numerical stability at high Reynolds numbers. In the streaming step, we employ high-order spectral-element discontinuous Galerkin discretizations using a tensor product basis of one-dimensional Lagrange interpolation polynomials based on Gauss-Lobatto-Legendre grids. Our scheme is cost-effective with a fully diagonal mass matrix, advancing time integration with the fourth-order Runge-Kutta method. We present a consistent treatment for imposing boundary conditions with a numerical flux in the discontinuous Galerkin approach. We show convergence studies for Couette flows and demonstrate two benchmark cases with lid-driven cavity flows for Re = 400-5000 and flows around an impulsively started cylinder for Re = 550-9500. Computational results are compared with those of other theoretical and computational work that used a multigrid method, a vortex method, and a spectral element model.

  2. A spectral-element discontinuous Galerkin lattice Boltzmann method for incompressible flows.

    SciTech Connect

    Min, M.; Lee, T.; Mathematics and Computer Science; City Univ. of New York

    2011-01-01

    We present a spectral-element discontinuous Galerkin lattice Boltzmann method for solving nearly incompressible flows. Decoupling the collision step from the streaming step offers numerical stability at high Reynolds numbers. In the streaming step, we employ high-order spectral-element discontinuous Galerkin discretizations using a tensor product basis of one-dimensional Lagrange interpolation polynomials based on Gauss-Lobatto-Legendre grids. Our scheme is cost-effective with a fully diagonal mass matrix, advancing time integration with the fourth-order Runge-Kutta method. We present a consistent treatment for imposing boundary conditions with a numerical flux in the discontinuous Galerkin approach. We show convergence studies for Couette flows and demonstrate two benchmark cases with lid-driven cavity flows for Re = 400-5000 and flows around an impulsively started cylinder for Re = 550-9500. Computational results are compared with those of other theoretical and computational work that used a multigrid method, a vortex method, and a spectral element model.

  3. A Hierarchical WENO Reconstructed Discontinuous Galerkin Method for Computing Shock Waves

    NASA Astrophysics Data System (ADS)

    Xia, Y.; Frisbey, M.; Luo, H.

    The discontinuous Galerkin (DG) methods[1] have recently become popular for the solution of systems of conservation laws because of their several attractive features such as easy extension to and compact stencil for higher-order (> 2nd) approximation, flexibility in handling arbitrary types of grids for complex geometries, and amenability to parallelization and hp-adaptation. However, the DG Methods have their own share weaknesses. In particular, how to effectively control spurious oscillations in the presence of strong discontinuities, and how to reduce the computing costs and storage requirements for the DGM remain the two most challenging and unresolved issues in the DGM.

  4. A discontinuous Galerkin method for two-dimensional PDE models of Asian options

    NASA Astrophysics Data System (ADS)

    Hozman, J.; Tichý, T.; Cvejnová, D.

    2016-06-01

    In our previous research we have focused on the problem of plain vanilla option valuation using discontinuous Galerkin method for numerical PDE solution. Here we extend a simple one-dimensional problem into two-dimensional one and design a scheme for valuation of Asian options, i.e. options with payoff depending on the average of prices collected over prespecified horizon. The algorithm is based on the approach combining the advantages of the finite element methods together with the piecewise polynomial generally discontinuous approximations. Finally, an illustrative example using DAX option market data is provided.

  5. High-order nodal discontinuous Galerkin methods for the Maxwell eigenvalue problem.

    PubMed

    Hesthaven, J S; Warburton, T

    2004-03-15

    The Maxwell eigenvalue problem is known to pose difficulties for standard numerical methods, predominantly due to its large null space. As an alternative to the widespread use of Galerkin finite-element methods based on curl-conforming elements, we propose to use high-order nodal elements in a discontinuous element scheme. We consider both two- and three-dimensional problems and show the former to be without problems in a wide range of cases. Numerical experiments suggest the validity of this for general problems. For the three-dimensional eigenproblem, we encounter difficulties with a naive formulation of the scheme and propose minor modifications, intimately related to the discontinuous nature of the formulation, to overcome these concerns. We conclude by connecting the findings to time domain solution of Maxwell's equations. The discussion, analysis, and numerous computational experiments suggest that using discontinuous element schemes for solving Maxwell's equation in the frequency- or time-domain present a high-order accurate, efficient and robust alternative to classical Galerkin finite-element methods.

  6. An Iterative Phase-Space Explicit Discontinuous Galerkin Method for Stellar Radiative Transfer in Extended Atmospheres

    SciTech Connect

    de Almeida, V.F.

    2004-01-28

    A phase-space discontinuous Galerkin (PSDG) method is presented for the solution of stellar radiative transfer problems. It allows for greater adaptivity than competing methods without sacrificing generality. The method is extensively tested on a spherically symmetric, static, inverse-power-law scattering atmosphere. Results for different sizes of atmospheres and intensities of scattering agreed with asymptotic values. The exponentially decaying behavior of the radiative field in the diffusive-transparent transition region and the forward peaking behavior at the surface of extended atmospheres were accurately captured. The integrodifferential equation of radiation transfer is solved iteratively by alternating between the radiative pressure equation and the original equation with the integral term treated as an energy density source term. In each iteration, the equations are solved via an explicit, flux-conserving, discontinuous Galerkin method. Finite elements are ordered in wave fronts perpendicularly to the characteristic curves so that elemental linear algebraic systems are solved quickly by sweeping the phase space element by element. Two implementations of a diffusive boundary condition at the origin are demonstrated wherein the finite discontinuity in the radiative intensity is accurately captured by the proposed method. This allows for a consistent mechanism to preserve photon luminosity. The method was proved to be robust and fast, and a case is made for the adequacy of parallel processing. In addition to classical two-dimensional plots, results of normalized radiative intensity were mapped onto a log-polar surface exhibiting all distinguishing features of the problem studied.

  7. An iterative phase-space explicit discontinuous Galerkin method for stellar radiative transfer in extended atmospheres

    NASA Astrophysics Data System (ADS)

    de Almeida, Valmor F.

    2017-07-01

    A phase-space discontinuous Galerkin (PSDG) method is presented for the solution of stellar radiative transfer problems. It allows for greater adaptivity than competing methods without sacrificing generality. The method is extensively tested on a spherically symmetric, static, inverse-power-law scattering atmosphere. Results for different sizes of atmospheres and intensities of scattering agreed with asymptotic values. The exponentially decaying behavior of the radiative field in the diffusive-transparent transition region, and the forward peaking behavior at the surface of extended atmospheres were accurately captured. The integrodifferential equation of radiation transfer is solved iteratively by alternating between the radiative pressure equation and the original equation with the integral term treated as an energy density source term. In each iteration, the equations are solved via an explicit, flux-conserving, discontinuous Galerkin method. Finite elements are ordered in wave fronts perpendicular to the characteristic curves so that elemental linear algebraic systems are solved quickly by sweeping the phase space element by element. Two implementations of a diffusive boundary condition at the origin are demonstrated wherein the finite discontinuity in the radiation intensity is accurately captured by the proposed method. This allows for a consistent mechanism to preserve photon luminosity. The method was proved to be robust and fast, and a case is made for the adequacy of parallel processing. In addition to classical two-dimensional plots, results of normalized radiation intensity were mapped onto a log-polar surface exhibiting all distinguishing features of the problem studied.

  8. An HP Adaptive Discontinuous Galerkin Method for Hyperbolic Conservation Laws. Ph.D. Thesis

    NASA Technical Reports Server (NTRS)

    Bey, Kim S.

    1994-01-01

    This dissertation addresses various issues for model classes of hyperbolic conservation laws. The basic approach developed in this work employs a new family of adaptive, hp-version, finite element methods based on a special discontinuous Galerkin formulation for hyperbolic problems. The discontinuous Galerkin formulation admits high-order local approximations on domains of quite general geometry, while providing a natural framework for finite element approximations and for theoretical developments. The use of hp-versions of the finite element method makes possible exponentially convergent schemes with very high accuracies in certain cases; the use of adaptive hp-schemes allows h-refinement in regions of low regularity and p-enrichment to deliver high accuracy, while keeping problem sizes manageable and dramatically smaller than many conventional approaches. The use of discontinuous Galerkin methods is uncommon in applications, but the methods rest on a reasonable mathematical basis for low-order cases and has local approximation features that can be exploited to produce very efficient schemes, especially in a parallel, multiprocessor environment. The place of this work is to first and primarily focus on a model class of linear hyperbolic conservation laws for which concrete mathematical results, methodologies, error estimates, convergence criteria, and parallel adaptive strategies can be developed, and to then briefly explore some extensions to more general cases. Next, we provide preliminaries to the study and a review of some aspects of the theory of hyperbolic conservation laws. We also provide a review of relevant literature on this subject and on the numerical analysis of these types of problems.

  9. A weighted Runge-Kutta discontinuous Galerkin method for wavefield modelling

    NASA Astrophysics Data System (ADS)

    He, Xijun; Yang, Dinghui; Wu, Hao

    2015-03-01

    In this paper, we propose a weighted Runge-Kutta (RK) discontinuous Galerkin (WRKDG) method for wavefield modelling. For this method, we first transform the seismic wave equations in 2-D heterogeneous anisotropic media into a first-order hyperbolic system, and then combine the discontinuous Galerkin method (DGM) with a weighted RK time discretization. The time discretization is based on an implicit diagonal RK method and an explicit technique, which changes the implicit RK method into an explicit one. In addition, we introduce a weighting factor in the process. Linear and quadratic polynomials for spatial basis functions are typically employed. We investigate the properties of the method in great detail, including the stability criteria and numerical dispersion relations for solving the 2-D acoustic equations. Our analysis indicates that the stability condition for the WRKDG method is more relaxed compared with the classic total variation diminishing (TVD) RK discontinuous Galerkin (RKDG) method, resulting in a 1.7 times superiority for P1 element and is about as efficient as TVD RKDG method for P2 element in computational efficiency. We also demonstrate that the WRKDG method can suppress numerical dispersion more efficiently than the staggered-grid (SG) method on the same grid. The WRKDG method is applied to simulate the wavefields in a large velocity contrast model, a 2-D homogeneous transversely isotropic (TI) model, a fluid-filled fracture model, and a 2-D SEG/EAGE salt dome model. Regular rectangular and irregular triangular elements are used. The numerical results show that the WRKDG method can effectively suppress numerical dispersion and provide accurate information on the wavefield on a coarse mesh. Therefore, the method evidently reduces the scale of the problem and increases computational efficiency. In addition, promising numerical tests show that the WRKDG method combines well with split perfectly matched layer boundary conditions.

  10. Energy Stable Flux Formulas For The Discontinuous Galerkin Discretization Of First Order Nonlinear Conservation Laws

    NASA Technical Reports Server (NTRS)

    Barth, Timothy; Charrier, Pierre; Mansour, Nagi N. (Technical Monitor)

    2001-01-01

    We consider the discontinuous Galerkin (DG) finite element discretization of first order systems of conservation laws derivable as moments of the kinetic Boltzmann equation. This includes well known conservation law systems such as the Euler For the class of first order nonlinear conservation laws equipped with an entropy extension, an energy analysis of the DG method for the Cauchy initial value problem is developed. Using this DG energy analysis, several new variants of existing numerical flux functions are derived and shown to be energy stable.

  11. The Discontinuous Galerkin Method for the Multiscale Modeling of Dynamics of Crystalline Solids

    DTIC Science & Technology

    2007-08-26

    can be used to provide the equation of state, and therefore close the continuum equations . 3 Numerical methodology 3.1 Macroscale solver: the DG method...Hamilton-Jacobi equations . In the standard RKDG method, we seek the solution in the finite dimen- sional polynomial space V̄ 7,k h = { v = (v1, v2, v3...Comput. Methods. Appl. Mech. En- grg., 193(17–20):1645–1669, 2004. [49] J. Yan and C.-W. Shu. A local discontinuous Galerkin method for KdV type equations

  12. LES of a bluff-body stabilized premixed flame using discontinuous Galerkin scheme

    NASA Astrophysics Data System (ADS)

    Lv, Yu; Ihme, Matthias; Stanford University Team

    2016-11-01

    This talk focuses on the development of a high-order discontinuous Galerkin (DG) method for application to chemically reacting flows. To enable these simulations, several algorithmic aspects are addressed, including the time-integration of multi-step chemical reactions, the incorporation of detailed thermo-viscous transport properties, and the stabilization of high-order solution representation. This DG solver is applied in implicit LES of a turbulent bluff-body stabilized propane/air premixed flame. The simulation results for cold-flow and reacting conditions are reported and compared to experimental data.

  13. Modeling of thin heterogeneous sheets in the discontinuous Galerkin method for 3D transient scattering problems

    NASA Astrophysics Data System (ADS)

    Boubekeur, Mohamed; Kameni Ntichi, Abelin; Pichon, Lionel

    2016-02-01

    This paper presents a modeling of heterogeneous sheets in the time domain discontinuous Galerkin method. An homogenization model combined to a sheet interface condition is used to avoid the mesh of these sheets in order to study the transient response of heterogeneous enclosures. The validation of this approach is based on a comparison with the case when the sheet is meshed. To illustrate the efficiency of the interface condition, the simulation of a 3D cavity is performed. Contribution to the topical issue "Numelec 2015 - Elected submissions", edited by Adel Razek

  14. On the discontinuous Galerkin method for numerical pricing of basket spread options with the average strike

    NASA Astrophysics Data System (ADS)

    Hozman, J.; Tichý, T.

    2017-07-01

    The paper is based on the results from our recent research on path-dependent multi-asset options. Here we focus on options, payoff of which depends on the difference of the spread of two underlying assets at expiry and their average spread during the life of the option. The main idea uses a concept of the dimensional reduction to the PDE model with only two spatial variables describing this option pricing problem. Then the numerical option pricing scheme arising from the discontinuous Galerkin method is developed. Finally, a simple numerical result is presented on real market data.

  15. Analysis of Preconditioning and Relaxation Operators for the Discontinuous Galerkin Method Applied to Diffusion

    NASA Technical Reports Server (NTRS)

    Atkins, H. L.; Shu, Chi-Wang

    2001-01-01

    The explicit stability constraint of the discontinuous Galerkin method applied to the diffusion operator decreases dramatically as the order of the method is increased. Block Jacobi and block Gauss-Seidel preconditioner operators are examined for their effectiveness at accelerating convergence. A Fourier analysis for methods of order 2 through 6 reveals that both preconditioner operators bound the eigenvalues of the discrete spatial operator. Additionally, in one dimension, the eigenvalues are grouped into two or three regions that are invariant with order of the method. Local relaxation methods are constructed that rapidly damp high frequencies for arbitrarily large time step.

  16. Testing Refinement Criteria in Adaptive Discontinuous Galerkin Simulations of Dry Atmospheric Convection

    DTIC Science & Technology

    2011-12-22

    Testing refinement criteria in adaptive Discontinuous Galerkin simulations of dry atmospheric convection Andreas Müllera,∗, Jörn Behrensb, Francis X...mainz.de (Andreas Müller), joern.behrens@zmaw.de (Jörn Behrens), fxgirald@nps.edu ( Francis X. Giraldo), vwirth@uni-mainz.de (Volkmar Wirth) Preprint...formulation and accuracy, Mon. Weather Rev. 120 (1992) 1675–1706. [3] D. P. Bacon , N. N. Ahmad, Z. Boybeyi, T. J. Dunn, M. S. Hall, P. C. S. Lee, R. A

  17. Comparison between Adaptive and Uniform Discontinuous Galerkin Simulations in Dry 2D Bubble Experiments

    DTIC Science & Technology

    2012-11-08

    Comparison between adaptive and uniform discontinuous Galerkin simulations in dry 2D bubble experiments Andreas Müllera,∗, Jörn Behrensb, Francis X...joern.behrens@zmaw.de (Jörn Behrens), fxgirald@nps.edu ( Francis X. Giraldo), vwirth@uni-mainz.de (Volkmar Wirth) Accepted by Journal of Computational...Mon. Weather Rev. 120 (1992) 1675–1706. [3] D. P. Bacon , N. N. Ahmad, Z. Boybeyi, T. J. Dunn, M. S. Hall, P. C. S. Lee, R. A. Sarma, M. D. Turner, K. T

  18. A free energy satisfying discontinuous Galerkin method for one-dimensional Poisson-Nernst-Planck systems

    NASA Astrophysics Data System (ADS)

    Liu, Hailiang; Wang, Zhongming

    2017-01-01

    We design an arbitrary-order free energy satisfying discontinuous Galerkin (DG) method for solving time-dependent Poisson-Nernst-Planck systems. Both the semi-discrete and fully discrete DG methods are shown to satisfy the corresponding discrete free energy dissipation law for positive numerical solutions. Positivity of numerical solutions is enforced by an accuracy-preserving limiter in reference to positive cell averages. Numerical examples are presented to demonstrate the high resolution of the numerical algorithm and to illustrate the proven properties of mass conservation, free energy dissipation, as well as the preservation of steady states.

  19. Numerical modelling of MPA-CVD reactors with the discontinuous Galerkin finite element method

    NASA Astrophysics Data System (ADS)

    Houston, Paul; Sime, Nathan

    2017-07-01

    In this article we develop a fully self consistent mathematical model describing the formation of a hydrogen plasma in a microwave power assisted chemical vapour deposition (MPA-CVD) reactor employed for the manufacture of synthetic diamond. The underlying multi-physics model includes constituent equations for the background gas mass average velocity, gas temperature, electromagnetic field energy and plasma density. The proposed mathematical model is numerically approximated based on exploiting the discontinuous Galerkin finite element method. We demonstrate the practical performance of this computational approach on a variety of CVD reactor geometries for a range of operating conditions.

  20. Towards Robust Discontinuous Galerkin Methods for General Relativistic Neutrino Radiation Transport

    NASA Astrophysics Data System (ADS)

    Endeve, E.; Hauck, C. D.; Xing, Y.; Mezzacappa, A.

    2015-10-01

    With an eye towards simulating neutrino transport in core-collapse supernovae, we have developed a conservative, robust, and high-order numerical method for solving the general relativistic phase space advection problem in stationary spacetimes. The method achieves high-order accuracy using Discontinuous Galerkin discretization and Runge-Kutta time integration. For robustness, care is taken to ensure that the physical bounds on the phase space distribution function are preserved; i.e., f ∈ [0,1]. We briefly describe the bound-preserving scheme, and present results from numerical experiments in spherical symmetry adopting the Schwarzschild metric, which demonstrate that the method preserves the bounds on the distribution function.

  1. Hamiltonian discontinuous Galerkin FEM for linear, stratified (in)compressible Euler equations: internal gravity waves

    NASA Astrophysics Data System (ADS)

    van Oers, Alexander M.; Maas, Leo R. M.; Bokhove, Onno

    2017-02-01

    The linear equations governing internal gravity waves in a stratified ideal fluid possess a Hamiltonian structure. A discontinuous Galerkin finite element method has been developed in which this Hamiltonian structure is discretized, resulting in conservation of discrete analogs of phase space and energy. This required (i) the discretization of the Hamiltonian structure using alternating flux functions and symplectic time integration, (ii) the discretization of a divergence-free velocity field using Dirac's theory of constraints and (iii) the handling of large-scale computational demands due to the 3-dimensional nature of internal gravity waves and, in confined, symmetry-breaking fluid domains, possibly its narrow zones of attraction.

  2. An efficient discontinuous Galerkin finite element method with nested domain decomposition for simulations of microresistivity imaging

    NASA Astrophysics Data System (ADS)

    Chen, Jiefu

    2015-03-01

    A discontinuous Galerkin finite element method is employed to study the responses of microresistivity imaging tools used in the oil and gas exploration industry. The multiscale structure of an imaging problem is decomposed into several nested subdomains based on its geometric characteristics. Each subdomain is discretized independently, and numerical flux is used to couple all subdomains together. The nested domain decomposition scheme will lead to a block tridiagonal linear system, and the block Thomas algorithm is utilized here to eliminate the subdomain based iteration in the step of solving the linear system. Numerical results demonstrate the validity and efficiency of this method.

  3. The ALE Discontinuous Galerkin Method for the Simulatio of Air Flow Through Pulsating Human Vocal Folds

    NASA Astrophysics Data System (ADS)

    Feistauer, Miloslav; Kučera, Václav; Prokopová, Jaroslav; Horáček, Jaromír

    2010-09-01

    The aim of this work is the simulation of viscous compressible flows in human vocal folds during phonation. The computational domain is a bounded subset of IR2, whose geometry mimics the shape of the human larynx. During phonation, parts of the solid impermeable walls are moving in a prescribed manner, thus simulating the opening and closing of the vocal chords. As the governing equations we take the compressible Navier-Stokes equations in ALE form. Space semidiscretization is carried out by the discontinuous Galerkin method combined with a linearized semi-implicit approach. Numerical experiments are performed with the resulting scheme.

  4. A new troubled-cell indicator for discontinuous Galerkin methods for hyperbolic conservation laws

    NASA Astrophysics Data System (ADS)

    Fu, Guosheng; Shu, Chi-Wang

    2017-10-01

    We introduce a new troubled-cell indicator for the discontinuous Galerkin (DG) methods for solving hyperbolic conservation laws. This indicator can be defined on unstructured meshes for high order DG methods and depends only on data from the target cell and its immediate neighbors. It is able to identify shocks without PDE sensitive parameters to tune. Extensive one- and two-dimensional simulations on the hyperbolic systems of Euler equations indicate the good performance of this new troubled-cell indicator coupled with a simple minmod-type TVD limiter for the Runge-Kutta DG (RKDG) methods.

  5. An hp -local discontinuous Galerkin method for some quasilinear elliptic boundary value problems of nonmonotone type

    NASA Astrophysics Data System (ADS)

    Gudi, Thirupathi; Nataraj, Neela; Pani, Amiya K.

    2008-06-01

    In this paper, an hp -local discontinuous Galerkin method is applied to a class of quasilinear elliptic boundary value problems which are of nonmonotone type. On hp -quasiuniform meshes, using the Brouwer fixed point theorem, it is shown that the discrete problem has a solution, and then using Lipschitz continuity of the discrete solution map, uniqueness is also proved. A priori error estimates in broken H^1 norm and L^2 norm which are optimal in h , suboptimal in p are derived. These results are exactly the same as in the case of linear elliptic boundary value problems. Numerical experiments are provided to illustrate the theoretical results.

  6. Development of a Perfectly Matched Layer Technique for a Discontinuous-Galerkin Spectral-Element Method

    NASA Technical Reports Server (NTRS)

    Garai, Anirban; Diosady, Laslo T.; Murman, Scott M.; Madavan, Nateri K.

    2016-01-01

    The perfectly matched layer (PML) technique is developed in the context of a high- order spectral-element Discontinuous-Galerkin (DG) method. The technique is applied to a range of test cases and is shown to be superior compared to other approaches, such as those based on using characteristic boundary conditions and sponge layers, for treating the inflow and outflow boundaries of computational domains. In general, the PML technique improves the quality of the numerical results for simulations of practical flow configurations, but it also exhibits some instabilities for large perturbations. A preliminary analysis that attempts to understand the source of these instabilities is discussed.

  7. A Discontinuous Petrov-Galerkin Methodology for Adaptive Solutions to the Incompressible Navier-Stokes Equations

    SciTech Connect

    Roberts, Nathan V.; Demkowiz, Leszek; Moser, Robert

    2015-11-15

    The discontinuous Petrov-Galerkin methodology with optimal test functions (DPG) of Demkowicz and Gopalakrishnan [18, 20] guarantees the optimality of the solution in an energy norm, and provides several features facilitating adaptive schemes. Whereas Bubnov-Galerkin methods use identical trial and test spaces, Petrov-Galerkin methods allow these function spaces to differ. In DPG, test functions are computed on the fly and are chosen to realize the supremum in the inf-sup condition; the method is equivalent to a minimum residual method. For well-posed problems with sufficiently regular solutions, DPG can be shown to converge at optimal rates—the inf-sup constants governing the convergence are mesh-independent, and of the same order as those governing the continuous problem [48]. DPG also provides an accurate mechanism for measuring the error, and this can be used to drive adaptive mesh refinements. We employ DPG to solve the steady incompressible Navier-Stokes equations in two dimensions, building on previous work on the Stokes equations, and focusing particularly on the usefulness of the approach for automatic adaptivity starting from a coarse mesh. We apply our approach to a manufactured solution due to Kovasznay as well as the lid-driven cavity flow, backward-facing step, and flow past a cylinder problems.

  8. A Hybrid Discontinuous/Continuous Galerkin Approach to Non-Hydrostatic Euler Model for Water Waves

    NASA Astrophysics Data System (ADS)

    Zhu, L.; Chen, Q. J.; Wan, X.

    2016-02-01

    Coastal communities and ecosystems in near-shore areas are prone to the impacts of hurricanes, winter storms, tsunamis, and chronic sea-level rise. Numerical modeling has become an indispensable tool in coastal engineering and science. Rapid advances in computational technology allow solving the Euler equations or Navier-Stokes equations for non-hydrostatic free surface waves directly for practical applications. This work presents a non-hydrostatic Euler model for fully dispersive nonlinear water waves, and its application to wave propagation from deep sea to the shoreline. The σ-coordinate maps the irregular physical domain to the regular computation domain. A numerical scheme based on a hybrid of discontinuous Galerkin (DG) method and continuous Galerkin (CG) method is proposed. This scheme retains the advantages of Galerkin methods such as high-order approximation, hp-adaptivity (element size and polynomial order, respectively), easy handling complex geometry, etc., and reduces the computational cost by coupling the CG and DG method. The numerical algorithm is split into hydrostatic and non-hydrostatic phases. In the hydrostatic phase, the DG method is employed to compute the mass and momentum flux explicitly. In the following non-hydrostatic phase, the CG method is adopted to solve the Poisson equation to obtain pressure correction. Model validation is carried out by conducting numerical test cases of wave propagation over a submerged bar, wave shoaling, breaking and run-up. The numerical results are compared with analytical solutions and experimental data. Detailed results will be presented at the conference.

  9. Non-conformal and parallel discontinuous Galerkin time domain method for Maxwell's equations: EM analysis of IC packages

    NASA Astrophysics Data System (ADS)

    Dosopoulos, Stylianos; Zhao, Bo; Lee, Jin-Fa

    2013-04-01

    In this article, we present an Interior Penalty discontinuous Galerkin Time Domain (IPDGTD) method on non-conformal meshes. The motivation for a non-conformal IPDGTD comes from the fact there are applications with very complicated geometries (for example, IC packages) where a conformal mesh may be very difficult to obtain. Therefore, the ability to handle non-conformal meshes really comes in handy. In the proposed approach, we first decompose the computational domain into non-overlapping subdomains. Afterward, each sub-domain is meshed independently resulting in non-conformal domain interfaces, but simultaneously providing great flexibility in the meshing process. The non-conformal triangulations at sub-domain interfaces can be naturally supported within the IPDGTD framework. Moreover, a MPI parallelization together with a local time-stepping strategy is applied to significantly increase the efficiency of the method. Furthermore, a general balancing strategy is described. Through a practical example with multi-scale features, it is shown that the proposed balancing strategy leads to better use of the available computational resources and reduces substantially the total simulation time. Finally, numerical results are included to validate the accuracy and demonstrate the flexibilities of the proposed non-conformal IPDGTD.

  10. Discontinuous Galerkin method with Gaussian artificial viscosity on graphical processing units for nonlinear acoustics

    NASA Astrophysics Data System (ADS)

    Tripathi, Bharat B.; Marchiano, Régis; Baskar, Sambandam; Coulouvrat, François

    2015-10-01

    Propagation of acoustical shock waves in complex geometry is a topic of interest in the field of nonlinear acoustics. For instance, simulation of Buzz Saw Noice requires the treatment of shock waves generated by the turbofan through the engines of aeroplanes with complex geometries and wall liners. Nevertheless, from a numerical point of view it remains a challenge. The two main hurdles are to take into account the complex geometry of the domain and to deal with the spurious oscillations (Gibbs phenomenon) near the discontinuities. In this work, first we derive the conservative hyperbolic system of nonlinear acoustics (up to quadratic nonlinear terms) using the fundamental equations of fluid dynamics. Then, we propose to adapt the classical nodal discontinuous Galerkin method to develop a high fidelity solver for nonlinear acoustics. The discontinuous Galerkin method is a hybrid of finite element and finite volume method and is very versatile to handle complex geometry. In order to obtain better performance, the method is parallelized on Graphical Processing Units. Like other numerical methods, discontinuous Galerkin method suffers with the problem of Gibbs phenomenon near the shock, which is a numerical artifact. Among the various ways to manage these spurious oscillations, we choose the method of parabolic regularization. Although, the introduction of artificial viscosity into the system is a popular way of managing shocks, we propose a new approach of introducing smooth artificial viscosity locally in each element, wherever needed. Firstly, a shock sensor using the linear coefficients of the spectral solution is used to locate the position of the discontinuities. Then, a viscosity coefficient depending on the shock sensor is introduced into the hyperbolic system of equations, only in the elements near the shock. The viscosity is applied as a two-dimensional Gaussian patch with its shape parameters depending on the element dimensions, referred here as Element

  11. Solving 3D relativistic hydrodynamical problems with weighted essentially nonoscillatory discontinuous Galerkin methods

    NASA Astrophysics Data System (ADS)

    Bugner, Marcus; Dietrich, Tim; Bernuzzi, Sebastiano; Weyhausen, Andreas; Brügmann, Bernd

    2016-10-01

    Discontinuous Galerkin (DG) methods coupled to weighted essentially nonoscillatory (WENO) algorithms allow high order convergence for smooth problems and for the simulation of discontinuities and shocks. In this work, we investigate WENO-DG algorithms in the context of numerical general relativity, in particular for general relativistic hydrodynamics. We implement the standard WENO method at different orders, a compact (simple) WENO scheme, as well as an alternative subcell evolution algorithm. To evaluate the performance of the different numerical schemes, we study nonrelativistic, special relativistic, and general relativistic test beds. We present the first three-dimensional simulations of general relativistic hydrodynamics, albeit for a fixed spacetime background, within the framework of WENO-DG methods. The most important test bed is a single Tolman-Oppenheimer-Volkoff star in three dimensions, showing that long term stable simulations of single isolated neutron stars can be obtained with WENO-DG methods.

  12. Dynamic Rupture Modeling in Three Dimensions on Unstructured Meshes Using a Discontinuous Galerkin Method

    NASA Astrophysics Data System (ADS)

    Pelties, C.; Käser, M.

    2010-12-01

    We will present recent developments concerning the extensions of the ADER-DG method to solve three dimensional dynamic rupture problems on unstructured tetrahedral meshes. The simulation of earthquake rupture dynamics and seismic wave propagation using a discontinuous Galerkin (DG) method in 2D was recently presented by J. de la Puente et al. (2009). A considerable feature of this study regarding spontaneous rupture problems was the combination of the DG scheme and a time integration method using Arbitrarily high-order DERivatives (ADER) to provide high accuracy in space and time with the discretization on unstructured meshes. In the resulting discrete velocity-stress formulation of the elastic wave equations variables are naturally discontinuous at the interfaces between elements. The so-called Riemann problem can then be solved to obtain well defined values of the variables at the discontinuity itself. This is in particular valid for the fault at which a certain friction law has to be evaluated. Hence, the fault’s geometry is honored by the computational mesh. This way, complex fault planes can be modeled adequately with small elements while fast mesh coarsening is possible with increasing distance from the fault. Due to the strict locality of the scheme using only direct neighbor communication, excellent parallel behavior can be observed. A further advantage of the scheme is that it avoids spurious high-frequency contributions in the slip rate spectra and therefore does not require artificial Kelvin-Voigt damping or filtering of synthetic seismograms. In order to test the accuracy of the ADER-DG method the Southern California Earthquake Center (SCEC) benchmark for spontaneous rupture simulations was employed. Reference: J. de la Puente, J.-P. Ampuero, and M. Käser (2009), Dynamic rupture modeling on unstructured meshes using a discontinuous Galerkin method, JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114, B10302, doi:10.1029/2008JB006271

  13. Elastic wave propagation in variable media using a discontinuous Galerkin method.

    SciTech Connect

    Ober, Curtis Curry; Smith, Thomas Michael; Collis, Samuel Scott; Overfelt, James Robert; Schwaiger, Hans

    2010-04-01

    Motivated by the needs of seismic inversion and building on our prior experience for fluid-dynamics systems, we present a high-order discontinuous Galerkin (DG) Runge-Kutta method applied to isotropic, linearized elasto-dynamics. Unlike other DG methods recently presented in the literature, our method allows for inhomogeneous material variations within each element that enables representation of realistic earth models - a feature critical for future use in seismic inversion. Likewise, our method supports curved elements and hybrid meshes that include both simplicial and nonsimplicial elements. We demonstrate the capabilities of this method through a series of numerical experiments including hybrid mesh discretizations of the Marmousi2 model as well as a modified Marmousi2 model with a oscillatory ocean bottom that is exactly captured by our discretization. A discontinuous Galerkin method for solving the equations of linear isotropic elasticity has been presented. The formulation is designed to accommodate variation of media parameters within elements, curved elements and unstructured heterogeneous meshes. We have demonstrated that each of these important features of the formulation can produce results that are significantly different from formulations that do not possess these capabilities suggesting that each of these capabilities may be important for effective full waveform inversion of elastic medium.

  14. A Reconstructed Discontinuous Galerkin Method for the Compressible Euler Equations on Arbitrary Grids

    SciTech Connect

    Hong Luo; Luquing Luo; Robert Nourgaliev; Vincent Mousseau

    2009-06-01

    A reconstruction-based discontinuous Galerkin (DG) method is presented for the solution of the compressible Euler equations on arbitrary grids. By taking advantage of handily available and yet invaluable information, namely the derivatives, in the context of the discontinuous Galerkin methods, a solution polynomial of one degree higher is reconstructed using a least-squares method. The stencils used in the reconstruction involve only the van Neumann neighborhood (face-neighboring cells) and are compact and consistent with the underlying DG method. The resulting DG method can be regarded as an improvement of a recovery-based DG method in the sense that it shares the same nice features as the recovery-based DG method, such as high accuracy and efficiency, and yet overcomes some of its shortcomings such as a lack of flexibility, compactness, and robustness. The developed DG method is used to compute a variety of flow problems on arbitrary meshes to demonstrate the accuracy and efficiency of the method. The numerical results indicate that this reconstructed DG method is able to obtain a third-order accurate solution at a slightly higher cost than its second-order DG method and provide an increase in performance over the third order DG method in terms of computing time and storage requirement.

  15. Discontinuous Galerkin methods with plane waves for time-harmonic problems

    SciTech Connect

    Gabard, Gwenael . E-mail: gabard@soton.ac.uk

    2007-08-10

    A general framework for discontinuous Galerkin methods in the frequency domain with numerical flux is presented. The main feature of the method is the use of plane waves instead of polynomials to approximate the solution in each element. The method is formulated for a general system of linear hyperbolic equations and is applied to problems of aeroacoustic propagation by solving the two-dimensional linearized Euler equations. It is found that the method requires only a small number of elements per wavelength to obtain accurate solutions and that it is more efficient than high-order DRP schemes. In addition, the conditioning of the method is found to be high but not critical in practice. It is shown that the Ultra-Weak Variational Formulation is in fact a subset of the present discontinuous Galerkin method. A special extension of the method is devised in order to deal with singular solutions generated by point sources like monopoles or dipoles. Aeroacoustic problems with non-uniform flows are also considered and results are presented for the sound radiated from a two-dimensional jet.

  16. A space-time hybridizable discontinuous Galerkin method for incompressible flows on deforming domains

    NASA Astrophysics Data System (ADS)

    Rhebergen, Sander; Cockburn, Bernardo

    2012-06-01

    We present the first space-time hybridizable discontinuous Galerkin (HDG) finite element method for the incompressible Navier-Stokes and Oseen equations. Major advantages of a space-time formulation are its excellent capabilities of dealing with moving and deforming domains and grids and its ability to achieve higher-order accurate approximations in both time and space by simply increasing the order of polynomial approximation in the space-time elements. Our formulation is related to the HDG formulation for incompressible flows introduced recently in, e.g., [N.C. Nguyen, J. Peraire, B. Cockburn, A hybridizable discontinuous Galerkin method for Stokes flow, Comput. Methods Appl. Mech. Eng. 199 (2010) 582-597]. However, ours is inspired in typical DG formulations for compressible flows which allow for a more straightforward implementation. Another difference is the use of polynomials of fixed total degree with space-time hexahedral and quadrilateral elements, instead of simplicial elements. We present numerical experiments in order to assess the quality of the performance of the methods on deforming domains and to experimentally investigate the behavior of the convergence rates of each component of the solution with respect to the polynomial degree of the approximations in both space and time.

  17. Numerical solution of the Navier-Stokes equations by discontinuous Galerkin method

    NASA Astrophysics Data System (ADS)

    Krasnov, M. M.; Kuchugov, P. A.; E Ladonkina, M.; E Lutsky, A.; Tishkin, V. F.

    2017-02-01

    Detailed unstructured grids and numerical methods of high accuracy are frequently used in the numerical simulation of gasdynamic flows in areas with complex geometry. Galerkin method with discontinuous basis functions or Discontinuous Galerkin Method (DGM) works well in dealing with such problems. This approach offers a number of advantages inherent to both finite-element and finite-difference approximations. Moreover, the present paper shows that DGM schemes can be viewed as Godunov method extension to piecewise-polynomial functions. As is known, DGM involves significant computational complexity, and this brings up the question of ensuring the most effective use of all the computational capacity available. In order to speed up the calculations, operator programming method has been applied while creating the computational module. This approach makes possible compact encoding of mathematical formulas and facilitates the porting of programs to parallel architectures, such as NVidia CUDA and Intel Xeon Phi. With the software package, based on DGM, numerical simulations of supersonic flow past solid bodies has been carried out. The numerical results are in good agreement with the experimental ones.

  18. A high order characteristic discontinuous Galerkin scheme for advection on unstructured meshes

    NASA Astrophysics Data System (ADS)

    Lee, D.; Lowrie, R.; Petersen, M.; Ringler, T.; Hecht, M.

    2016-11-01

    A new characteristic discontinuous Galerkin (CDG) advection scheme is presented. In contrast to standard discontinuous Galerkin schemes, the test functions themselves follow characteristics in order to ensure conservation and the edges of each element are also traced backwards along characteristics in order to create a swept region, which is integrated in order to determine the mass flux across the edge. Both the accuracy and performance of the scheme are greatly improved by the use of large Courant-Friedrichs-Lewy numbers for a shear flow test case and the scheme is shown to scale sublinearly with the number of tracers being advected, outperforming a standard flux corrected transport scheme for 10 or more tracers with a linear basis. Moreover the CDG scheme may be run to arbitrarily high order spatial accuracy and on unstructured grids, and is shown to give the correct order of error convergence for piecewise linear and quadratic bases on regular quadrilateral and hexahedral planar grids. Using a modal Taylor series basis, the scheme may be made monotone while preserving conservation with the use of a standard slope limiter, although this reduces the formal accuracy of the scheme to first order. The second order scheme is roughly as accurate as the incremental remap scheme with nonlocal gradient reconstruction at half the horizontal resolution. The scheme is being developed for implementation within the Model for Prediction Across Scales (MPAS) Ocean model, an unstructured grid finite volume ocean model.

  19. Local Discontinuous Galerkin Approximations And Variable Step Size, Variable Order Time Integration For Richards' Equation

    NASA Astrophysics Data System (ADS)

    Li, H.; Farthing, M. W.; Dawson, C. N.; Miller, C. T.

    2004-12-01

    Numerical simulation of Richards' equation continues to be difficult. It is highly nonlinear under common constitutive relations and exhibits sharp fronts in both the pressure head and volume fraction for many problems of interest. For a number of multiphase flow problems, the use of variable order and variable step size temporal discretizations has shown some advantages. However, the spatial discretizations commonly used for variably saturated flow are dominated by nonadaptive, low-order finite difference and finite element methods. Discontinuous Galerkin (DG) finite element methods have received significant attention in a number of fields for hyperbolic PDE's and, more recently, for elliptic and parabolic problems. DG approaches like the local discontinuous Galerkin (LDG) method are appealing for modeling subsurface flow since they can lead to velocity fields that are locally mass-conservative without the need for auxiliary variables or alternative meshes. DG discretizations are also inherently local and so better-suited for unstructured meshes and h-p adaption strategies than traditional methods. While some work has been done recently for multiphase subsurface flow, there are a range of issues related to the performance of DG methods for highly nonlinear parabolic problems like Richards' equation that have not been investigated fully. In this work, we consider the combination of higher order adaptive time integration with an LDG spatial discretization for Richards' equation. We compare this approach to standard low-order methods for a series of test problems and consider a number of issues including the methods' relative accuracy and computational efficiency.

  20. Baroclinic waves on the β plane using low-order Discontinuous Galerkin discretisation

    NASA Astrophysics Data System (ADS)

    Orgis, Thomas; Läuter, Matthias; Handorf, Dörthe; Dethloff, Klaus

    2017-06-01

    A classic Discontinuous Galerkin Method with low-order polynomials and explicit as well as semi-implicit time-stepping is applied to an atmospheric model employing the Euler equations on the β plane. The method, which was initially proposed without regard for the source terms and their balance with the pressure gradient that dominates atmospheric dynamics, needs to be adapted to be able to keep the combined geostrophic and hydrostatic balance in three spatial dimensions. This is achieved inside the discretisation through a polynomial mapping of both source and flux terms without imposing filters between time steps. After introduction and verification of this balancing, the realistic development of barotropic and baroclinic waves in the model is demonstrated, including the formation of a retrograde Rossby wave pattern. A prerequesite is the numerical solution of the thermal wind equation to construct geostrophically balanced initial states in z coordinates with arbitrary prescribed zonal wind profile, offering a new set of test cases for atmospheric models employing z coordinates. The resulting simulations demonstrate that the balanced low-order Discontinuous Galerkin discretisation with polynomial degrees down to k = 1 can be a viable option for atmospheric modelling.

  1. Lyapunov exponents and adaptive mesh refinement for high-speed flows using a discontinuous Galerkin scheme

    NASA Astrophysics Data System (ADS)

    Moura, R. C.; Silva, A. F. C.; Bigarella, E. D. V.; Fazenda, A. L.; Ortega, M. A.

    2016-08-01

    This paper proposes two important improvements to shock-capturing strategies using a discontinuous Galerkin scheme, namely, accurate shock identification via finite-time Lyapunov exponent (FTLE) operators and efficient shock treatment through a point-implicit discretization of a PDE-based artificial viscosity technique. The advocated approach is based on the FTLE operator, originally developed in the context of dynamical systems theory to identify certain types of coherent structures in a flow. We propose the application of FTLEs in the detection of shock waves and demonstrate the operator's ability to identify strong and weak shocks equally well. The detection algorithm is coupled with a mesh refinement procedure and applied to transonic and supersonic flows. While the proposed strategy can be used potentially with any numerical method, a high-order discontinuous Galerkin solver is used in this study. In this context, two artificial viscosity approaches are employed to regularize the solution near shocks: an element-wise constant viscosity technique and a PDE-based smooth viscosity model. As the latter approach is more sophisticated and preferable for complex problems, a point-implicit discretization in time is proposed to reduce the extra stiffness introduced by the PDE-based technique, making it more competitive in terms of computational cost.

  2. The Elastoplast Discontinuous Galerkin (EDG) method for the Navier-Stokes equations

    NASA Astrophysics Data System (ADS)

    Borrel, M.; Ryan, J.

    2012-01-01

    The present work details the Elastoplast (this name is a translation from the French "sparadrap", a concept first applied by Yves Morchoisne for Spectral methods [1]) Discontinuous Galerkin (EDG) method to solve the compressible Navier-Stokes equations. This method was first presented in 2009 at the ICOSAHOM congress with some Cartesian grid applications. We focus here on unstructured grid applications for which the EDG method seems very attractive. As in the Recovery method presented by van Leer and Nomura in 2005 for diffusion, jumps across element boundaries are locally eliminated by recovering the solution on an overlapping cell. In the case of Recovery, this cell is the union of the two neighboring cells and the Galerkin basis is twice as large as the basis used for one element. In our proposed method the solution is rebuilt through an L2 projection of the discontinuous interface solution on a small rectangular overlapping interface element, named Elastoplast, with an orthogonal basis of the same order as the one in the neighboring cells. Comparisons on 1D and 2D scalar diffusion problems in terms of accuracy and stability with other viscous DG schemes are first given. Then, 2D results on acoustic problems, vortex problems and boundary layer problems both on Cartesian or unstructured triangular grids illustrate stability, precision and versatility of this method.

  3. The direct Discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids

    NASA Astrophysics Data System (ADS)

    Yang, Xiaoquan; Cheng, Jian; Liu, Tiegang; Luo, Hong

    2015-11-01

    The direct discontinuous Galerkin (DDG) method based on a traditional discontinuous Galerkin (DG) formulation is extended and implemented for solving the compressible Navier-Stokes equations on arbitrary grids. Compared to the widely used second Bassi-Rebay (BR2) scheme for the discretization of diffusive fluxes, the DDG method has two attractive features: first, it is simple to implement as it is directly based on the weak form, and therefore there is no need for any local or global lifting operator; second, it can deliver comparable results, if not better than BR2 scheme, in a more efficient way with much less CPU time. Two approaches to perform the DDG flux for the Navier- Stokes equations are presented in this work, one is based on conservative variables, the other is based on primitive variables. In the implementation of the DDG method for arbitrary grid, the definition of mesh size plays a critical role as the formation of viscous flux explicitly depends on the geometry. A variety of test cases are presented to demonstrate the accuracy and efficiency of the DDG method for discretizing the viscous fluxes in the compressible Navier-Stokes equations on arbitrary grids.

  4. High performance computing aspects of a dimension independent semi-Lagrangian discontinuous Galerkin code

    NASA Astrophysics Data System (ADS)

    Einkemmer, Lukas

    2016-05-01

    The recently developed semi-Lagrangian discontinuous Galerkin approach is used to discretize hyperbolic partial differential equations (usually first order equations). Since these methods are conservative, local in space, and able to limit numerical diffusion, they are considered a promising alternative to more traditional semi-Lagrangian schemes (which are usually based on polynomial or spline interpolation). In this paper, we consider a parallel implementation of a semi-Lagrangian discontinuous Galerkin method for distributed memory systems (so-called clusters). Both strong and weak scaling studies are performed on the Vienna Scientific Cluster 2 (VSC-2). In the case of weak scaling we observe a parallel efficiency above 0.8 for both two and four dimensional problems and up to 8192 cores. Strong scaling results show good scalability to at least 512 cores (we consider problems that can be run on a single processor in reasonable time). In addition, we study the scaling of a two dimensional Vlasov-Poisson solver that is implemented using the framework provided. All of the simulations are conducted in the context of worst case communication overhead; i.e., in a setting where the CFL (Courant-Friedrichs-Lewy) number increases linearly with the problem size. The framework introduced in this paper facilitates a dimension independent implementation of scientific codes (based on C++ templates) using both an MPI and a hybrid approach to parallelization. We describe the essential ingredients of our implementation.

  5. h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems

    NASA Astrophysics Data System (ADS)

    Botti, L.; Colombo, A.; Bassi, F.

    2017-10-01

    In this work we exploit agglomeration based h-multigrid preconditioners to speed-up the iterative solution of discontinuous Galerkin discretizations of the Stokes and Navier-Stokes equations. As a distinctive feature h-coarsened mesh sequences are generated by recursive agglomeration of a fine grid, admitting arbitrarily unstructured grids of complex domains, and agglomeration based discontinuous Galerkin discretizations are employed to deal with agglomerated elements of coarse levels. Both the expense of building coarse grid operators and the performance of the resulting multigrid iteration are investigated. For the sake of efficiency coarse grid operators are inherited through element-by-element L2 projections, avoiding the cost of numerical integration over agglomerated elements. Specific care is devoted to the projection of viscous terms discretized by means of the BR2 dG method. We demonstrate that enforcing the correct amount of stabilization on coarse grids levels is mandatory for achieving uniform convergence with respect to the number of levels. The numerical solution of steady and unsteady, linear and non-linear problems is considered tackling challenging 2D test cases and 3D real life computations on parallel architectures. Significant execution time gains are documented.

  6. An iterative phase-space explicit discontinuous Galerkin method for stellar radiative transfer in extended atmospheres

    DOE PAGES

    de Almeida, Valmor F

    2017-04-19

    In this work, a phase-space discontinuous Galerkin (PSDG) method is presented for the solution of stellar radiative transfer problems. It allows for greater adaptivity than competing methods without sacrificing generality. The method is extensively tested on a spherically symmetric, static, inverse-power-law scattering atmosphere. Results for different sizes of atmospheres and intensities of scattering agreed with asymptotic values. The exponentially decaying behavior of the radiative field in the diffusive-transparent transition region, and the forward peaking behavior at the surface of extended atmospheres were accurately captured. The integrodifferential equation of radiation transfer is solved iteratively by alternating between the radiative pressure equationmore » and the original equation with the integral term treated as an energy density source term. In each iteration, the equations are solved via an explicit, flux-conserving, discontinuous Galerkin method. Finite elements are ordered in wave fronts perpendicular to the characteristic curves so that elemental linear algebraic systems are solved quickly by sweeping the phase space element by element. Two implementations of a diffusive boundary condition at the origin are demonstrated wherein the finite discontinuity in the radiation intensity is accurately captured by the proposed method. This allows for a consistent mechanism to preserve photon luminosity. The method was proved to be robust and fast, and a case is made for the adequacy of parallel processing. In addition to classical two-dimensional plots, results of normalized radiation intensity were mapped onto a log-polar surface exhibiting all distinguishing features of the problem studied.« less

  7. Coupling p-multigrid to geometric multigrid for discontinuous Galerkin formulations of the convection-diffusion equation

    NASA Astrophysics Data System (ADS)

    Mascarenhas, Brendan S.; Helenbrook, Brian T.; Atkins, Harold L.

    2010-05-01

    An improved p-multigrid algorithm for discontinuous Galerkin (DG) discretizations of convection-diffusion problems is presented. The general p-multigrid algorithm for DG discretizations involves a restriction from the p=1 to p=0 discontinuous polynomial solution spaces. This restriction is problematic and has limited the efficiency of the p-multigrid method. For purely diffusive problems, Helenbrook and Atkins have demonstrated rapid convergence using a method that restricts from a discontinuous to continuous polynomial solution space at p=1. It is shown that this method is not directly applicable to the convection-diffusion (CD) equation because it results in a central-difference discretization for the convective term. To remedy this, ideas from the streamwise upwind Petrov-Galerkin (SUPG) formulation are used to devise a transition from the discontinuous to continuous space at p=1 that yields an upwind discretization. The results show that the new method converges rapidly for all Peclet numbers.

  8. On high-order accurate weighted essentially non-oscillatory and discontinuous Galerkin schemes for compressible turbulence simulations.

    PubMed

    Shu, Chi-Wang

    2013-01-13

    In this article, we give a brief overview on high-order accurate shock capturing schemes with the aim of applications in compressible turbulence simulations. The emphasis is on the basic methodology and recent algorithm developments for two classes of high-order methods: the weighted essentially non-oscillatory and discontinuous Galerkin methods.

  9. Discontinuous Galerkin finite element method for solving population density functions of cortical pyramidal and thalamic neuronal populations.

    PubMed

    Huang, Chih-Hsu; Lin, Chou-Ching K; Ju, Ming-Shaung

    2015-02-01

    Compared with the Monte Carlo method, the population density method is efficient for modeling collective dynamics of neuronal populations in human brain. In this method, a population density function describes the probabilistic distribution of states of all neurons in the population and it is governed by a hyperbolic partial differential equation. In the past, the problem was mainly solved by using the finite difference method. In a previous study, a continuous Galerkin finite element method was found better than the finite difference method for solving the hyperbolic partial differential equation; however, the population density function often has discontinuity and both methods suffer from a numerical stability problem. The goal of this study is to improve the numerical stability of the solution using discontinuous Galerkin finite element method. To test the performance of the new approach, interaction of a population of cortical pyramidal neurons and a population of thalamic neurons was simulated. The numerical results showed good agreement between results of discontinuous Galerkin finite element and Monte Carlo methods. The convergence and accuracy of the solutions are excellent. The numerical stability problem could be resolved using the discontinuous Galerkin finite element method which has total-variation-diminishing property. The efficient approach will be employed to simulate the electroencephalogram or dynamics of thalamocortical network which involves three populations, namely, thalamic reticular neurons, thalamocortical neurons and cortical pyramidal neurons.

  10. Error Analysis of p-Version Discontinuous Galerkin Method for Heat Transfer in Built-up Structures

    NASA Technical Reports Server (NTRS)

    Kaneko, Hideaki; Bey, Kim S.

    2004-01-01

    The purpose of this paper is to provide an error analysis for the p-version of the discontinuous Galerkin finite element method for heat transfer in built-up structures. As a special case of the results in this paper, a theoretical error estimate for the numerical experiments recently conducted by James Tomey is obtained.

  11. Advanced Discontinuous Galerkin Algorithms and First Open-Field Line Turbulence Simulations

    NASA Astrophysics Data System (ADS)

    Hammett, G. W.; Hakim, A.; Shi, E. L.

    2016-10-01

    New versions of Discontinuous Galerkin (DG) algorithms have interesting features that may help with challenging problems of higher-dimensional kinetic problems. We are developing the gyrokinetic code Gkeyll based on DG. DG also has features that may help with the next generation of Exascale computers. Higher-order methods do more FLOPS to extract more information per byte, thus reducing memory and communications costs (which are a bottleneck at exascale). DG uses efficient Gaussian quadrature like finite elements, but keeps the calculation local for the kinetic solver, also reducing communication. Sparse grid methods might further reduce the cost significantly in higher dimensions. The inner product norm can be chosen to preserve energy conservation with non-polynomial basis functions (such as Maxwellian-weighted bases), which can be viewed as a Petrov-Galerkin method. This allows a full- F code to benefit from similar Gaussian quadrature as used in popular δf gyrokinetic codes. Consistent basis functions avoid high-frequency numerical modes from electromagnetic terms. We will show our first results of 3 x + 2 v simulations of open-field line/SOL turbulence in a simple helical geometry (like Helimak/TORPEX), with parameters from LAPD, TORPEX, and NSTX. Supported by the Max-Planck/Princeton Center for Plasma Physics, the SciDAC Center for the Study of Plasma Microturbulence, and DOE Contract DE-AC02-09CH11466.

  12. Features of Discontinuous Galerkin Algorithms in Gkeyll, and Exponentially-Weighted Basis Functions

    NASA Astrophysics Data System (ADS)

    Hammett, G. W.; Hakim, A.; Shi, E. L.

    2016-10-01

    There are various versions of Discontinuous Galerkin (DG) algorithms that have interesting features that could help with challenging problems of higher-dimensional kinetic problems (such as edge turbulence in tokamaks and stellarators). We are developing the gyrokinetic code Gkeyll based on DG methods. Higher-order methods do more FLOPS to extract more information per byte, thus reducing memory and communication costs (which are a bottleneck for exascale computing). The inner product norm can be chosen to preserve energy conservation with non-polynomial basis functions (such as Maxwellian-weighted bases), which alternatively can be viewed as a Petrov-Galerkin method. This allows a full- F code to benefit from similar Gaussian quadrature employed in popular δf continuum gyrokinetic codes. We show some tests for a 1D Spitzer-Härm heat flux problem, which requires good resolution for the tail. For two velocity dimensions, this approach could lead to a factor of 10 or more speedup. Supported by the Max-Planck/Princeton Center for Plasma Physics, the SciDAC Center for the Study of Plasma Microturbulence, and DOE Contract DE-AC02-09CH11466.

  13. Conservative discontinuous Galerkin discretizations of the 2D incompressible Euler equation

    NASA Astrophysics Data System (ADS)

    Waelbroeck, Francois; Michoski, Craig; Bernard, Tess

    2016-10-01

    Discontinuous Galerkin (DG) methods provide local high-order adaptive numerical schemes for the solution of convection-diffusion problems. They combine the advantages of finite element and finite volume methods. In particular, DG methods automatically ensure the conservation of all first-order invariants provided that single-valued fluxes are prescribed at inter-element boundaries. For the 2D incompressible Euler equation, this implies that the discretized fluxes globally obey Gauss' and Stokes' laws exactly, and that they conserve total vorticity. Liu and Shu have shown that combining a continuous Galerkin (CG) solution of Poisson's equation with a central DG flux for the convection term leads to an algorithm that conserves the principal two quadratic invariants, namely the energy and enstrophy. Here, we present a discretization that applies the DG method to Poisson's equation as well as to the vorticity equation while maintaining conservation of the quadratic invariants. Using a DG algorithm for Poisson's equation can be advantageous when solving problems with mixed Dirichlet-Neuman boundary conditions such as for the injection of fluid through a slit (Bickley jet) or during compact toroid injection for tokamak startup.

  14. Efficient construction of unified continuous and discontinuous Galerkin formulations for the 3D Euler equations

    NASA Astrophysics Data System (ADS)

    Abdi, Daniel S.; Giraldo, Francis X.

    2016-09-01

    A unified approach for the numerical solution of the 3D hyperbolic Euler equations using high order methods, namely continuous Galerkin (CG) and discontinuous Galerkin (DG) methods, is presented. First, we examine how classical CG that uses a global storage scheme can be constructed within the DG framework using constraint imposition techniques commonly used in the finite element literature. Then, we implement and test a simplified version in the Non-hydrostatic Unified Model of the Atmosphere (NUMA) for the case of explicit time integration and a diagonal mass matrix. Constructing CG within the DG framework allows CG to benefit from the desirable properties of DG such as, easier hp-refinement, better stability etc. Moreover, this representation allows for regional mixing of CG and DG depending on the flow regime in an area. The different flavors of CG and DG in the unified implementation are then tested for accuracy and performance using a suite of benchmark problems representative of cloud-resolving scale, meso-scale and global-scale atmospheric dynamics. The value of our unified approach is that we are able to show how to carry both CG and DG methods within the same code and also offer a simple recipe for modifying an existing CG code to DG and vice versa.

  15. Analysis of p-multigrid solution schemes for discontinuous Galerkin discretizations of flow problems

    NASA Astrophysics Data System (ADS)

    Mascarenhas, Brendan S.

    p-multigrid is a 'multigrid-like' algorithm used to obtain solutions to high-order hp-finite element discretizations. In this method convergence is accelerated by using coarse levels constructed by reducing the order, p, of the approximating polynomial. We have investigated p-multigrid coupled with preconditioned block relaxation schemes to obtain the steady-state solution to discontinuous Galerkin (DG) discretizations of the Euler equations. Block-diagonal, -line, and sweeping preconditioners, and also the alternate direction implicit (ADI), and the incomplete lower-upper (ILU(0)) preconditioners are considered. Relaxation schemes that approximately-invert (AI) the steady-state stiffness matrix and implicit psuedo time-advancing (ITA) schemes are Fourier analyzed and compared. In general, for orders of approximating polynomial p ≥ 2, the AI schemes perform better than the similarly preconditioned ITA schemes. The results show that p-multigrid iterations of the AI-ILU(0) scheme with under-relaxation o = 1/2 converge fastest and are the most robust of the schemes studied. Similar to prior observations by Helenbrook and Atkins p-multigrid was observed to behave anomalously when p transitions from 1 to 0. Using ideas from Helenbrook and Atkins correction for diffusion, and the streamwise upwind Petrov-Galerkin (SUPG) formulation, this anomalous behavior is corrected for the 1D convection equation. The correction is then extended to the 1D convection-diffusion equation.

  16. Tests of Maxwellian-Weighted Basis Functions in a Discontinuous Galerkin Kinetic Code

    NASA Astrophysics Data System (ADS)

    Hammett, G. W.; Hakim, A.; Shi, E. L.

    2013-10-01

    Discontinuous Galerkin (DG) algorithms have been very actively studied and used in the applied math and computational fluid dynamics communities in the past decade. They combine certain attractive properties of finite element methods (like high accuracy per interpolation point) and finite volume methods (like locality of calculation for parallel computers and flexibility for limiters). Higher-order methods also have more floating point operations per data point, and so can be more efficient on modern computers that are often bandwidth limited. The flexibility of DG allows one to consider various types of Maxwellian-weighted basis functions while preserving important conservation properties of the underlying system. One can think of this either as a modified inner-product norm or a Petrov-Galerkin approach. Here we explore some ways of using Maxwellian-Weighted Basis functions and test them on paradigm problems using the Gkeyll code, which is being developed for edge gyrokinetic simulations. In addition to the formal order of accuracy in the asymptotic limit as a grid is refined, we are also interested in robust reasonable solutions on coarser grids. This work was supported by the Max-Planck/Princeton Center for Plasma Physics and DOE Contract DE-AC02-09CH11466.

  17. Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations

    SciTech Connect

    Rhebergen, S. Bokhove, O. Vegt, J.J.W. van der

    2008-01-10

    We present space- and space-time discontinuous Galerkin finite element (DGFEM) formulations for systems containing nonconservative products, such as occur in dispersed multiphase flow equations. The main criterium we pose on the weak formulation is that if the system of nonconservative partial differential equations can be transformed into conservative form, then the formulation must reduce to that for conservative systems. Standard DGFEM formulations cannot be applied to nonconservative systems of partial differential equations. We therefore introduce the theory of weak solutions for nonconservative products into the DGFEM formulation leading to the new question how to define the path connecting left and right states across a discontinuity. The effect of different paths on the numerical solution is investigated and found to be small. We also introduce a new numerical flux that is able to deal with nonconservative products. Our scheme is applied to two different systems of partial differential equations. First, we consider the shallow water equations, where topography leads to nonconservative products, in which the known, possibly discontinuous, topography is formally taken as an unknown in the system. Second, we consider a simplification of a depth-averaged two-phase flow model which contains more intrinsic nonconservative products.

  18. A Reconstruction Approach to High-Order Schemes Including Discontinuous Galerkin for Diffusion

    NASA Technical Reports Server (NTRS)

    Huynh, H. T.

    2009-01-01

    We introduce a new approach to high-order accuracy for the numerical solution of diffusion problems by solving the equations in differential form using a reconstruction technique. The approach has the advantages of simplicity and economy. It results in several new high-order methods including a simplified version of discontinuous Galerkin (DG). It also leads to new definitions of common value and common gradient quantities at each interface shared by the two adjacent cells. In addition, the new approach clarifies the relations among the various choices of new and existing common quantities. Fourier stability and accuracy analyses are carried out for the resulting schemes. Extensions to the case of quadrilateral meshes are obtained via tensor products. For the two-point boundary value problem (steady state), it is shown that these schemes, which include most popular DG methods, yield exact common interface quantities as well as exact cell average solutions for nearly all cases.

  19. A GPU-accelerated adaptive discontinuous Galerkin method for level set equation

    NASA Astrophysics Data System (ADS)

    Karakus, A.; Warburton, T.; Aksel, M. H.; Sert, C.

    2016-01-01

    This paper presents a GPU-accelerated nodal discontinuous Galerkin method for the solution of two- and three-dimensional level set (LS) equation on unstructured adaptive meshes. Using adaptive mesh refinement, computations are localised mostly near the interface location to reduce the computational cost. Small global time step size resulting from the local adaptivity is avoided by local time-stepping based on a multi-rate Adams-Bashforth scheme. Platform independence of the solver is achieved with an extensible multi-threading programming API that allows runtime selection of different computing devices (GPU and CPU) and different threading interfaces (CUDA, OpenCL and OpenMP). Overall, a highly scalable, accurate and mass conservative numerical scheme that preserves the simplicity of LS formulation is obtained. Efficiency, performance and local high-order accuracy of the method are demonstrated through distinct numerical test cases.

  20. A discontinuous Galerkin method for gravity-driven viscous fingering instabilities in porous media

    SciTech Connect

    Scovazzi, G.; Gerstenberger, A.; Collis, S. S.

    2013-01-01

    We present a new approach to the simulation of gravity-driven viscous fingering instabilities in porous media flow. These instabilities play a very important role during carbon sequestration processes in brine aquifers. Our approach is based on a nonlinear implementation of the discontinuous Galerkin method, and possesses a number of key features. First, the method developed is inherently high order, and is therefore well suited to study unstable flow mechanisms. Secondly, it maintains high-order accuracy on completely unstructured meshes. The combination of these two features makes it a very appealing strategy in simulating the challenging flow patterns and very complex geometries of actual reservoirs and aquifers. This article includes an extensive set of verification studies on the stability and accuracy of the method, and also features a number of computations with unstructured grids and non-standard geometries.

  1. On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier-Stokes equations

    NASA Astrophysics Data System (ADS)

    Zhang, Xiangxiong

    2017-01-01

    We construct a local Lax-Friedrichs type positivity-preserving flux for compressible Navier-Stokes equations, which can be easily extended to multiple dimensions for generic forms of equations of state, shear stress tensor and heat flux. With this positivity-preserving flux, any finite volume type schemes including discontinuous Galerkin (DG) schemes with strong stability preserving Runge-Kutta time discretizations satisfy a weak positivity property. With a simple and efficient positivity-preserving limiter, high order explicit Runge-Kutta DG schemes are rendered preserving the positivity of density and internal energy without losing local conservation or high order accuracy. Numerical tests suggest that the positivity-preserving flux and the positivity-preserving limiter do not induce excessive artificial viscosity, and the high order positivity-preserving DG schemes without other limiters can produce satisfying non-oscillatory solutions when the nonlinear diffusion in compressible Navier-Stokes equations is accurately resolved.

  2. A Discrete Analysis of Non-reflecting Boundary Conditions for Discontinuous Galerkin Method

    NASA Technical Reports Server (NTRS)

    Hu, Fang Q.; Atkins, Harold L.

    2003-01-01

    We present a discrete analysis of non-reflecting boundary conditions for the discontinuous Galerkin method. The boundary conditions considered in this paper include the recently proposed Perfectly Matched Layer absorbing boundary condition for the linearized Euler equation and two non-reflecting boundary conditions based on the characteristic decomposition of the flux on the boundary. The analyses for the three boundary conditions are carried out in a unifled way. In each case, eigensolutions of the discrete system are obtained and applied to compute the numerical reflection coefficients of a specified out-going wave. The dependencies of the reflections at the boundary on the out-going wave angle and frequency as well as the mesh sizes arc? studied. Comparisons with direct numerical simulation results are also presented.

  3. A discontinuous Galerkin finite element discretization of the Euler equations for compressible and incompressible fluids

    SciTech Connect

    Pesch, L. Vegt, J.J.W. van der

    2008-05-10

    Using the generalized variable formulation of the Euler equations of fluid dynamics, we develop a numerical method that is capable of simulating the flow of fluids with widely differing thermodynamic behavior: ideal and real gases can be treated with the same method as an incompressible fluid. The well-defined incompressible limit relies on using pressure primitive or entropy variables. In particular entropy variables can provide numerical methods with attractive properties, e.g. fulfillment of the second law of thermodynamics. We show how a discontinuous Galerkin finite element discretization previously used for compressible flow with an ideal gas equation of state can be extended for general fluids. We also examine which components of the numerical method have to be changed or adapted. Especially, we investigate different possibilities of solving the nonlinear algebraic system with a pseudo-time iteration. Numerical results highlight the applicability of the method for various fluids.

  4. Optimal numerical parameterization of discontinuous Galerkin method applied to wave propagation problems

    SciTech Connect

    Chevaugeon, Nicolas . E-mail: chevaugeon@gce.ucl.ac.be; Hillewaert, Koen; Gallez, Xavier; Ploumhans, Paul; Remacle, Jean-Francois . E-mail: remacle@gce.ucl.ac.be

    2007-04-10

    This paper deals with the high-order discontinuous Galerkin (DG) method for solving wave propagation problems. First, we develop a one-dimensional DG scheme and numerically compute dissipation and dispersion errors for various polynomial orders. An optimal combination of time stepping scheme together with the high-order DG spatial scheme is presented. It is shown that using a time stepping scheme with the same formal accuracy as the DG scheme is too expensive for the range of wave numbers that is relevant for practical applications. An efficient implementation of a high-order DG method in three dimensions is presented. Using 1D convergence results, we further show how to adequately choose elementary polynomial orders in order to equi-distribute a priori the discretization error. We also show a straightforward manner to allow variable polynomial orders in a DG scheme. We finally propose some numerical examples in the field of aero-acoustics.

  5. Evaluation of shielding effectiveness of composite wall with a time domain discontinuous Galerkin method

    NASA Astrophysics Data System (ADS)

    Kameni Ntichi, Abelin; Modave, Axel; Boubekeur, Mohamed; Preault, Valentin; Pichon, Lionel; Geuzaine, Christophe

    2013-11-01

    This article presents a time domain discontinuous Galerkin method applied for solving the con-servative form of Maxwells' equations and computing the radiated fields in electromagnetic compatibility problems. The results obtained in homogeneous media for the transverse magnetic waves are validated in two cases. We compare our solution to an analytical solution of Maxwells' equations based on characteristic method. Our results on shielding effectiveness of a conducting wall are same as those obtained from analytical expression in frequency domain. The propagation in heterogeneous medium is explored. The shielding effectiveness of a composite wall partially filled by circular conductives inclusions is computed. The proposed results are in conformity with the classical predictive homogenization rules. Contribution to the Topical Issue "Numelec 2012", Edited by Adel Razek.

  6. Simulation of the flow between rotating disks with Discontinuous Galerkin method

    NASA Astrophysics Data System (ADS)

    Marek, Maciej

    2011-12-01

    In this work Discontinuous Galerkin (DG) method is applied to the simulation of the incompressible, viscous flow between rotating disks. The code is based on the projection method and accepts unstructured meshes with hexahedral elements. It is explicit in time and allows for arbitrary partition of the computational domain between processors in parallel computing (communication patterns are automatically assigned). Two cases (configurations) are considered: flat disks (structured mesh) and disk with a step placed at the center of the cavity (unstructured mesh). In both configurations the upper disk is rotating, the other one and the remaining walls are stationary. The results for the flat disks are compared to DNS data and very good agreement is obtained. The second case shows capability of the new approach with handling complex geometries.

  7. A Discontinuous Galerkin Time Domain Framework for Periodic Structures Subject to Oblique Excitation

    NASA Astrophysics Data System (ADS)

    Miller, Nicholas C.; Baczewski, Andrew D.; Albrecht, John D.; Shanker, Balasubramaniam

    2014-08-01

    A nodal Discontinuous Galerkin (DG) method is derived for the analysis of time-domain (TD) scattering from doubly periodic PEC/dielectric structures under oblique interrogation. Field transformations are employed to elaborate a formalism that is free from any issues with causality that are common when applying spatial periodic boundary conditions simultaneously with incident fields at arbitrary angles of incidence. An upwind numerical flux is derived for the transformed variables, which retains the same form as it does in the original Maxwell problem for domains without explicitly imposed periodicity. This, in conjunction with the amenability of the DG framework to non-conformal meshes, provides a natural means of accurately solving the first order TD Maxwell equations for a number of periodic systems of engineering interest. Results are presented that substantiate the accuracy and utility of our method.

  8. Current sheets in the Discontinuous Galerkin Time-Domain method: an application to graphene

    NASA Astrophysics Data System (ADS)

    Werra, Julia F. M.; Wolff, Christian; Matyssek, Christian; Busch, Kurt

    2015-05-01

    We describe the treatment of thin conductive sheets within the Discontinuous Galerkin Time-Domain (DGTD) method for solving the Maxwell equations and apply this approach to the efficient computation of the optical properties of graphene-based systems. In particular, we show that a thin conductive sheet can be handled by incorporating the associated jump conditions of the electromagnetic field into the numerical flux of the DGTD approach. This results in a flexible and efficient numerical scheme that can be applied to a number of systems. Specifically, we show how to treat individual graphene sheets on substrates as well as finite stacks of alternating graphene and dielectric layers by modeling the dispersive and dissipative properties of graphene via a two-term critical-point model for its electrostatically doped conductivity.

  9. A Discontinuous Galerkin ALE Method for Compressible Viscous Flows in Moving Domains

    NASA Astrophysics Data System (ADS)

    Lomtev, I.; Kirby, R. M.; Karniadakis, G. E.

    1999-10-01

    We present a matrix-free discontinuous Galerkin method for simulating compressible viscous flows in two- and three-dimensional moving domains. To this end, we solve the Navier-Stokes equations in an arbitrary Lagrangian Eulerian (ALE) framework. Spatial discretization is based on standard structured and unstructured grids but using an orthogonal hierarchical spectral basis. The method is third-order accurate in time and converges exponentially fast in space for smooth solutions. A novelty of the method is the use of a force-directed algorithm from graph theory that requires no matrix inversion to efficiently update the grid while minimizing distortions. We present several simulations using the new method, including validation with published results from a pitching airfoil, and new results for flow past a three-dimensional wing subject to large flapping insect-like motion.

  10. A discontinuous Galerkin front tracking method for two-phase flows with surface tension

    SciTech Connect

    Nguyen, V.-T.; Peraire, J.; Cheong, K.B.; Persson, P.-O.

    2008-12-28

    A Discontinuous Galerkin method for solving hyperbolic systems of conservation laws involving interfaces is presented. The interfaces are represented by a collection of element boundaries and their position is updated using an arbitrary Lagrangian-Eulerian method. The motion of the interfaces and the numerical fluxes are obtained by solving a Riemann problem. As the interface is propagated, a simple and effective remeshing technique based on distance functions regenerates the grid to preserve its quality. Compared to other interface capturing techniques, the proposed approach avoids smearing of the jumps across the interface which leads to an improvement in accuracy. Numerical results are presented for several typical two-dimensional interface problems, including flows with surface tension.

  11. Adaptive entropy-constrained discontinuous Galerkin method for simulation of turbulent flows

    NASA Astrophysics Data System (ADS)

    Lv, Yu; Ihme, Matthias

    2015-11-01

    A robust and adaptive computational framework will be presented for high-fidelity simulations of turbulent flows based on the discontinuous Galerkin (DG) scheme. For this, an entropy-residual based adaptation indicator is proposed to enable adaptation in polynomial and physical space. The performance and generality of this entropy-residual indicator is evaluated through direct comparisons with classical indicators. In addition, a dynamic load balancing procedure is developed to improve computational efficiency. The adaptive framework is tested by considering a series of turbulent test cases, which include homogeneous isotropic turbulence, channel flow and flow-over-a-cylinder. The accuracy, performance and scalability are assessed, and the benefit of this adaptive high-order method is discussed. The funding from NSF CAREER award is greatly acknowledged.

  12. High-order discontinuous Galerkin method for applications to multicomponent and chemically reacting flows

    NASA Astrophysics Data System (ADS)

    Lv, Yu; Ihme, Matthias

    2017-06-01

    This article focuses on the development of a discontinuous Galerkin (DG) method for simulations of multicomponent and chemically reacting flows. Compared to aerodynamic flow applications, in which DG methods have been successfully employed, DG simulations of chemically reacting flows introduce challenges that arise from flow unsteadiness, combustion, heat release, compressibility effects, shocks, and variations in thermodynamic properties. To address these challenges, algorithms are developed, including an entropy-bounded DG method, an entropy-residual shock indicator, and a new formulation of artificial viscosity. The performance and capabilities of the resulting DG method are demonstrated in several relevant applications, including shock/bubble interaction, turbulent combustion, and detonation. It is concluded that the developed DG method shows promising performance in application to multicomponent reacting flows. The paper concludes with a discussion of further research needs to enable the application of DG methods to more complex reacting flows.

  13. Energy Stable Space-Time Discontinuous Galerkin Approximations of the 2-Fluid Plasma Equations

    NASA Astrophysics Data System (ADS)

    Rossmanith, James; Barth, Tim

    2010-11-01

    Energy stable variants of the space-time discontinuous Galerkin (DG) finite element method are developed that approximate the ideal two-fluid plasma equations. Using standard symmetrization techniques, the two-fluid plasma equations are symmeterized via convex entropy function and the introduction of entropy variables. Using these entropy variables, the source term coupling in the two-fluid plasma equations is shown to have iso-energetic properties so that the source term neither creates nor removes energy from the system. Finite-dimensional approximation spaces utilizing entropy variables are utilized in the DG discretization yielding provable nonlinear stability and exact preservation of this iso-energetic source term property. Numerical results for the two-fluid approximation of magnetic reconnection are presented verifying and assessing properties of the present method.

  14. Discontinuous Galerkin method for coupled problems of compressible flow and elastic structures

    NASA Astrophysics Data System (ADS)

    Kosík, A.; Feistauer, M.; Hadrava, M.; Horáček, J.

    2013-10-01

    This paper is concerned with the numerical simulation of the interaction of 2D compressible viscous flow and an elastic structure. We consider the model of dynamical linear elasticity. Each individual problem is discretized in space by the discontinuous Galerkin method (DGM). For the time discretization we can use either the BDF (backward difference formula) method or also the DGM. The time dependence of the domain occupied by the fluid is given by the deformation of the elastic structure adjacent to the flow domain. It is treated with the aid of the Arbitrary Lagrangian-Eulerian (ALE) method. The fluid-structure interaction, given by transient conditions, is realized by an iterative process. The developed method is applied to the simulation of the biomechanical problem containing the onset of the voice production.

  15. A high-order discontinuous Galerkin method for unsteady advection-diffusion problems

    NASA Astrophysics Data System (ADS)

    Borker, Raunak; Farhat, Charbel; Tezaur, Radek

    2017-03-01

    A high-order discontinuous Galerkin method with Lagrange multipliers is presented for the solution of unsteady advection-diffusion problems in the high Péclet number regime. It operates directly on the second-order form of the governing equation and does not require any stabilization. Its spatial basis functions are chosen among the free-space solutions of the homogeneous form of the partial differential equation obtained after time-discretization. It also features Lagrange multipliers for enforcing a weak continuity of the approximated solution across the element interface boundaries. This leads to a system of differential-algebraic equations which are time-integrated by an implicit family of schemes. The numerical stability of these schemes and the well-posedness of the overall discretization method are supported by a theoretical analysis. The performance of this method is demonstrated for various high Péclet number constant-coefficient model flow problems.

  16. Assessment of a high-order accurate Discontinuous Galerkin method for turbomachinery flows

    NASA Astrophysics Data System (ADS)

    Bassi, F.; Botti, L.; Colombo, A.; Crivellini, A.; Franchina, N.; Ghidoni, A.

    2016-04-01

    In this work the capabilities of a high-order Discontinuous Galerkin (DG) method applied to the computation of turbomachinery flows are investigated. The Reynolds averaged Navier-Stokes equations coupled with the two equations k-ω turbulence model are solved to predict the flow features, either in a fixed or rotating reference frame, to simulate the fluid flow around bodies that operate under an imposed steady rotation. To ensure, by design, the positivity of all thermodynamic variables at a discrete level, a set of primitive variables based on pressure and temperature logarithms is used. The flow fields through the MTU T106A low-pressure turbine cascade and the NASA Rotor 37 axial compressor have been computed up to fourth-order of accuracy and compared to the experimental and numerical data available in the literature.

  17. DNS of Flows over Periodic Hills using a Discontinuous-Galerkin Spectral-Element Method

    NASA Technical Reports Server (NTRS)

    Diosady, Laslo T.; Murman, Scott M.

    2014-01-01

    Direct numerical simulation (DNS) of turbulent compressible flows is performed using a higher-order space-time discontinuous-Galerkin finite-element method. The numerical scheme is validated by performing DNS of the evolution of the Taylor-Green vortex and turbulent flow in a channel. The higher-order method is shown to provide increased accuracy relative to low-order methods at a given number of degrees of freedom. The turbulent flow over a periodic array of hills in a channel is simulated at Reynolds number 10,595 using an 8th-order scheme in space and a 4th-order scheme in time. These results are validated against previous large eddy simulation (LES) results. A preliminary analysis provides insight into how these detailed simulations can be used to improve Reynoldsaveraged Navier-Stokes (RANS) modeling

  18. A hybridizable discontinuous Galerkin method for solving nonlocal optical response models

    NASA Astrophysics Data System (ADS)

    Li, Liang; Lanteri, Stéphane; Mortensen, N. Asger; Wubs, Martijn

    2017-10-01

    We propose Hybridizable Discontinuous Galerkin (HDG) methods for solving the frequency-domain Maxwell's equations coupled to the Nonlocal Hydrodynamic Drude (NHD) and Generalized Nonlocal Optical Response (GNOR) models, which are employed to describe the optical properties of nano-plasmonic scatterers and waveguides. Brief derivations for both the NHD model and the GNOR model are presented. The formulations of the HDG method for the 2D TM mode are given, in which we introduce two hybrid variables living only on the skeleton of the mesh. The local field solutions are expressed in terms of the hybrid variables in each element. Two conservativity conditions are globally enforced to make the problem solvable and to guarantee the continuity of the tangential component of the electric field and the normal component of the current density. Numerical results show that the proposed HDG methods converge at optimal rate. We benchmark our implementation and demonstrate that the HDG method has the potential to solve complex nanophotonic problems.

  19. Discontinuous Galerkin method for Krauseʼs consensus models and pressureless Euler equations

    NASA Astrophysics Data System (ADS)

    Yang, Yang; Wei, Dongming; Shu, Chi-Wang

    2013-11-01

    In this paper, we apply discontinuous Galerkin (DG) methods to solve two model equations: Krause's consensus models and pressureless Euler equations. These two models are used to describe the collisions of particles, and the distributions can be identified as density functions. If the particles are placed at a single point, then the density function turns out to be a δ-function and is difficult to be well approximated numerically. In this paper, we use DG method to approximate such a singularity and demonstrate the good performance of the scheme. Since the density functions are always positive, we apply a positivity-preserving limiter to them. Moreover, for pressureless Euler equations, the velocity satisfies the maximum principle. We also construct special limiters to fulfill this requirement.

  20. A Realizability-Preserving Discontinuous Galerkin Method for the $M_1$ Model of Radiative Transfer

    SciTech Connect

    Frank, Martin; Olbrant, Edgar; Hauck, Cory D

    2012-01-01

    The M{sub 1} model for radiative transfer coupled to a material energy equation in planar geometry is studied in this paper. For this model to be well-posed, its moment variables must fulfill certain realizability conditions. Our main focus is the design and implementation of an explicit Runge-Kutta discontinuous Galerkin method which, under a more restrictive CFL condition, guarantees the realizability of the moment variables and the positivity of the material temperature. An analytical proof for our realizability-preserving scheme, which also includes a slope-limiting technique, is provided and confirmed by various numerical examples. Among other things, we present accuracy tests showing convergence up to fourth-order, compare our results with an analytical solution in a Riemann problem, and consider a Marshak wave problem.

  1. Local discontinuous Galerkin method for a nonlinear time-fractional fourth-order partial differential equation

    NASA Astrophysics Data System (ADS)

    Du, Yanwei; Liu, Yang; Li, Hong; Fang, Zhichao; He, Siriguleng

    2017-09-01

    In this article, a fully discrete local discontinuous Galerkin (LDG) method with high-order temporal convergence rate is presented and developed to look for the numerical solution of nonlinear time-fractional fourth-order partial differential equation (PDE). In the temporal direction, for approximating the fractional derivative with order α ∈ (0 , 1), the weighted and shifted Grünwald difference (WSGD) scheme with second-order convergence rate is introduced and for approximating the integer time derivative, two step backward Euler method with second-order convergence rate is used. For the spatial direction, the LDG method is used. For the numerical theories, the stability is derived and a priori error results are proved. Further, some error results and convergence rates are calculated by numerical procedure to illustrate the effectiveness of proposed method.

  2. A discontinuous Galerkin method for simulating the effects of arbitrary discrete fractures on elastic wave propagation

    NASA Astrophysics Data System (ADS)

    Zhan, Qiwei; Sun, Qingtao; Ren, Qiang; Fang, Yuan; Wang, Hua; Liu, Qing Huo

    2017-08-01

    We develop a non-conformal mesh discontinuous Galerkin (DG) pseudospectral time domain (PSTD) method for 3-D elastic wave scattering problems with arbitrary fracture inclusions. In contrast to directly meshing the exact thin-layer fracture, we use the linear-slip model, one kind of transmission boundary condition, for the DG scheme. Intrinsically, we can efficiently impose a jump-boundary condition by defining a new numerical flux for the surface integration in the DG framework. This transmission boundary condition in the DG-PSTD method significantly reduces the computational cost. 3-D DG simulations and accurate waveform comparisons validate our results for arbitrary discrete fractures. Numerical results indicate that fractures have a significant influence on wave propagation.

  3. A discontinuous Galerkin method for gravity-driven viscous fingering instabilities in porous media

    NASA Astrophysics Data System (ADS)

    Scovazzi, G.; Gerstenberger, A.; Collis, S. S.

    2013-01-01

    We present a new approach to the simulation of gravity-driven viscous fingering instabilities in porous media flow. These instabilities play a very important role during carbon sequestration processes in brine aquifers. Our approach is based on a nonlinear implementation of the discontinuous Galerkin method, and possesses a number of key features. First, the method developed is inherently high order, and is therefore well suited to study unstable flow mechanisms. Secondly, it maintains high-order accuracy on completely unstructured meshes. The combination of these two features makes it a very appealing strategy in simulating the challenging flow patterns and very complex geometries of actual reservoirs and aquifers. This article includes an extensive set of verification studies on the stability and accuracy of the method, and also features a number of computations with unstructured grids and non-standard geometries.

  4. High-Order Discontinuous Galerkin Solution of Low-Re Viscous Flows

    NASA Astrophysics Data System (ADS)

    Lu, Hongqiang

    In this paper, the BR2 high-order Discontinuous Galerkin (DG) method is used to discretize the 2D Navier-Stokes (N-S) equations. The nonlinear discrete system is solved using a Newton method. Both preconditioned GMRES methods and block Gauss-Seidel method can be used to solve the resulting sparse linear system at each nonlinear step in low-order cases. In order to save memory and accelerate the convergence in high-order cases, a linear p-multigrid is developed based on the Taylor basis instead of the GMRES method and the block Gauss-Seidel method. Numerical results indicate that highly accurate solutions can be obtained on very coarse grids when using high order schemes and the linear p-multigrid works well when the implicit backward Euler method is employed to improve the robustness.

  5. Energy conserving discontinuous Galerkin spectral element method for the Vlasov-Poisson system

    NASA Astrophysics Data System (ADS)

    Madaule, Éric; Restelli, Marco; Sonnendrücker, Eric

    2014-12-01

    We propose a new, energy conserving, spectral element, discontinuous Galerkin method for the approximation of the Vlasov-Poisson system in arbitrary dimension, using Cartesian grids. The method is derived from the one proposed in [4], with two modifications: energy conservation is obtained by a suitable projection operator acting on the solution of the Poisson problem, rather than by solving multiple Poisson problems, and all the integrals appearing in the finite element formulation are approximated with Gauss-Lobatto quadrature, thereby yielding a spectral element formulation. The resulting method has the following properties: exact energy conservation (up to errors introduced by the time discretization), stability (thanks to the use of upwind numerical fluxes), high order accuracy and high locality. For the time discretization, we consider both Runge-Kutta methods and exponential integrators, and show results for 1D and 2D cases (2D and 4D in phase space, respectively).

  6. Coupling discontinuous Galerkin methods and retarded potentials for transient wave propagation on unbounded domains

    NASA Astrophysics Data System (ADS)

    Abboud, Toufic; Joly, Patrick; Rodríguez, Jerónimo; Terrasse, Isabelle

    2011-07-01

    This work deals with the numerical simulation of wave propagation on unbounded domains with localized heterogeneities. To do so, we propose to combine a discretization based on a discontinuous Galerkin method in space and explicit finite differences in time on the regions containing heterogeneities with the retarded potential method to account the unbounded nature of the computational domain. The coupling formula enforces a discrete energy identity ensuring the stability under the usual CFL condition in the interior. Moreover, the scheme allows to use a smaller time step in the interior domain yielding to quasi-optimal discretization parameters for both methods. The aliasing phenomena introduced by the local time stepping are reduced by a post-processing by averaging in time obtaining a stable and second order consistent (in time) coupling algorithm. We compute the numerical rate of convergence of the method for an academic problem. The numerical results show the feasibility of the whole discretization procedure.

  7. A multidimensional discontinuous Galerkin modeling framework for overland flow and channel routing

    NASA Astrophysics Data System (ADS)

    West, Dustin W.; Kubatko, Ethan J.; Conroy, Colton J.; Yaufman, Mariah; Wood, Dylan

    2017-04-01

    In this paper, we present the development and application of a new multidimensional, unstructured-mesh model for simulating coupled overland/open-channel flows in the kinematic wave approximation regime. The modeling approach makes use of discontinuous Galerkin (DG) finite element spatial discretizations of variable polynomial degree p, paired with explicit Runge-Kutta time steppers, and is supported by advancements made to an automatic mesh generation tool, ADMESH +, that is used to construct constrained triangulations for channel routing. The developed modeling framework is applied to a set of four test cases, where numerical results are found to compare well with known analytic solutions, experimental data and results from another well-established (structured, finite difference) model within the area of application. The numerical results obtained demonstrate the accuracy and robustness of the developed modeling framework and highlight the potential benefits of using p (polynomial) refinement for hydrological simulations.

  8. A discontinuous Galerkin method for numerical pricing of European options under Heston stochastic volatility

    NASA Astrophysics Data System (ADS)

    Hozman, J.; Tichý, T.

    2016-12-01

    The paper is based on the results from our recent research on multidimensional option pricing problems. We focus on European option valuation when the price movement of the underlying asset is driven by a stochastic volatility following a square root process proposed by Heston. The stochastic approach incorporates a new additional spatial variable into this model and makes it very robust, i.e. it provides a framework to price a variety of options that is closer to reality. The main topic is to present the numerical scheme arising from the concept of discontinuous Galerkin methods and applicable to the Heston option pricing model. The numerical results are presented on artificial benchmarks as well as on reference market data.

  9. Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws

    NASA Astrophysics Data System (ADS)

    Chen, Tianheng; Shu, Chi-Wang

    2017-09-01

    It is well known that semi-discrete high order discontinuous Galerkin (DG) methods satisfy cell entropy inequalities for the square entropy for both scalar conservation laws (Jiang and Shu (1994) [39]) and symmetric hyperbolic systems (Hou and Liu (2007) [36]), in any space dimension and for any triangulations. However, this property holds only for the square entropy and the integrations in the DG methods must be exact. It is significantly more difficult to design DG methods to satisfy entropy inequalities for a non-square convex entropy, and/or when the integration is approximated by a numerical quadrature. In this paper, we develop a unified framework for designing high order DG methods which will satisfy entropy inequalities for any given single convex entropy, through suitable numerical quadrature which is specific to this given entropy. Our framework applies from one-dimensional scalar cases all the way to multi-dimensional systems of conservation laws. For the one-dimensional case, our numerical quadrature is based on the methodology established in Carpenter et al. (2014) [5] and Gassner (2013) [19]. The main ingredients are summation-by-parts (SBP) operators derived from Legendre Gauss-Lobatto quadrature, the entropy conservative flux within elements, and the entropy stable flux at element interfaces. We then generalize the scheme to two-dimensional triangular meshes by constructing SBP operators on triangles based on a special quadrature rule. A local discontinuous Galerkin (LDG) type treatment is also incorporated to achieve the generalization to convection-diffusion equations. Extensive numerical experiments are performed to validate the accuracy and shock capturing efficacy of these entropy stable DG methods.

  10. The Unfitted Discontinuous Galerkin Method for Solving the EEG Forward Problem.

    PubMed

    Nusing, Andreas; Wolters, Carsten H; Brinck, Heinrich; Engwer, Christian

    2016-12-01

    The purpose of this study is to introduce and evaluate the unfitted discontinuous Galerkin finite element method (UDG-FEM) for solving the electroencephalography (EEG) forward problem. This new approach for source analysis does not use a geometry conforming volume triangulation, but instead uses a structured mesh that does not resolve the geometry. The geometry is described using level set functions and is incorporated implicitly in its mathematical formulation. As no triangulation is necessary, the complexity of a simulation pipeline and the need for manual interaction for patient-specific simulations can be reduced and is comparable with that of the FEM for hexahedral meshes. In addition, it maintains conservation laws on a discrete level. Here, we present the theory for UDG-FEM forward modeling, its verification using quasi-analytical solutions in multilayer sphere models and an evaluation in a comparison with a discontinuous Galerkin (DG-FEM) method on hexahedral and on conforming tetrahedral meshes. We furthermore apply the UDG-FEM forward approach in a realistic head model simulation study. The results show convergence to the quasi-analytical solution and indicate a good accuracy of UDG-FEM. UDG-FEM performs comparable or even better than DG-FEM on a conforming tetrahedral mesh while providing a less complex simulation pipeline. When compared to DG-FEM on hexahedral meshes, an overall better accuracy is achieved. The UDG-FEM approach is an accurate, flexible, and promising method to solve the EEG forward problem. This study shows the first application of the UDG-FEM approach to the EEG forward problem.

  11. High-order triangle-based discontinuous Galerkin methods for hyperbolic equations on a rotating sphere

    SciTech Connect

    Giraldo, Francis X. . E-mail: giraldo@nrlmry.navy.mil

    2006-05-20

    High-order triangle-based discontinuous Galerkin (DG) methods for hyperbolic equations on a rotating sphere are presented. The DG method can be characterized as the fusion of finite elements with finite volumes. This DG formulation uses high-order Lagrange polynomials on the triangle using nodal sets up to 15th order. The finite element-type area integrals are evaluated using order 2N Gauss cubature rules. This leads to a full mass matrix which, unlike for continuous Galerkin (CG) methods such as the spectral element (SE) method presented in Giraldo and Warburton [A nodal triangle-based spectral element method for the shallow water equations on the sphere, J. Comput. Phys. 207 (2005) 129-150], is small, local and efficient to invert. Two types of finite volume-type flux integrals are studied: a set based on Gauss-Lobatto quadrature points (order 2N - 1) and a set based on Gauss quadrature points (order 2N). Furthermore, we explore conservation and advection forms as well as strong and weak forms. Seven test cases are used to compare the different methods including some with scale contractions and shock waves. All three strong forms performed extremely well with the strong conservation form with 2N integration being the most accurate of the four DG methods studied. The strong advection form with 2N integration performed extremely well even for flows with shock waves. The strong conservation form with 2N - 1 integration yielded results almost as good as those with 2N while being less expensive. All the DG methods performed better than the SE method for almost all the test cases, especially for those with strong discontinuities. Finally, the DG methods required less computing time than the SE method due to the local nature of the mass matrix.

  12. Discontinuous Galerkin Method with Numerical Roe Flux for Spherical Shallow Water Equations

    NASA Astrophysics Data System (ADS)

    Yi, T.; Choi, S.; Kang, S.

    2013-12-01

    In developing the dynamic core of a numerical weather prediction model with discontinuous Galerkin method, a numerical flux at the boundaries of grid elements plays a vital role since it preserves the local conservation properties and has a significant impact on the accuracy and stability of numerical solutions. Due to these reasons, we developed the numerical Roe flux based on an approximate Riemann problem for spherical shallow water equations in Cartesian coordinates [1] to find out its stability and accuracy. In order to compare the performance with its counterpart flux, we used the Lax-Friedrichs flux, which has been used in many dynamic cores such as NUMA [1], CAM-DG [2] and MCore [3] because of its simplicity. The Lax-Friedrichs flux is implemented by a flux difference between left and right states plus the maximum characteristic wave speed across the boundaries of elements. It has been shown that the Lax-Friedrichs flux with the finite volume method is more dissipative and unstable than other numerical fluxes such as HLLC, AUSM+ and Roe. The Roe flux implemented in this study is based on the decomposition of flux difference over the element boundaries where the nonlinear equations are linearized. It is rarely used in dynamic cores due to its complexity and thus computational expensiveness. To compare the stability and accuracy of the Roe flux with the Lax-Friedrichs, two- and three-dimensional test cases are performed on a plane and cubed-sphere, respectively, with various numbers of element and polynomial order. For the two-dimensional case, the Gaussian bell is simulated on the plane with two different numbers of elements at the fixed polynomial orders. In three-dimensional cases on the cubed-sphere, we performed the test cases of a zonal flow over an isolated mountain and a Rossby-Haurwitz wave, of which initial conditions are the same as those of Williamson [4]. This study presented that the Roe flux with the discontinuous Galerkin method is less

  13. Almost Optimal Interior Penalty Discontinuous Approximations of Symmetric Elliptic Problems on Non-Matching Grids

    SciTech Connect

    Lazarov, R D; Pasciak, J E; Schoberl, J; Vassilevski, P S

    2001-08-08

    We consider an interior penalty discontinuous approximation for symmetric elliptic problems of second order on non-matching grids in this paper. The main result is an almost optimal error estimate for the interior penalty approximation of the original problem based on the partition of the domain into a finite number of subdomains. Further, an error analysis for the finite element approximation of the penalty formulation is given. Finally, numerical experiments on a series of model second order problems are presented.

  14. Direct discontinuous Galerkin method and its variations for second order elliptic equations

    SciTech Connect

    Huang, Hongying; Chen, Zheng; Li, Jin; Yan, Jue

    2016-08-23

    In this study, we study direct discontinuous Galerkin method (Liu and Yan in SIAM J Numer Anal 47(1):475–698, 2009) and its variations (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010; Vidden and Yan in J Comput Math 31(6):638–662, 2013; Yan in J Sci Comput 54(2–3):663–683, 2013) for 2nd order elliptic problems. A priori error estimate under energy norm is established for all four methods. Optimal error estimate under L2 norm is obtained for DDG method with interface correction (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010) and symmetric DDG method (Vidden and Yan in J Comput Math 31(6):638–662, 2013). A series of numerical examples are carried out to illustrate the accuracy and capability of the schemes. Numerically we obtain optimal (k+1)th order convergence for DDG method with interface correction and symmetric DDG method on nonuniform and unstructured triangular meshes. An interface problem with discontinuous diffusion coefficients is investigated and optimal (k+1)th order accuracy is obtained. Peak solutions with sharp transitions are captured well. Highly oscillatory wave solutions of Helmholz equation are well resolved.

  15. Direct discontinuous Galerkin method and its variations for second order elliptic equations

    DOE PAGES

    Huang, Hongying; Chen, Zheng; Li, Jin; ...

    2016-08-23

    In this study, we study direct discontinuous Galerkin method (Liu and Yan in SIAM J Numer Anal 47(1):475–698, 2009) and its variations (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010; Vidden and Yan in J Comput Math 31(6):638–662, 2013; Yan in J Sci Comput 54(2–3):663–683, 2013) for 2nd order elliptic problems. A priori error estimate under energy norm is established for all four methods. Optimal error estimate under L2 norm is obtained for DDG method with interface correction (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010) and symmetric DDG method (Vidden and Yan in J Comput Math 31(6):638–662,more » 2013). A series of numerical examples are carried out to illustrate the accuracy and capability of the schemes. Numerically we obtain optimal (k+1)th order convergence for DDG method with interface correction and symmetric DDG method on nonuniform and unstructured triangular meshes. An interface problem with discontinuous diffusion coefficients is investigated and optimal (k+1)th order accuracy is obtained. Peak solutions with sharp transitions are captured well. Highly oscillatory wave solutions of Helmholz equation are well resolved.« less

  16. General relativistic monopole magnetosphere of neutron stars: a pseudo-spectral discontinuous Galerkin approach

    NASA Astrophysics Data System (ADS)

    Pétri, J.

    2015-03-01

    The close vicinity of neutron stars remains poorly constrained by observations. Although plenty of data are available for the peculiar class of pulsars we are still unable to deduce the underlying plasma distribution in their magnetosphere. In the present paper, we try to unravel the magnetospheric structure starting from basic physics principles and reasonable assumptions about the magnetosphere. Beginning with the monopole force-free case, we compute accurate general relativistic solutions for the electromagnetic field around a slowly rotating magnetized neutron star. Moreover, here we address this problem by including the important effect of plasma screening. This is achieved by solving the time-dependent Maxwell equations in a curved space-time following the 3+1 formalism. We improved our previous numerical code based on pseudo-spectral methods in order to allow for possible discontinuities in the solution. Our algorithm based on a multidomain decomposition of the simulation box belongs to the discontinuous Galerkin finite element methods. We performed several sets of simulations to look for the general relativistic force-free monopole and split monopole solutions. Results show that our code is extremely powerful in handling extended domains of hundredth of light cylinder radii rL. The code has been validated against known exact analytical monopole solutions in flat space-time. We also present semi-analytical calculations for the general relativistic vacuum monopole.

  17. Multiple time scales and pressure forcing in discontinuous Galerkin approximations to layered ocean models

    NASA Astrophysics Data System (ADS)

    Higdon, Robert L.

    2015-08-01

    This paper addresses some issues involving the application of discontinuous Galerkin (DG) methods to ocean circulation models having a generalized vertical coordinate. These issues include the following. (1) Determine the pressure forcing at cell edges, where the dependent variables can be discontinuous. In principle, this could be accomplished by solving a Riemann problem for the full system, but some ideas related to barotropic-baroclinic time splitting can be used to reduce the Riemann problem to a much simpler system of lower dimension. Such splittings were originally developed in order to address the multiple time scales that are present in the system. (2) Adapt the general idea of barotropic-baroclinic splitting to a DG implementation. A significant step is enforcing consistency between the numerical solution of the layer equations and the numerical solution of the vertically-integrated barotropic equations. The method used here has the effect of introducing a type of time filtering into the forcing for the layer equations, which are solved with a long time step. (3) Test these ideas in a model problem involving geostrophic adjustment in a multi-layer fluid. In certain situations, the DG formulation can give significantly better results than those obtained with a standard finite difference formulation.

  18. Direct discontinuous Galerkin method and its variations for second order elliptic equations

    SciTech Connect

    Huang, Hongying; Chen, Zheng; Li, Jin; Yan, Jue

    2016-08-23

    In this study, we study direct discontinuous Galerkin method (Liu and Yan in SIAM J Numer Anal 47(1):475–698, 2009) and its variations (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010; Vidden and Yan in J Comput Math 31(6):638–662, 2013; Yan in J Sci Comput 54(2–3):663–683, 2013) for 2nd order elliptic problems. A priori error estimate under energy norm is established for all four methods. Optimal error estimate under L2 norm is obtained for DDG method with interface correction (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010) and symmetric DDG method (Vidden and Yan in J Comput Math 31(6):638–662, 2013). A series of numerical examples are carried out to illustrate the accuracy and capability of the schemes. Numerically we obtain optimal (k+1)th order convergence for DDG method with interface correction and symmetric DDG method on nonuniform and unstructured triangular meshes. An interface problem with discontinuous diffusion coefficients is investigated and optimal (k+1)th order accuracy is obtained. Peak solutions with sharp transitions are captured well. Highly oscillatory wave solutions of Helmholz equation are well resolved.

  19. Runge-Kutta central discontinuous Galerkin BGK method for the Navier-Stokes equations

    NASA Astrophysics Data System (ADS)

    Ren, Tan; Hu, Jun; Xiong, Tao; Qiu, Jing-Mei

    2014-10-01

    In this paper, we propose a Runge-Kutta (RK) central discontinuous Galerkin (CDG) gas-kinetic BGK method for the Navier-Stokes equations. The proposed method is based on the CDG method defined on two sets of overlapping meshes to avoid discontinuous solutions at cell interfaces, as well as the gas-kinetic BGK model to evaluate fluxes for both convection and diffusion terms. Redundant representation of the numerical solution in the CDG method offers great convenience in the design of gas-kinetic BGK fluxes. Specifically, the evaluation of fluxes at cell interfaces of one set of computational mesh is right inside the cells of the staggered mesh, hence the corresponding particle distribution function for flux evaluation is much simpler than that in existing gas-kinetic BGK methods. As a central scheme, the proposed CDG-BGK has doubled the memory requirement as the corresponding DG scheme; on the other hand, for the convection part, the CFL time step constraint of the CDG method for numerical stability is relatively large compared with that for the DG method. Numerical boundary conditions have to be treated with special care. Numerical examples for 1D and 2D viscous flow simulations are presented to validate the accuracy and robustness of the proposed RK CDG-BGK method.

  20. Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations

    SciTech Connect

    Xing, Yulong; Zhang, Xiangxiong; Shu, Chi-wang

    2010-01-01

    Shallow water equations with a non-flat bottom topography have been widely used to model flows in rivers and coastal areas. An important difficulty arising in these simulations is the appearance of dry areas where no water is present, as standard numerical methods may fail in the presence of these areas. These equations also have still water steady state solutions in which the flux gradients are nonzero but exactly balanced by the source term. In this paper we propose a high order discontinuous Galerkin method which can maintain the still water steady state exactly, and at the same time can preserve the non-negativity of the water height without loss of mass conservation. A simple positivity-preserving limiter, valid under suitable CFL condition, will be introduced in one dimension and then extended to two dimensions with rectangular meshes. Numerical tests are performed to verify the positivity-preserving property, well-balanced property, high order accuracy, and good resolution for smooth and discontinuous solutions.

  1. A parallel code base on discontinuous Galerkin method on three dimensional unstructured meshes for MHD equations

    NASA Astrophysics Data System (ADS)

    Li, Xujing; Zheng, Weiying

    2016-10-01

    A new parallel code based on discontinuous Galerkin (DG) method for hyperbolic conservation laws on three dimensional unstructured meshes is developed recently. This code can be used for simulations of MHD equations, which are very important in magnetic confined plasma research. The main challenges in MHD simulations in fusion include the complex geometry of the configurations, such as plasma in tokamaks, the possibly discontinuous solutions and large scale computing. Our new developed code is based on three dimensional unstructured meshes, i.e. tetrahedron. This makes the code flexible to arbitrary geometries. Second order polynomials are used on each element and HWENO type limiter are applied. The accuracy tests show that our scheme reaches the desired three order accuracy and the nonlinear shock test demonstrate that our code can capture the sharp shock transitions. Moreover, One of the advantages of DG compared with the classical finite element methods is that the matrices solved are localized on each element, making it easy for parallelization. Several simulations including the kink instabilities in toroidal geometry will be present here. Chinese National Magnetic Confinement Fusion Science Program 2015GB110003.

  2. A full vectorial generalized discontinuous Galerkin beam propagation method (GDG-BPM) for nonsmooth electromagnetic fields in waveguides

    SciTech Connect

    Fan Kai; Cai Wei Ji Xia

    2008-07-20

    In this paper, we propose a new full vectorial generalized discontinuous Galerkin beam propagation method (GDG-BPM) to accurately handle the discontinuities in electromagnetic fields associated with wave propagations in inhomogeneous optical waveguides. The numerical method is a combination of the traditional beam propagation method (BPM) with a newly developed generalized discontinuous Galerkin (GDG) method [K. Fan, W. Cai, X. Ji, A generalized discontinuous Galerkin method (GDG) for Schroedinger equations with nonsmooth solutions, J. Comput. Phys. 227 (2008) 2387-2410]. The GDG method is based on a reformulation, using distributional variables to account for solution jumps across material interfaces, of Schroedinger equations resulting from paraxial approximations of vector Helmholtz equations. Four versions of the GDG-BPM are obtained for either the electric or magnetic field components. Modeling of wave propagations in various optical fibers using the full vectorial GDG-BPM is included. Numerical results validate the high order accuracy and the flexibility of the method for various types of interface jump conditions.

  3. Seismic waves in heterogeneous material: subcell resolution of the discontinuous Galerkin method

    NASA Astrophysics Data System (ADS)

    Castro, Cristóbal E.; Käser, Martin; Brietzke, Gilbert B.

    2010-07-01

    We present an important extension of the arbitrary high-order discontinuous Galerkin (DG) finite-element method to model 2-D elastic wave propagation in highly heterogeneous material. In this new approach we include space-variable coefficients to describe smooth or discontinuous material variations inside each element using the same numerical approximation strategy as for the velocity-stress variables in the formulation of the elastic wave equation. The combination of the DG method with a time integration scheme based on the solution of arbitrary accuracy derivatives Riemann problems still provides an explicit, one-step scheme which achieves arbitrary high-order accuracy in space and time. Compared to previous formulations the new scheme contains two additional terms in the form of volume integrals. We show that the increasing computational cost per element can be overcompensated due to the improved material representation inside each element as coarser meshes can be used which reduces the total number of elements and therefore computational time to reach a desired error level. We confirm the accuracy of the proposed scheme performing convergence tests and several numerical experiments considering smooth and highly heterogeneous material. As the approximation of the velocity and stress variables in the wave equation and of the material properties in the model can be chosen independently, we investigate the influence of the polynomial material representation on the accuracy of the synthetic seismograms with respect to computational cost. Moreover, we study the behaviour of the new method on strong material discontinuities, in the case where the mesh is not aligned with such a material interface. In this case second-order linear material approximation seems to be the best choice, with higher-order intra-cell approximation leading to potential instable behaviour. For all test cases we validate our solution against the well-established standard fourth-order finite

  4. A high order discontinuous Galerkin method for elastic wave propagation in arbitrary heterogeneous media

    NASA Astrophysics Data System (ADS)

    Glinsky, Nathalie; Mercerat, Diego

    2013-04-01

    High-order numerical methods allow accurate simulations of ground motion using unstructured and relatively coarse meshes. In realistic media (sedimentary basins for example), we have to include strong variations of the material properties. For such configurations, the hypothesis that material properties are set constant in each element of the mesh can be a severe limitation since we need to use very fine meshes resulting in very small time steps for explicit time integration schemes. Moreover, smooth models are approximated by piecewise constant materials. For these reasons, we present an improvement of a nodal discontinuous Galerkin method (DG) allowing non constant material properties in the elements of the mesh for a better approximation of arbitrary heterogeneous media. We consider an isotropic, linearly elastic two-dimensional medium (characterized by ?, ? and μ) and solve the first-order velocity-stress system. As the stress tensor is symmetrical, let W? = (?V,?-?)t contain the velocity vector ?-V = (vx,vy)t and the stress components ?-? = (?xx,?yy,?xy)t, then, the system writes t ?-W + Ax (?,?,μ) ^xW?- + Ay(?,?,μ) ^yW ?- = 0, where Ax and Ay are 5x5 matrices depending of the material properties. We apply a discontinuous Galerkin method based on centered fluxes and a leap-frog time scheme to this system. We consider a bounded polyhedral domain discretized by triangles. The approximation of W?- is defined locally on each element by considering the Lagrange nodal interpolants. The system is multiplied by a test function t and integrated on each element Ti. To avoid computing extra terms, related to the variable properties within Ti, we introduce a change of variables on the stress components ( )t ?-? = (?xx,?yy,?xy)t ? ??? = 1(?xx + ?yy), 1 (?xx - ?yy),?xy 2 2 which allows writing the system in a pseudo-conservative form in the variable ?-W? = (?-V,???)t ? (?,?,μ) ^tW ?? + ?Ax x ??W + A?y y ??W = 0 , where the constant matrices Ãx and Ãy do not depend

  5. High-order Hybridized Discontinuous Galerkin (HDG) method for wave propagation simulation in complex geophysical media (elastic, acoustic and hydro-acoustic); an unifying framework to couple continuous Spectral Element and Discontinuous Galerkin Methods

    NASA Astrophysics Data System (ADS)

    Terrana, Sebastien; Vilotte, Jean-Pierre; Guillot, Laurent; Mariotti, Christian

    2015-04-01

    Today seismological observation systems combine broadband seismic receivers, hydrophones and micro-barometers antenna that provide complementary observations of source-radiated waves in heterogeneous and complex geophysical media. Exploiting these observations requires accurate and multi-physics - elastic, hydro-acoustic, infrasonic - wave simulation methods. A popular approach is the Spectral Element Method (SEM) (Chaljub et al, 2006) which is high-order accurate (low dispersion error), very flexible to parallelization and computationally attractive due to efficient sum factorization technique and diagonal mass matrix. However SEMs suffer from lack of flexibility in handling complex geometry and multi-physics wave propagation. High-order Discontinuous Galerkin Methods (DGMs), i.e. Dumbser et al (2006), Etienne et al. (2010), Wilcox et al (2010), are recent alternatives that can handle complex geometry, space-and-time adaptativity, and allow efficient multi-physics wave coupling at interfaces. However, DGMs are more memory demanding and less computationally attractive than SEMs, especially when explicit time stepping is used. We propose a new class of higher-order Hybridized Discontinuous Galerkin Spectral Elements (HDGSEM) methods for spatial discretization of wave equations, following the unifying framework for hybridization of Cockburn et al (2009) and Nguyen et al (2011), which allows for a single implementation of conforming and non-conforming SEMs. When used with energy conserving explicit time integration schemes, HDGSEM is flexible to handle complex geometry, computationally attractive and has significantly less degrees of freedom than classical DGMs, i.e., the only coupled unknowns are the single-valued numerical traces of the velocity field on the element's faces. The formulation can be extended to model fractional energy loss at interfaces between elastic, acoustic and hydro-acoustic media. Accuracy and performance of the HDGSEM are illustrated and

  6. Discontinuous Galerkin methods for dispersive shallow water models in closed basins: Spurious eddies and their removal using curved boundary methods

    NASA Astrophysics Data System (ADS)

    Steinmoeller, D. T.; Stastna, M.; Lamb, K. G.

    2016-11-01

    Discontinuous Galerkin methods offer a promising methodology for treating nearly hyperbolic systems such as dispersion-modified shallow water equations in complicated basins. Use of straight-edged triangular elements can lead to the generation of spurious eddies when wave fronts propagate around sharp, re-entrant obstacles such as headlands. While these eddies may be removed by adding strong artificial dissipation (e.g., eddy viscosity), for nearly inviscid simulations that focus on wave phenomena this approach is not reasonable. We demonstrate that the moderate order Discontinuous Galerkin methodology may be extended to curved triangular elements provided that the integral formulations are computed with high-order quadrature and cubature rules. Simulations with the new technique do not exhibit spurious eddy generation in idealized complex domains or real-world basins as exemplified by Pinehurst Lake, Alberta, Canada.

  7. A mass and momentum flux-form high-order discontinuous Galerkin shallow water model on the cubed-sphere

    NASA Astrophysics Data System (ADS)

    Bao, Lei; Nair, Ramachandran D.; Tufo, Henry M.

    2014-08-01

    A well-balanced discontinuous Galerkin (DG) flux-form shallow-water (SW) model on the sphere is developed and compared with a nodal DG SW model cast in the vector-invariant form for accuracy and conservation properties. A second-order diffusion scheme based on the local discontinuous Galerkin (LDG) method is added to the viscous version of the SW model and tested for conservation behaviors. The inviscid flux-form SW model is found to have better conservation of total energy and zonal angular momentum while the vector-invariant form provides better ability of conserving potential enstrophy. The inviscid flux-form tends to generate spurious vorticity but the LDG scheme combined with a well-balanced treatment can effectively eliminate the small-scale noise and generate smooth and accurate results.

  8. A priori error estimates for an hp-version of the discontinuous Galerkin method for hyperbolic conservation laws

    NASA Technical Reports Server (NTRS)

    Bey, Kim S.; Oden, J. Tinsley

    1993-01-01

    A priori error estimates are derived for hp-versions of the finite element method for discontinuous Galerkin approximations of a model class of linear, scalar, first-order hyperbolic conservation laws. These estimates are derived in a mesh dependent norm in which the coefficients depend upon both the local mesh size h(sub K) and a number p(sub k) which can be identified with the spectral order of the local approximations over each element.

  9. Kershaw closures for linear transport equations in slab geometry II: High-order realizability-preserving discontinuous-Galerkin schemes

    NASA Astrophysics Data System (ADS)

    Schneider, Florian

    2016-10-01

    This paper provides a generalization of the realizability-preserving discontinuous-Galerkin scheme given in [3] to general full-moment models that can be closed analytically. It is applied to the class of Kershaw closures, which are able to provide a cheap closure of the moment problem. This results in an efficient algorithm for the underlying linear transport equation. The efficiency of high-order methods is demonstrated using numerical convergence tests and non-smooth benchmark problems.

  10. A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media

    SciTech Connect

    Wilcox, Lucas C.; Stadler, Georg; Burstedde, Carsten; Ghattas, Omar

    2010-12-10

    We introduce a high-order discontinuous Galerkin (dG) scheme for the numerical solution of three-dimensional (3D) wave propagation problems in coupled elastic-acoustic media. A velocity-strain formulation is used, which allows for the solution of the acoustic and elastic wave equations within the same unified framework. Careful attention is directed at the derivation of a numerical flux that preserves high-order accuracy in the presence of material discontinuities, including elastic-acoustic interfaces. Explicit expressions for the 3D upwind numerical flux, derived as an exact solution for the relevant Riemann problem, are provided. The method supports h-non-conforming meshes, which are particularly effective at allowing local adaptation of the mesh size to resolve strong contrasts in the local wavelength, as well as dynamic adaptivity to track solution features. The use of high-order elements controls numerical dispersion, enabling propagation over many wave periods. We prove consistency and stability of the proposed dG scheme. To study the numerical accuracy and convergence of the proposed method, we compare against analytical solutions for wave propagation problems with interfaces, including Rayleigh, Lamb, Scholte, and Stoneley waves as well as plane waves impinging on an elastic-acoustic interface. Spectral rates of convergence are demonstrated for these problems, which include a non-conforming mesh case. Finally, we present scalability results for a parallel implementation of the proposed high-order dG scheme for large-scale seismic wave propagation in a simplified earth model, demonstrating high parallel efficiency for strong scaling to the full size of the Jaguar Cray XT5 supercomputer.

  11. A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media

    NASA Astrophysics Data System (ADS)

    Wilcox, Lucas C.; Stadler, Georg; Burstedde, Carsten; Ghattas, Omar

    2010-12-01

    We introduce a high-order discontinuous Galerkin (dG) scheme for the numerical solution of three-dimensional (3D) wave propagation problems in coupled elastic-acoustic media. A velocity-strain formulation is used, which allows for the solution of the acoustic and elastic wave equations within the same unified framework. Careful attention is directed at the derivation of a numerical flux that preserves high-order accuracy in the presence of material discontinuities, including elastic-acoustic interfaces. Explicit expressions for the 3D upwind numerical flux, derived as an exact solution for the relevant Riemann problem, are provided. The method supports h-non-conforming meshes, which are particularly effective at allowing local adaptation of the mesh size to resolve strong contrasts in the local wavelength, as well as dynamic adaptivity to track solution features. The use of high-order elements controls numerical dispersion, enabling propagation over many wave periods. We prove consistency and stability of the proposed dG scheme. To study the numerical accuracy and convergence of the proposed method, we compare against analytical solutions for wave propagation problems with interfaces, including Rayleigh, Lamb, Scholte, and Stoneley waves as well as plane waves impinging on an elastic-acoustic interface. Spectral rates of convergence are demonstrated for these problems, which include a non-conforming mesh case. Finally, we present scalability results for a parallel implementation of the proposed high-order dG scheme for large-scale seismic wave propagation in a simplified earth model, demonstrating high parallel efficiency for strong scaling to the full size of the Jaguar Cray XT5 supercomputer.

  12. A Discontinuous Galerkin Method Compatible with the BSSN Formulation of the Einstein Equations

    NASA Astrophysics Data System (ADS)

    Miller, Jonah; Schnetter, Erik

    2017-01-01

    The BSSN formulation of the Einstein equations has repeatedly demonstrated its robustness. The formulation is not only stable but allows for puncture-type evolutions of black hole systems. Discontinuous Galerkin Finite Element (DGFE) methods offer a mathematically beautiful, computationally efficient, and highly parallelizable way to solve hyperbolic PDEs. These properties make them highly desirable for numerical relativity. To-date no one has been able to solve the full (3+1)-dimensional BSSN equations using DGFE methods. This is partly because DGFE discretization often occurs at the level of the equations, not the derivative operator, and partly because DGFE methods are traditionally formulated for manifestly flux-conservative systems. By discretizing the derivative operator, we generalize a particular flavor of DGFE methods, Local DG methods, to solve arbitrary second-order hyperbolic equations. This generalization allows us to solve the BSSN equations. The authors acknowledge support from the Natural Sciences and Engineering Research Council of Canada and from the National Science Foundation of the USA (OCI 0905046, PHY 1212401).

  13. De-Aliasing Through Over-Integration Applied to the Flux Reconstruction and Discontinuous Galerkin Methods

    NASA Technical Reports Server (NTRS)

    Spiegel, Seth C.; Huynh, H. T.; DeBonis, James R.

    2015-01-01

    High-order methods are quickly becoming popular for turbulent flows as the amount of computer processing power increases. The flux reconstruction (FR) method presents a unifying framework for a wide class of high-order methods including discontinuous Galerkin (DG), Spectral Difference (SD), and Spectral Volume (SV). It offers a simple, efficient, and easy way to implement nodal-based methods that are derived via the differential form of the governing equations. Whereas high-order methods have enjoyed recent success, they have been known to introduce numerical instabilities due to polynomial aliasing when applied to under-resolved nonlinear problems. Aliasing errors have been extensively studied in reference to DG methods; however, their study regarding FR methods has mostly been limited to the selection of the nodal points used within each cell. Here, we extend some of the de-aliasing techniques used for DG methods, primarily over-integration, to the FR framework. Our results show that over-integration does remove aliasing errors but may not remove all instabilities caused by insufficient resolution (for FR as well as DG).

  14. DG-FTLE: Lagrangian coherent structures with high-order discontinuous-Galerkin methods

    NASA Astrophysics Data System (ADS)

    Nelson, Daniel A.; Jacobs, Gustaaf B.

    2015-08-01

    We present an algorithm for the computation of finite-time Lyapunov exponent (FTLE) fields using discontinuous-Galerkin (dG) methods in two dimensions. The algorithm is designed to compute FTLE fields simultaneously with the time integration of dG-based flow solvers of conservation laws. Fluid tracers are initialized at Gauss-Lobatto quadrature nodes within an element. The deformation gradient tensor, defined by the deformation of the Lagrangian flow map in finite time, is determined per element with high-order dG operators. Multiple flow maps are constructed from a particle trace that is released at a single initial time by mapping and interpolating the flow map formed by the locations of the fluid tracers after finite time integration to a unit square master element and to the quadrature nodes within the element, respectively. The interpolated flow maps are used to compute forward-time and backward-time FTLE fields at several times using dG operators. For a large finite integration time, the interpolation is increasingly poorly conditioned because of the excessive subdomain deformation. The conditioning can be used in addition to the FTLE to quantify the deformation of the flow field and identify subdomains with material lines that define Lagrangian coherent structures. The algorithm is tested on three benchmarks: an analytical spatially periodic gyre flow, a vortex advected by a uniform inviscid flow, and the viscous flow around a square cylinder. In these cases, the algorithm is shown to have spectral convergence.

  15. A nodal discontinuous Galerkin method for reverse-time migration on GPU clusters

    NASA Astrophysics Data System (ADS)

    Modave, A.; St-Cyr, A.; Mulder, W. A.; Warburton, T.

    2015-11-01

    Improving both accuracy and computational performance of numerical tools is a major challenge for seismic imaging and generally requires specialized implementations to make full use of modern parallel architectures. We present a computational strategy for reverse-time migration (RTM) with accelerator-aided clusters. A new imaging condition computed from the pressure and velocity fields is introduced. The model solver is based on a high-order discontinuous Galerkin time-domain (DGTD) method for the pressure-velocity system with unstructured meshes and multirate local time stepping. We adopted the MPI+X approach for distributed programming where X is a threaded programming model. In this work we chose OCCA, a unified framework that makes use of major multithreading languages (e.g. CUDA and OpenCL) and offers the flexibility to run on several hardware architectures. DGTD schemes are suitable for efficient computations with accelerators thanks to localized element-to-element coupling and the dense algebraic operations required for each element. Moreover, compared to high-order finite-difference schemes, the thin halo inherent to DGTD method reduces the amount of data to be exchanged between MPI processes and storage requirements for RTM procedures. The amount of data to be recorded during simulation is reduced by storing only boundary values in memory rather than on disk and recreating the forward wavefields. Computational results are presented that indicate that these methods are strong scalable up to at least 32 GPUs for a three-dimensional RTM case.

  16. A Kinetic Vlasov Model for Plasma Simulation Using Discontinuous Galerkin Method on Many-Core Architectures

    NASA Astrophysics Data System (ADS)

    Reddell, Noah

    Advances are reported in the three pillars of computational science achieving a new capability for understanding dynamic plasma phenomena outside of local thermodynamic equilibrium. A continuum kinetic model for plasma based on the Vlasov-Maxwell system for multiple particle species is developed. Consideration is added for boundary conditions in a truncated velocity domain and supporting wall interactions. A scheme to scale the velocity domain for multiple particle species with different temperatures and particle mass while sharing one computational mesh is described. A method for assessing the degree to which the kinetic solution differs from a Maxwell-Boltzmann distribution is introduced and tested on a thoroughly studied test case. The discontinuous Galerkin numerical method is extended for efficient solution of hyperbolic conservation laws in five or more particle phase-space dimensions using tensor-product hypercube elements with arbitrary polynomial order. A scheme for velocity moment integration is integrated as required for coupling between the plasma species and electromagnetic waves. A new high performance simulation code WARPM is developed to efficiently implement the model and numerical method on emerging many-core supercomputing architectures. WARPM uses the OpenCL programming model for computational kernels and task parallelism to overlap computation with communication. WARPM single-node performance and parallel scaling efficiency are analyzed with bottlenecks identified guiding future directions for the implementation. The plasma modeling capability is validated against physical problems with analytic solutions and well established benchmark problems.

  17. Runge-Kutta discontinuous Galerkin methods for the special relativistic magnetohydrodynamics

    NASA Astrophysics Data System (ADS)

    Zhao, Jian; Tang, Huazhong

    2017-08-01

    This paper develops PK-based non-central and central Runge-Kutta discontinuous Galerkin (DG) methods with WENO limiter for the one- and two-dimensional special relativistic magnetohydrodynamical (RMHD) equations, K = 1 , 2 , 3. The non-central DG methods are locally divergence-free, while the central DG are ;exactly; divergence-free but have to find two approximate solutions defined on mutually dual meshes. The adaptive WENO limiter first identifies the ;troubled; cells by using a modified TVB minmod function, and then uses the WENO technique and the cell average values of the DG solutions in the neighboring cells as well as the original cell averages of the ;troubled; cells to locally reconstruct a new polynomial of degree (2 K + 1) inside the ;troubled; cells instead of the DG solution. The WENO limiting procedure does not destroy the locally or ;exactly; divergence-free property of magnetic field and is only employed for finite ;troubled; cells so that the computational cost can be as little as possible. Several test problems in one and two dimensions are solved by using our non-central and central Runge-Kutta DG methods with WENO limiter. The numerical results demonstrate that our methods are stable, accurate, and robust in resolving complex wave structures.

  18. On Formulations of Discontinuous Galerkin and Related Methods for Conservation Laws

    NASA Technical Reports Server (NTRS)

    Huynh, H. T.

    2014-01-01

    A formulation for the discontinuous Galerkin (DG) method that leads to solutions using the differential form of the equation (as opposed to the standard integral form) is presented. The formulation includes (a) a derivative calculation that involves only data within each cell with no data interaction among cells, and (b) for each cell, corrections to this derivative that deal with the jumps in fluxes at the cell boundaries and allow data across cells to interact. The derivative with no interaction is obtained by a projection, but for nodal-type methods, evaluating this derivative by interpolation at the nodal points is more economical. The corrections are derived using the approximate (Dirac) delta functions. The formulation results in a family of schemes: different approximate delta functions give rise to different methods. It is shown that the current formulation is essentially equivalent to the flux reconstruction (FR) formulation. Due to the use of approximate delta functions, an energy stability proof simpler than that of Vincent, Castonguay, and Jameson (2011) for a family of schemes is derived. Accuracy and stability of resulting schemes are discussed via Fourier analyses. Similar to FR, the current formulation provides a unifying framework for high-order methods by recovering the DG, spectral difference (SD), and spectral volume (SV) schemes. It also yields stable, accurate, and economical methods.

  19. Dynamic mesh adaptation for front evolution using discontinuous Galerkin based weighted condition number relaxation

    NASA Astrophysics Data System (ADS)

    Greene, Patrick T.; Schofield, Samuel P.; Nourgaliev, Robert

    2017-04-01

    A new mesh smoothing method designed to cluster cells near a dynamically evolving interface is presented. The method is based on weighted condition number mesh relaxation with the weight function computed from a level set representation of the interface. The weight function is expressed as a Taylor series based discontinuous Galerkin projection, which makes the computation of the derivatives of the weight function needed during the condition number optimization process a trivial matter. For cases when a level set is not available, a fast method for generating a low-order level set from discrete cell-centered fields, such as a volume fraction or index function, is provided. Results show that the low-order level set works equally well as the actual level set for mesh smoothing. Meshes generated for a number of interface geometries are presented, including cases with multiple level sets. Dynamic cases with moving interfaces show the new method is capable of maintaining a desired resolution near the interface with an acceptable number of relaxation iterations per time step, which demonstrates the method's potential to be used as a mesh relaxer for arbitrary Lagrangian Eulerian (ALE) methods.

  20. Dynamic mesh adaptation for front evolution using discontinuous Galerkin based weighted condition number relaxation

    NASA Astrophysics Data System (ADS)

    Greene, Patrick; Schofield, Sam; Nourgaliev, Robert

    2016-11-01

    A new mesh smoothing method designed to cluster cells near a dynamically evolving interface is presented. The method is based on weighted condition number mesh relaxation with the weight function being computed from a level set representation of the interface. The weight function is expressed as a Taylor series based discontinuous Galerkin (DG) projection, which makes the computation of the derivatives of the weight function needed during the condition number optimization process a trivial matter. For cases when a level set is not available, a fast method for generating a low-order level set from discrete cell-centered fields, such as a volume fraction or index function, is provided. Results show that the low-order level set works equally well for the weight function as the actual level set. The method retains the excellent smoothing capabilities of condition number relaxation, while providing a method for clustering mesh cells near regions of interest. Dynamic cases for moving interfaces are presented to demonstrate the method's potential usefulness as a mesh relaxer for arbitrary Lagrangian Eulerian (ALE) methods. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

  1. High-order Hybridized Discontinuous Galerkin methods for Large-Eddy Simulation

    NASA Astrophysics Data System (ADS)

    Fernandez, Pablo; Nguyen, Ngoc-Cuong; Peraire, Jaime

    2016-11-01

    With the increase in computing power, Large-Eddy Simulation emerges as a promising technique to improve both knowledge of complex flow physics and reliability of flow predictions. Most LES works, however, are limited to simple geometries and low Reynolds numbers due to high computational cost. While most existing LES codes are based on 2nd-order finite volume schemes, the efficient and accurate prediction of complex turbulent flows may require a paradigm shift in computational approach. This drives a growing interest in the development of Discontinuous Galerkin (DG) methods for LES. DG methods allow for high-order, conservative implementations on complex geometries, and offer opportunities for improved sub-grid scale modeling. Also, high-order DG methods are better-suited to exploit modern HPC systems. In the spirit of making them more competitive, researchers have recently developed the hybridized DG methods that result in reduced computational cost and memory footprint. In this talk we present an overview of high-order hybridized DG methods for LES. Numerical accuracy, computational efficiency, and SGS modeling issues are discussed. Numerical results up to Re=460k show rapid grid convergence and excellent agreement with experimental data at moderate computational cost.

  2. A Hamiltonian preserving discontinuous Galerkin method for the generalized Korteweg-de Vries equation

    NASA Astrophysics Data System (ADS)

    Liu, Hailiang; Yi, Nianyu

    2016-09-01

    The invariant preserving property is one of the guiding principles for numerical algorithms in solving wave equations, in order to minimize phase and amplitude errors after long time simulation. In this paper, we design, analyze and numerically validate a Hamiltonian preserving discontinuous Galerkin method for solving the Korteweg-de Vries (KdV) equation. For the generalized KdV equation, the semi-discrete formulation is shown to preserve both the first and the third conserved integrals, and approximately preserve the second conserved integral; for the linearized KdV equation, all the first three conserved integrals are preserved, and optimal error estimates are obtained for polynomials of even degree. The preservation properties are also maintained by the fully discrete DG scheme. Our numerical experiments demonstrate both high accuracy of convergence and preservation of all three conserved integrals for the generalized KdV equation. We also show that the shape of the solution, after long time simulation, is well preserved due to the Hamiltonian preserving property.

  3. Hamiltonian discontinuous Galerkin FEM for linear, rotating incompressible Euler equations: Inertial waves

    SciTech Connect

    Nurijanyan, S.; Vegt, J.J.W. van der; Bokhove, O.

    2013-05-15

    A discontinuous Galerkin finite element method (DGFEM) has been developed and tested for the linear, three-dimensional, rotating incompressible Euler equations. These equations admit complicated wave solutions, which poses numerical challenges. These challenges concern: (i) discretisation of a divergence-free velocity field; (ii) discretisation of geostrophic boundary conditions combined with no-normal flow at solid walls; (iii) discretisation of the conserved, Hamiltonian dynamics of the inertial-waves; and, (iv) large-scale computational demands owing to the three-dimensional nature of inertial-wave dynamics and possibly its narrow zones of chaotic attraction. These issues have been resolved, for example: (i) by employing Dirac’s method of constrained Hamiltonian dynamics to our DGFEM for linear, compressible flows, thus enforcing the incompressibility constraints; (ii) by enforcing no-normal flow at solid walls in a weak form and geostrophic tangential flow along the wall; and, (iii) by applying a symplectic time discretisation. We compared our simulations with exact solutions of three-dimensional incompressible flows, in (non) rotating periodic and partly periodic cuboids (Poincaré waves). Additional verifications concerned semi-analytical eigenmode solutions in rotating cuboids with solid walls. Finally, a simulation in a tilted rotating tank, yielding more complicated wave dynamics, demonstrates the potential of our new method.

  4. Application of a discontinuous Galerkin time domain method to simulation of optical properties of dielectric particles.

    PubMed

    Tang, Guanglin; Panetta, R Lee; Yang, Ping

    2010-05-20

    We applied a discontinuous Galerkin time domain (DGTD) method, using a fourth-order Runga-Kutta time stepping of the Maxwell equations, to the simulation of the optical properties of dielectric particles in two-dimensional (2D) geometry. As examples of the numerical implementation of this method, the single-scattering properties of 2D circular and hexagonal particles are presented. In the case of circular particles, the scattering phase matrix was computed using the DGTD method and compared with the exact solution. For hexagonal particles, the DGTD method was used to compute single-scattering properties of randomly oriented 2D hexagonal ice crystals, and results were compared with those calculated using a geometric optics method. We consider both shortwave (visible) and longwave (infrared) cases, with particle size parameters 50 and 100. In the hexagonal case, scattering results are also presented as a function of both incident and scattering angles, revealing a structure apparently not reported before. Using the geometric optics method, we are able to interpret this structure in terms of contributions from varying numbers of internal reflections within the crystal.

  5. Recovery Discontinuous Galerkin Jacobian-free Newton-Krylov Method for all-speed flows

    SciTech Connect

    HyeongKae Park; Robert Nourgaliev; Vincent Mousseau; Dana Knoll

    2008-07-01

    There is an increasing interest to develop the next generation simulation tools for the advanced nuclear energy systems. These tools will utilize the state-of-art numerical algorithms and computer science technology in order to maximize the predictive capability, support advanced reactor designs, reduce uncertainty and increase safety margins. In analyzing nuclear energy systems, we are interested in compressible low-Mach number, high heat flux flows with a wide range of Re, Ra, and Pr numbers. Under these conditions, the focus is placed on turbulent heat transfer, in contrast to other industries whose main interest is in capturing turbulent mixing. Our objective is to develop singlepoint turbulence closure models for large-scale engineering CFD code, using Direct Numerical Simulation (DNS) or Large Eddy Simulation (LES) tools, requireing very accurate and efficient numerical algorithms. The focus of this work is placed on fully-implicit, high-order spatiotemporal discretization based on the discontinuous Galerkin method solving the conservative form of the compressible Navier-Stokes equations. The method utilizes a local reconstruction procedure derived from weak formulation of the problem, which is inspired by the recovery diffusion flux algorithm of van Leer and Nomura [?] and by the piecewise parabolic reconstruction [?] in the finite volume method. The developed methodology is integrated into the Jacobianfree Newton-Krylov framework [?] to allow a fully-implicit solution of the problem.

  6. Edge reconstruction in armchair phosphorene nanoribbons revealed by discontinuous Galerkin density functional theory.

    PubMed

    Hu, Wei; Lin, Lin; Yang, Chao

    2015-12-21

    With the help of our recently developed massively parallel DGDFT (Discontinuous Galerkin Density Functional Theory) methodology, we perform large-scale Kohn-Sham density functional theory calculations on phosphorene nanoribbons with armchair edges (ACPNRs) containing a few thousands to ten thousand atoms. The use of DGDFT allows us to systematically achieve a conventional plane wave basis set type of accuracy, but with a much smaller number (about 15) of adaptive local basis (ALB) functions per atom for this system. The relatively small number of degrees of freedom required to represent the Kohn-Sham Hamiltonian, together with the use of the pole expansion the selected inversion (PEXSI) technique that circumvents the need to diagonalize the Hamiltonian, results in a highly efficient and scalable computational scheme for analyzing the electronic structures of ACPNRs as well as their dynamics. The total wall clock time for calculating the electronic structures of large-scale ACPNRs containing 1080-10,800 atoms is only 10-25 s per self-consistent field (SCF) iteration, with accuracy fully comparable to that obtained from conventional planewave DFT calculations. For the ACPNR system, we observe that the DGDFT methodology can scale to 5000-50,000 processors. We use DGDFT based ab initio molecular dynamics (AIMD) calculations to study the thermodynamic stability of ACPNRs. Our calculations reveal that a 2 × 1 edge reconstruction appears in ACPNRs at room temperature.

  7. A nodal discontinuous Galerkin finite element method for nonlinear elastic wave propagation.

    PubMed

    Bou Matar, Olivier; Guerder, Pierre-Yves; Li, YiFeng; Vandewoestyne, Bart; Van Den Abeele, Koen

    2012-05-01

    A nodal discontinuous Galerkin finite element method (DG-FEM) to solve the linear and nonlinear elastic wave equation in heterogeneous media with arbitrary high order accuracy in space on unstructured triangular or quadrilateral meshes is presented. This DG-FEM method combines the geometrical flexibility of the finite element method, and the high parallelization potentiality and strongly nonlinear wave phenomena simulation capability of the finite volume method, required for nonlinear elastodynamics simulations. In order to facilitate the implementation based on a numerical scheme developed for electromagnetic applications, the equations of nonlinear elastodynamics have been written in a conservative form. The adopted formalism allows the introduction of different kinds of elastic nonlinearities, such as the classical quadratic and cubic nonlinearities, or the quadratic hysteretic nonlinearities. Absorbing layers perfectly matched to the calculation domain of the nearly perfectly matched layers type have been introduced to simulate, when needed, semi-infinite or infinite media. The developed DG-FEM scheme has been verified by means of a comparison with analytical solutions and numerical results already published in the literature for simple geometrical configurations: Lamb's problem and plane wave nonlinear propagation.

  8. An efficient parallel implementation of explicit multirate Runge–Kutta schemes for discontinuous Galerkin computations

    SciTech Connect

    Seny, Bruno Lambrechts, Jonathan; Toulorge, Thomas; Legat, Vincent; Remacle, Jean-François

    2014-01-01

    Although explicit time integration schemes require small computational efforts per time step, their efficiency is severely restricted by their stability limits. Indeed, the multi-scale nature of some physical processes combined with highly unstructured meshes can lead some elements to impose a severely small stable time step for a global problem. Multirate methods offer a way to increase the global efficiency by gathering grid cells in appropriate groups under local stability conditions. These methods are well suited to the discontinuous Galerkin framework. The parallelization of the multirate strategy is challenging because grid cells have different workloads. The computational cost is different for each sub-time step depending on the elements involved and a classical partitioning strategy is not adequate any more. In this paper, we propose a solution that makes use of multi-constraint mesh partitioning. It tends to minimize the inter-processor communications, while ensuring that the workload is almost equally shared by every computer core at every stage of the algorithm. Particular attention is given to the simplicity of the parallel multirate algorithm while minimizing computational and communication overheads. Our implementation makes use of the MeTiS library for mesh partitioning and the Message Passing Interface for inter-processor communication. Performance analyses for two and three-dimensional practical applications confirm that multirate methods preserve important computational advantages of explicit methods up to a significant number of processors.

  9. Mixed meshless local Petrov-Galerkin collocation method for modeling of material discontinuity

    NASA Astrophysics Data System (ADS)

    Jalušić, Boris; Sorić, Jurica; Jarak, Tomislav

    2017-01-01

    A mixed meshless local Petrov-Galerkin (MLPG) collocation method is proposed for solving the two-dimensional boundary value problem of heterogeneous structures. The heterogeneous structures are defined by partitioning the total material domain into subdomains with different linear-elastic isotropic properties which define homogeneous materials. The discretization and approximation of unknown field variables is done for each homogeneous material independently, therein the interface of the homogeneous materials is discretized with overlapping nodes. For the approximation, the moving least square method with the imposed interpolation condition is utilized. The solution for the entire heterogeneous structure is obtained by enforcing displacement continuity and traction reciprocity conditions at the nodes representing the interface boundary. The accuracy and numerical efficiency of the proposed mixed MLPG collocation method is demonstrated by numerical examples. The obtained results are compared with a standard fully displacement (primal) meshless approach as well as with available analytical and numerical solutions. Excellent agreement of the solutions is exhibited and a more robust, superior and stable modeling of material discontinuity is achieved using the mixed method.

  10. A New Discontinuous Galerkin Method for Convection-Diffusion Problems: The Gradient-Recovery DG Method

    NASA Astrophysics Data System (ADS)

    Johnson, Philip; Johnsen, Eric

    2016-11-01

    The Discontinuous Galerkin (DG) numerical method, while well-suited for hyperbolic PDE systems such as the Euler equations, is not naturally competitive for convection-diffusion systems, such as the Navier-Stokes equations. Where the DG weak form of the Euler equations depends only on the field variables for calculation of numerical fluxes, the traditional form of the Navier-Stokes equations requires calculation of the gradients of field variables for flux calculations. It is this latter task for which the standard DG discretization is ill-suited, and several approaches have been proposed to treat the issue. The most popular strategy for handling diffusion is the "mixed" approach, where the solution gradient is constructed from the primal as an auxiliary. We designed a new mixed approach, called Gradient-Recovery DG; it uses the Recovery concept of Van Leer & Nomura with the mixed approach to produce a scheme with excellent stability, high accuracy, and unambiguous implementation when compared to typical mixed approach concepts. In addition to describing the scheme, we will perform analysis with comparison to other DG approaches for diffusion. Gas dynamics examples will be presented to demonstrate the scheme's capabilities.

  11. A direct discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids

    NASA Astrophysics Data System (ADS)

    Cheng, Jian; Yang, Xiaoquan; Liu, Xiaodong; Liu, Tiegang; Luo, Hong

    2016-12-01

    A Direct Discontinuous Galerkin (DDG) method is developed for solving the compressible Navier-Stokes equations on arbitrary grids in the framework of DG methods. The DDG method, originally introduced for scalar diffusion problems on structured grids, is extended to discretize viscous and heat fluxes in the Navier-Stokes equations. Two approaches of implementing the DDG method to compute numerical diffusive fluxes for the Navier-Stokes equations are presented: one is based on the conservative variables, and the other is based on the primitive variables. The importance of the characteristic cell size used in the DDG formulation on unstructured grids is examined. The numerical fluxes on the boundary by the DDG method are discussed. A number of test cases are presented to assess the performance of the DDG method for solving the compressible Navier-Stokes equations. Based on our numerical results, we observe that DDG method can achieve the designed order of accuracy and is able to deliver the same accuracy as the widely used BR2 method at a significantly reduced cost, clearly demonstrating that the DDG method provides an attractive alternative for solving the compressible Navier-Stokes equations on arbitrary grids owning to its simplicity in implementation and its efficiency in computational cost.

  12. GPU performance analysis of a nodal discontinuous Galerkin method for acoustic and elastic models

    NASA Astrophysics Data System (ADS)

    Modave, A.; St-Cyr, A.; Warburton, T.

    2016-06-01

    Finite element schemes based on discontinuous Galerkin methods possess features amenable to massively parallel computing accelerated with general purpose graphics processing units (GPUs). However, the computational performance of such schemes strongly depends on their implementation. In the past, several implementation strategies have been proposed. They are based exclusively on specialized compute kernels tuned for each operation, or they can leverage BLAS libraries that provide optimized routines for basic linear algebra operations. In this paper, we present and analyze up-to-date performance results for different implementations, tested in a unified framework on a single NVIDIA GTX980 GPU. We show that specialized kernels written with a one-node-per-thread strategy are competitive for polynomial bases up to the fifth and seventh degrees for acoustic and elastic models, respectively. For higher degrees, a strategy that makes use of the NVIDIA cuBLAS library provides better results, able to reach a net arithmetic throughput 35.7% of the theoretical peak value.

  13. Optimal Runge-Kutta schemes for discontinuous Galerkin space discretizations applied to wave propagation problems

    NASA Astrophysics Data System (ADS)

    Toulorge, T.; Desmet, W.

    2012-02-01

    We study the performance of methods of lines combining discontinuous Galerkin spatial discretizations and explicit Runge-Kutta time integrators, with the aim of deriving optimal Runge-Kutta schemes for wave propagation applications. We review relevant Runge-Kutta methods from literature, and consider schemes of order q from 3 to 4, and number of stages up to q + 4, for optimization. From a user point of view, the problem of the computational efficiency involves the choice of the best combination of mesh and numerical method; two scenarios are defined. In the first one, the element size is totally free, and a 8-stage, fourth-order Runge-Kutta scheme is found to minimize a cost measure depending on both accuracy and stability. In the second one, the elements are assumed to be constrained to such a small size by geometrical features of the computational domain, that accuracy is disregarded. We then derive one 7-stage, third-order scheme and one 8-stage, fourth-order scheme that maximize the stability limit. The performance of the three new schemes is thoroughly analyzed, and the benefits are illustrated with two examples. For each of these Runge-Kutta methods, we provide the coefficients for a 2 N-storage implementation, along with the information needed by the user to employ them optimally.

  14. A time-domain Discontinuous Galerkin method for mechanical resonator quality factor computations

    NASA Astrophysics Data System (ADS)

    Govindjee, Sanjay; Persson, Per-Olof

    2012-08-01

    Numerical simulations are becoming increasingly important in the design of micromechanical resonators, in particular for the prediction of complex frequency response in high quality devices where damping is controlled by anchor losses. Frequency based approaches have been shown to predict these accurately, however, they require the solution of eigenvalue problems or the inversion of Helmholtz-type operators which are known to be very difficult for large-scale iterative solvers. We propose using a time-domain approach instead, where a broadband input signal is propagated through the system with a local explicit time-stepper. This is achieved using a new high-order Discontinuous Galerkin (DG) discretization for the linear elasticity equations, in particular a second-order formulation with Compact DG fluxes and a Runge-Kutta time integrator, where the block-diagonal mass matrices allow for efficient, stable, and accurate time stepping. Our solver scales well on distributed parallel computers, even in three spatial dimension and for large problem sizes. The resulting output signal is analyzed using a well-known filter diagonalization method, which is capable of finding accurate frequencies and quality factors for as little as a hundred periods of data. We validate the properties of our scheme on model problems, and demonstrate the feasibility of our proposed analysis process on two high quality factor disk resonators, using an axisymmetric formulation as well as full three dimensional simulations which is shown to scale well.

  15. Validation of an Adaptive Triangular Discontinuous Galerkin Shallow Water Model for the 2011 Tohoku Tsunami

    NASA Astrophysics Data System (ADS)

    Vater, Stefan; Behrens, Jörn

    2016-04-01

    We apply a tsunami simulation framework, which is based on depth-integrated hydrodynamic model equations, to the 2011 Tohoku tsunami event. While this model has been previously validated for analytic test cases and laboratory experiments, here it is applied to earthquake sources which are based on seismic inversion. Simulated wave heights and runup at the coast are compared to actual measurements. The discretization is based on a second-order Runge-Kutta discontinuous Galerkin (RKDG) scheme on triangular grids and features a robust wetting and drying scheme for the simulation of inundation events at the coast. Adaptive mesh refinement enables the efficient computation of large domains, while at the same time it allows for high local resolution and geometric accuracy. This work is part of the ASCETE (Advanced Simulation of Coupled Earthquake and Tsunami Events) project, which aims at an improved understanding of the coupling between the earthquake and the generated tsunami event. In this course, a coupled simulation framework has been developed which couples physics-based rupture generation with the presented hydrodynamic tsunami propagation and inundation model.

  16. A curved boundary treatment for discontinuous Galerkin schemes solving time dependent problems

    NASA Astrophysics Data System (ADS)

    Zhang, Xiangxiong

    2016-03-01

    For problems defined in a two-dimensional domain Ω with boundary conditions specified on a curve Γ, we consider discontinuous Galerkin (DG) schemes with high order polynomial basis functions on a geometry fitting triangular mesh. It is well known that directly imposing the given boundary conditions on a piecewise segment approximation boundary Γh will render any finite element method to be at most second order accurate. Unless the boundary conditions can be accurately transferred from Γ to Γh, in general curvilinear element method should be used to obtain high order accuracy. We discuss a simple boundary treatment which can be implemented as a modified DG scheme defined on triangles adjacent to Γh. Even though integration along the curve is still necessary, integrals over any curved element are avoided. If the domain Ω is convex, or if Ω is nonconvex and the true solutions can be smoothly extended to the exterior of Ω, the modified DG scheme is high order accurate. In these cases, numerical tests on first order and second order partial differential equations including hyperbolic systems and the scalar wave equation suggest that it is as accurate as the full curvilinear DG scheme.

  17. 3D unstructured mesh ALE hydrodynamics with the upwind discontinuous galerkin method

    SciTech Connect

    Kershaw, D S; Milovich, J L; Prasad, M K; Shaw, M J; Shestakov, A I

    1999-05-07

    The authors describe a numerical scheme to solve 3D Arbitrary Lagrangian-Eulerian (ALE) hydrodynamics on an unstructured mesh using a discontinuous Galerkin method (DGM) and an explicit Runge-Kutta time discretization. Upwinding is achieved through Roe's linearized Riemann solver with the Harten-Hyman entropy fix. For stabilization, a 3D quadratic programming generalization of van Leer's 1D minmod slope limiter is used along with a Lapidus type artificial viscosity. This DGM scheme has been tested on a variety of hydrodynamic test problems and appears to be robust making it the basis for the integrated 3D inertial confinement fusion modeling code (ICF3D). For efficient code development, they use C++ object oriented programming to easily separate the complexities of an unstructured mesh from the basic physics modules. ICF3D is fully parallelized using domain decomposition and the MPI message passing library. It is fully portable. It runs on uniprocessor workstations and massively parallel platforms with distributed and shared memory.

  18. Non-conforming hybrid meshes for efficient 2-D wave propagation using the Discontinuous Galerkin Method

    NASA Astrophysics Data System (ADS)

    Hermann, Verena; Käser, Martin; Castro, Cristóbal E.

    2011-02-01

    We present a Discontinuous Galerkin finite element method using a high-order time integration technique for seismic wave propagation modelling on non-conforming hybrid meshes in two space dimensions. The scheme can be formulated to achieve the same approximation order in space and time and avoids numerical artefacts due to non-conforming mesh transitions or the change of the element type. A point-wise Gaussian integration along partially overlapping edges of adjacent elements is used to preserve the schemes accuracy while providing a higher flexibility in the problem-adapted mesh generation process. We describe the domain decomposition strategy of the parallel implementation and validate the performance of the new scheme by numerical convergence test and experiments with comparisons to independent reference solutions. The advantage of non-conforming hybrid meshes is the possibility to choose the mesh spacing proportional to the seismic velocity structure, which allows for simple refinement or coarsening methods even for regular quadrilateral meshes. For particular problems of strong material contrasts and geometrically thin structures, the scheme reduces the computational cost in the sense of memory and run-time requirements. The presented results promise to achieve a similar behaviour for an extension to three space dimensions where the coupling of tetrahedral and hexahedral elements necessitates non-conforming mesh transitions to avoid linking elements in form of pyramids.

  19. Preconditioning for modal discontinuous Galerkin methods for unsteady 3D Navier-Stokes equations

    NASA Astrophysics Data System (ADS)

    Birken, Philipp; Gassner, Gregor; Haas, Mark; Munz, Claus-Dieter

    2013-05-01

    We compare different block preconditioners in the context of parallel time adaptive higher order implicit time integration using Jacobian-free Newton-Krylov (JFNK) solvers for discontinuous Galerkin (DG) discretizations of the three dimensional time dependent Navier-Stokes equations. A special emphasis of this work is the performance for a relative high number of processors, i.e. with a low number of elements on the processor. For high order DG discretizations, a particular problem that needs to be addressed is the size of the blocks in the Jacobian. Thus, we propose a new class of preconditioners that exploits the hierarchy of modal basis functions and introduces a flexible order of the off-diagonal Jacobian blocks. While the standard preconditioners 'block Jacobi' (no off-blocks) and full symmetric Gauss-Seidel (full off-blocks) are included as special cases, the reduction of the off-block order results in the new scheme ROBO-SGS. This allows us to investigate the impact of the preconditioner's sparsity pattern with respect to the computational performance. Since the number of iterations is not well suited to judge the efficiency of a preconditioner, we additionally consider CPU time for the comparisons. We found that both block Jacobi and ROBO-SGS have good overall performance and good strong parallel scaling behavior.

  20. A hybridizable discontinuous Galerkin method for modeling fluid–structure interaction

    DOE PAGES

    Sheldon, Jason P.; Miller, Scott T.; Pitt, Jonathan S.

    2016-08-31

    This study presents a novel application of the hybridizable discontinuous Galerkin (HDG) finite element method to the multi-physics simulation of coupled fluid–structure interaction (FSI) problems. Recent applications of the HDG method have primarily been for single-physics problems including both solids and fluids, which are necessary building blocks for FSI modeling. Utilizing these established models, HDG formulations for linear elastostatics, a nonlinear elastodynamic model, and arbitrary Lagrangian–Eulerian Navier–Stokes are derived. The elasticity formulations are written in a Lagrangian reference frame, with the nonlinear formulation restricted to hyperelastic materials. With these individual solid and fluid formulations, the remaining challenge in FSI modelingmore » is coupling together their disparate mathematics on the fluid–solid interface. This coupling is presented, along with the resultant HDG FSI formulation. Verification of the component models, through the method of manufactured solutions, is performed and each model is shown to converge at the expected rate. The individual components, along with the complete FSI model, are then compared to the benchmark problems proposed by Turek and Hron [1]. The solutions from the HDG formulation presented in this work trend towards the benchmark as the spatial polynomial order and the temporal order of integration are increased.« less

  1. A hybridizable discontinuous Galerkin method for modeling fluid–structure interaction

    SciTech Connect

    Sheldon, Jason P.; Miller, Scott T.; Pitt, Jonathan S.

    2016-08-31

    This study presents a novel application of the hybridizable discontinuous Galerkin (HDG) finite element method to the multi-physics simulation of coupled fluid–structure interaction (FSI) problems. Recent applications of the HDG method have primarily been for single-physics problems including both solids and fluids, which are necessary building blocks for FSI modeling. Utilizing these established models, HDG formulations for linear elastostatics, a nonlinear elastodynamic model, and arbitrary Lagrangian–Eulerian Navier–Stokes are derived. The elasticity formulations are written in a Lagrangian reference frame, with the nonlinear formulation restricted to hyperelastic materials. With these individual solid and fluid formulations, the remaining challenge in FSI modeling is coupling together their disparate mathematics on the fluid–solid interface. This coupling is presented, along with the resultant HDG FSI formulation. Verification of the component models, through the method of manufactured solutions, is performed and each model is shown to converge at the expected rate. The individual components, along with the complete FSI model, are then compared to the benchmark problems proposed by Turek and Hron [1]. The solutions from the HDG formulation presented in this work trend towards the benchmark as the spatial polynomial order and the temporal order of integration are increased.

  2. Three discontinuous Galerkin schemes for the anisotropic heat conduction equation on non-aligned grids

    NASA Astrophysics Data System (ADS)

    Held, M.; Wiesenberger, M.; Stegmeir, A.

    2016-02-01

    We present and discuss three discontinuous Galerkin (dG) discretizations for the anisotropic heat conduction equation on non-aligned cylindrical grids. Our non-aligned scheme relies on a self-adjoint local dG (LDG) discretization of the elliptic operator. It conserves the energy exactly and converges with arbitrary order. The pollution by numerical perpendicular heat fluxes decreases with superconvergence rates. We compare this scheme with aligned schemes that are based on the flux-coordinate independent approach for the discretization of parallel derivatives. Here, the dG method provides the necessary interpolation. The first aligned discretization can be used in an explicit time-integrator. However, the scheme violates conservation of energy and shows up stagnating convergence rates for very high resolutions. We overcome this partly by using the adjoint of the parallel derivative operator to construct a second self-adjoint aligned scheme. This scheme preserves energy, but reveals unphysical oscillations in the numerical tests, which result in a decreased order of convergence. Both aligned schemes exhibit low numerical heat fluxes into the perpendicular direction and are superior for flute-modes with finite parallel gradients. We build our argumentation on various numerical experiments on all three schemes for a general axisymmetric magnetic field, which is closed by a comparison to the aligned finite difference (FD) schemes of Stegmeir et al. (2014) and Stegmeir et al. (submitted for publication).

  3. A discontinuous Galerkin conservative level set scheme for interface capturing in multiphase flows

    SciTech Connect

    Owkes, Mark Desjardins, Olivier

    2013-09-15

    The accurate conservative level set (ACLS) method of Desjardins et al. [O. Desjardins, V. Moureau, H. Pitsch, An accurate conservative level set/ghost fluid method for simulating turbulent atomization, J. Comput. Phys. 227 (18) (2008) 8395–8416] is extended by using a discontinuous Galerkin (DG) discretization. DG allows for the scheme to have an arbitrarily high order of accuracy with the smallest possible computational stencil resulting in an accurate method with good parallel scaling. This work includes a DG implementation of the level set transport equation, which moves the level set with the flow field velocity, and a DG implementation of the reinitialization equation, which is used to maintain the shape of the level set profile to promote good mass conservation. A near second order converging interface curvature is obtained by following a height function methodology (common amongst volume of fluid schemes) in the context of the conservative level set. Various numerical experiments are conducted to test the properties of the method and show excellent results, even on coarse meshes. The tests include Zalesak’s disk, two-dimensional deformation of a circle, time evolution of a standing wave, and a study of the Kelvin–Helmholtz instability. Finally, this novel methodology is employed to simulate the break-up of a turbulent liquid jet.

  4. An h-adaptive local discontinuous Galerkin method for the Navier-Stokes-Korteweg equations

    NASA Astrophysics Data System (ADS)

    Tian, Lulu; Xu, Yan; Kuerten, J. G. M.; van der Vegt, J. J. W.

    2016-08-01

    In this article, we develop a mesh adaptation algorithm for a local discontinuous Galerkin (LDG) discretization of the (non)-isothermal Navier-Stokes-Korteweg (NSK) equations modeling liquid-vapor flows with phase change. This work is a continuation of our previous research, where we proposed LDG discretizations for the (non)-isothermal NSK equations with a time-implicit Runge-Kutta method. To save computing time and to capture the thin interfaces more accurately, we extend the LDG discretization with a mesh adaptation method. Given the current adapted mesh, a criterion for selecting candidate elements for refinement and coarsening is adopted based on the locally largest value of the density gradient. A strategy to refine and coarsen the candidate elements is then provided. We emphasize that the adaptive LDG discretization is relatively simple and does not require additional stabilization. The use of a locally refined mesh in combination with an implicit Runge-Kutta time method is, however, non-trivial, but results in an efficient time integration method for the NSK equations. Computations, including cases with solid wall boundaries, are provided to demonstrate the accuracy, efficiency and capabilities of the adaptive LDG discretizations.

  5. A discontinuous Galerkin conservative level set scheme for interface capturing in multiphase flows

    NASA Astrophysics Data System (ADS)

    Owkes, Mark; Desjardins, Olivier

    2013-09-01

    The accurate conservative level set (ACLS) method of Desjardins et al. [O. Desjardins, V. Moureau, H. Pitsch, An accurate conservative level set/ghost fluid method for simulating turbulent atomization, J. Comput. Phys. 227 (18) (2008) 8395-8416] is extended by using a discontinuous Galerkin (DG) discretization. DG allows for the scheme to have an arbitrarily high order of accuracy with the smallest possible computational stencil resulting in an accurate method with good parallel scaling. This work includes a DG implementation of the level set transport equation, which moves the level set with the flow field velocity, and a DG implementation of the reinitialization equation, which is used to maintain the shape of the level set profile to promote good mass conservation. A near second order converging interface curvature is obtained by following a height function methodology (common amongst volume of fluid schemes) in the context of the conservative level set. Various numerical experiments are conducted to test the properties of the method and show excellent results, even on coarse meshes. The tests include Zalesak’s disk, two-dimensional deformation of a circle, time evolution of a standing wave, and a study of the Kelvin-Helmholtz instability. Finally, this novel methodology is employed to simulate the break-up of a turbulent liquid jet.

  6. Advanced Material Models for Nano-Plasmonic Systems via Discontinuous Galerkin Methods

    NASA Astrophysics Data System (ADS)

    Busch, Kurt

    2011-03-01

    Nano-Plasmonic systems offer a tremendous potential for the controlled delivery and extraction of electromagnetic energy to and from tiny objects such as molecules and quantum dots in their immediate vicinity. In view of the increasing sophistication of fabrication and spectroscopic characterization, quantitative computational approaches face challenges that go well beyond the usual description of metals as linear dispersive materials. These challenges include the development of material models that describe the (potentially) strongly nonlocal and nonlinear optical response of such metallic nano-structures themselves as well as the strongly modified light-matter interaction that is mediated by them. This talk reports on the progress of applying the Discontinuous-Galerkin Time-Domain (DGTD) method to the quantitative analysis of nano-plasmonic systems using advanced material models. This includes the efficient modeling of complex geometric features via curvilinear elements, the improvement of the time-stepping scheme via tailored low-storage Runge-Kutta schemes, and the incorporation of optically anisotropic media. In addition, this talk reports on recent results regarding the development and application of advanced material models that are based on a hydrodynamic description of the metal's conduction electrons. By coupling the Maxwell equations to this treatment of the free electrons as a plasma in a confined geometry one is able to capture nonlocal and nonlinear effects and to analyze their consequences.

  7. Simulation of linear aeracoustic propagation in lined ducts with discontinuous Galerkin method

    NASA Astrophysics Data System (ADS)

    Peyret, Christophe; Delorme, Philippe

    2004-05-01

    The simulation of the acoustic propagation inside a lined duct with nonuniform flow still presents problems when geometry is complex. To handle computations on a complex geometry, without consequential effort, unstructured meshes are required. Assuming irrotational flow and acoustic perturbation, a well-posed finite element method based on the potential equation is established. But, the effect of the thin boundary layer is then neglected, which is not relevant to the acoustical processes occurring near the lining. Recent works have focused on the tremendous interest of the Galerkin discontinuous method (GDM) to solve Euler's linearized equations. The GDM can handle computations on unstructured meshes and introduces low numerical dissipation. Very recent mathematical works have established, for the GDM, a well-posed boundary condition to simulate the lining effect. Results computed with the GDM are presented for a uniform cross section lined duct with a shear flow and are found to be in good agreement with the modal analysis results, thereby validating the boundary condition. To illustrate the flexibility of the method other applications dealing with instabilities, air-wing diffraction, and atmospheric propagation are also presented.

  8. Discontinuous Galerkin Methods for the Two-Moment Model of Radiation Transport

    NASA Astrophysics Data System (ADS)

    Endeve, Eirik; Hauck, Cory

    2016-03-01

    We are developing computational methods for simulation of radiation transport in astrophysical systems (e.g., neutrino transport in core-collapse supernovae). Here we consider the two-moment model of radiation transport, where the energy density E and flux F - angular moments of a phase space distribution function - approximates the radiation field in a computationally tractable manner. We aim to develop multi-dimensional methods that are (i) high-order accurate for computational efficiency, and (ii) robust in the sense that the solution remains in the realizable set R = { (E , F) | E >= 0 and E - | F | >= 0 } (i.e., E and F are consistent with moments of an underlying distribution). Our approach is based on the Runge-Kutta discontinuous Galerkin method, which has many attractive properties, including high-order accuracy on a compact stencil. We present the physical model and numerical method, and show results from a multi-dimensional implementation. Tests show that the method is high-order accurate and strictly preserves realizability of the moments.

  9. Robust and Accurate Shock Capturing Method for High-Order Discontinuous Galerkin Methods

    NASA Technical Reports Server (NTRS)

    Atkins, Harold L.; Pampell, Alyssa

    2011-01-01

    A simple yet robust and accurate approach for capturing shock waves using a high-order discontinuous Galerkin (DG) method is presented. The method uses the physical viscous terms of the Navier-Stokes equations as suggested by others; however, the proposed formulation of the numerical viscosity is continuous and compact by construction, and does not require the solution of an auxiliary diffusion equation. This work also presents two analyses that guided the formulation of the numerical viscosity and certain aspects of the DG implementation. A local eigenvalue analysis of the DG discretization applied to a shock containing element is used to evaluate the robustness of several Riemann flux functions, and to evaluate algorithm choices that exist within the underlying DG discretization. A second analysis examines exact solutions to the DG discretization in a shock containing element, and identifies a "model" instability that will inevitably arise when solving the Euler equations using the DG method. This analysis identifies the minimum viscosity required for stability. The shock capturing method is demonstrated for high-speed flow over an inviscid cylinder and for an unsteady disturbance in a hypersonic boundary layer. Numerical tests are presented that evaluate several aspects of the shock detection terms. The sensitivity of the results to model parameters is examined with grid and order refinement studies.

  10. Fully-Implicit Reconstructed Discontinuous Galerkin Method for Stiff Multiphysics Problems

    NASA Astrophysics Data System (ADS)

    Nourgaliev, Robert

    2015-11-01

    A new reconstructed Discontinuous Galerkin (rDG) method, based on orthogonal basis/test functions, is developed for fluid flows on unstructured meshes. Orthogonality of basis functions is essential for enabling robust and efficient fully-implicit Newton-Krylov based time integration. The method is designed for generic partial differential equations, including transient, hyperbolic, parabolic or elliptic operators, which are attributed to many multiphysics problems. We demonstrate the method's capabilities for solving compressible fluid-solid systems (in the low Mach number limit), with phase change (melting/solidification), as motivated by applications in Additive Manufacturing. We focus on the method's accuracy (in both space and time), as well as robustness and solvability of the system of linear equations involved in the linearization steps of Newton-based methods. The performance of the developed method is investigated for highly-stiff problems with melting/solidification, emphasizing the advantages from tight coupling of mass, momentum and energy conservation equations, as well as orthogonality of basis functions, which leads to better conditioning of the underlying (approximate) Jacobian matrices, and rapid convergence of the Krylov-based linear solver. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344, and funded by the LDRD at LLNL under project tracking code 13-SI-002.

  11. Fully-implicit orthogonal reconstructed Discontinuous Galerkin method for fluid dynamics with phase change

    NASA Astrophysics Data System (ADS)

    Nourgaliev, R.; Luo, H.; Weston, B.; Anderson, A.; Schofield, S.; Dunn, T.; Delplanque, J.-P.

    2016-01-01

    A new reconstructed Discontinuous Galerkin (rDG) method, based on orthogonal basis/test functions, is developed for fluid flows on unstructured meshes. Orthogonality of basis functions is essential for enabling robust and efficient fully-implicit Newton-Krylov based time integration. The method is designed for generic partial differential equations, including transient, hyperbolic, parabolic or elliptic operators, which are attributed to many multiphysics problems. We demonstrate the method's capabilities for solving compressible fluid-solid systems (in the low Mach number limit), with phase change (melting/solidification), as motivated by applications in Additive Manufacturing (AM). We focus on the method's accuracy (in both space and time), as well as robustness and solvability of the system of linear equations involved in the linearization steps of Newton-based methods. The performance of the developed method is investigated for highly-stiff problems with melting/solidification, emphasizing the advantages from tight coupling of mass, momentum and energy conservation equations, as well as orthogonality of basis functions, which leads to better conditioning of the underlying (approximate) Jacobian matrices, and rapid convergence of the Krylov-based linear solver.

  12. Recovery Discontinuous Galerkin Jacobian-Free Newton-Krylov Method for All-Speed Flows

    SciTech Connect

    HyeongKae Park; Robert Nourgaliev; Vincent Mousseau; Dana Knoll

    2008-07-01

    A novel numerical algorithm (rDG-JFNK) for all-speed fluid flows with heat conduction and viscosity is introduced. The rDG-JFNK combines the Discontinuous Galerkin spatial discretization with the implicit Runge-Kutta time integration under the Jacobian-free Newton-Krylov framework. We solve fully-compressible Navier-Stokes equations without operator-splitting of hyperbolic, diffusion and reaction terms, which enables fully-coupled high-order temporal discretization. The stability constraint is removed due to the L-stable Explicit, Singly Diagonal Implicit Runge-Kutta (ESDIRK) scheme. The governing equations are solved in the conservative form, which allows one to accurately compute shock dynamics, as well as low-speed flows. For spatial discretization, we develop a “recovery” family of DG, exhibiting nearly-spectral accuracy. To precondition the Krylov-based linear solver (GMRES), we developed an “Operator-Split”-(OS) Physics Based Preconditioner (PBP), in which we transform/simplify the fully-coupled system to a sequence of segregated scalar problems, each can be solved efficiently with Multigrid method. Each scalar problem is designed to target/cluster eigenvalues of the Jacobian matrix associated with a specific physics.

  13. Application of p-Multigrid to Discontinuous Galerkin Formulations of the Poisson Equation

    NASA Technical Reports Server (NTRS)

    Helenbrook, B. T.; Atkins, H. L.

    2006-01-01

    We investigate p-multigrid as a solution method for several different discontinuous Galerkin (DG) formulations of the Poisson equation. Different combinations of relaxation schemes and basis sets have been combined with the DG formulations to find the best performing combination. The damping factors of the schemes have been determined using Fourier analysis for both one and two-dimensional problems. One important finding is that when using DG formulations, the standard approach of forming the coarse p matrices separately for each level of multigrid is often unstable. To ensure stability the coarse p matrices must be constructed from the fine grid matrices using algebraic multigrid techniques. Of the relaxation schemes, we find that the combination of Jacobi relaxation with the spectral element basis is fairly effective. The results using this combination are p sensitive in both one and two dimensions, but reasonable convergence rates can still be achieved for moderate values of p and isotropic meshes. A competitive alternative is a block Gauss-Seidel relaxation. This actually out performs a more expensive line relaxation when the mesh is isotropic. When the mesh becomes highly anisotropic, the implicit line method and the Gauss-Seidel implicit line method are the only effective schemes. Adding the Gauss-Seidel terms to the implicit line method gives a significant improvement over the line relaxation method.

  14. A Reconstructed Discontinuous Galerkin Method for the Compressible Navier-Stokes Equations on Hybrid Grids

    SciTech Connect

    Xiaodong Liu; Lijun Xuan; Hong Luo; Yidong Xia

    2001-01-01

    A reconstructed discontinuous Galerkin (rDG(P1P2)) method, originally introduced for the compressible Euler equations, is developed for the solution of the compressible Navier- Stokes equations on 3D hybrid grids. In this method, a piecewise quadratic polynomial solution is obtained from the underlying piecewise linear DG solution using a hierarchical Weighted Essentially Non-Oscillatory (WENO) reconstruction. The reconstructed quadratic polynomial solution is then used for the computation of the inviscid fluxes and the viscous fluxes using the second formulation of Bassi and Reay (Bassi-Rebay II). The developed rDG(P1P2) method is used to compute a variety of flow problems to assess its accuracy, efficiency, and robustness. The numerical results demonstrate that the rDG(P1P2) method is able to achieve the designed third-order of accuracy at a cost slightly higher than its underlying second-order DG method, outperform the third order DG method in terms of both computing costs and storage requirements, and obtain reliable and accurate solutions to the large eddy simulation (LES) and direct numerical simulation (DNS) of compressible turbulent flows.

  15. Discontinuous Galerkin solution of the Navier-Stokes equations on deformable domains

    SciTech Connect

    Persson, P.-O.; Bonet, J.; Peraire, J.

    2009-01-13

    We describe a method for computing time-dependent solutions to the compressible Navier-Stokes equations on variable geometries. We introduce a continuous mapping between a fixed reference configuration and the time varying domain, By writing the Navier-Stokes equations as a conservation law for the independent variables in the reference configuration, the complexity introduced by variable geometry is reduced to solving a transformed conservation law in a fixed reference configuration, The spatial discretization is carried out using the Discontinuous Galerkin method on unstructured meshes of triangles, while the time integration is performed using an explicit Runge-Kutta method, For general domain changes, the standard scheme fails to preserve exactly the free-stream solution which leads to some accuracy degradation, especially for low order approximations. This situation is remedied by adding an additional equation for the time evolution of the transformation Jacobian to the original conservation law and correcting for the accumulated metric integration errors. A number of results are shown to illustrate the flexibility of the approach to handle high order approximations on complex geometries.

  16. Assessment of a Hybrid Continuous/Discontinuous Galerkin Finite Element Code for Geothermal Reservoir Simulations

    DOE PAGES

    Xia, Yidong; Podgorney, Robert; Huang, Hai

    2016-03-17

    FALCON (“Fracturing And Liquid CONvection”) is a hybrid continuous / discontinuous Galerkin finite element geothermal reservoir simulation code based on the MOOSE (“Multiphysics Object-Oriented Simulation Environment”) framework being developed and used for multiphysics applications. In the present work, a suite of verification and validation (“V&V”) test problems for FALCON was defined to meet the design requirements, and solved to the interests of enhanced geothermal system (“EGS”) design. Furthermore, the intent for this test problem suite is to provide baseline comparison data that demonstrates the performance of the FALCON solution methods. The simulation problems vary in complexity from singly mechanical ormore » thermo process, to coupled thermo-hydro-mechanical processes in geological porous media. Numerical results obtained by FALCON agreed well with either the available analytical solution or experimental data, indicating the verified and validated implementation of these capabilities in FALCON. Some form of solution verification has been attempted to identify sensitivities in the solution methods, where possible, and suggest best practices when using the FALCON code.« less

  17. Assessment of a Hybrid Continuous/Discontinuous Galerkin Finite Element Code for Geothermal Reservoir Simulations

    SciTech Connect

    Xia, Yidong; Podgorney, Robert; Huang, Hai

    2016-03-17

    FALCON (“Fracturing And Liquid CONvection”) is a hybrid continuous / discontinuous Galerkin finite element geothermal reservoir simulation code based on the MOOSE (“Multiphysics Object-Oriented Simulation Environment”) framework being developed and used for multiphysics applications. In the present work, a suite of verification and validation (“V&V”) test problems for FALCON was defined to meet the design requirements, and solved to the interests of enhanced geothermal system (“EGS”) design. Furthermore, the intent for this test problem suite is to provide baseline comparison data that demonstrates the performance of the FALCON solution methods. The simulation problems vary in complexity from singly mechanical or thermo process, to coupled thermo-hydro-mechanical processes in geological porous media. Numerical results obtained by FALCON agreed well with either the available analytical solution or experimental data, indicating the verified and validated implementation of these capabilities in FALCON. Some form of solution verification has been attempted to identify sensitivities in the solution methods, where possible, and suggest best practices when using the FALCON code.

  18. Fully-Implicit Orthogonal Reconstructed Discontinuous Galerkin for Fluid Dynamics with Phase Change

    SciTech Connect

    Nourgaliev, R.; Luo, H.; Weston, B.; Anderson, A.; Schofield, S.; Dunn, T.; Delplanque, J. -P.

    2015-11-11

    A new reconstructed Discontinuous Galerkin (rDG) method, based on orthogonal basis/test functions, is developed for fluid flows on unstructured meshes. Orthogonality of basis functions is essential for enabling robust and efficient fully-implicit Newton-Krylov based time integration. The method is designed for generic partial differential equations, including transient, hyperbolic, parabolic or elliptic operators, which are attributed to many multiphysics problems. We demonstrate the method’s capabilities for solving compressible fluid-solid systems (in the low Mach number limit), with phase change (melting/solidification), as motivated by applications in Additive Manufacturing (AM). We focus on the method’s accuracy (in both space and time), as well as robustness and solvability of the system of linear equations involved in the linearization steps of Newton-based methods. The performance of the developed method is investigated for highly-stiff problems with melting/solidification, emphasizing the advantages from tight coupling of mass, momentum and energy conservation equations, as well as orthogonality of basis functions, which leads to better conditioning of the underlying (approximate) Jacobian matrices, and rapid convergence of the Krylov-based linear solver.

  19. Fully-Implicit Orthogonal Reconstructed Discontinuous Galerkin for Fluid Dynamics with Phase Change

    DOE PAGES

    Nourgaliev, R.; Luo, H.; Weston, B.; ...

    2015-11-11

    A new reconstructed Discontinuous Galerkin (rDG) method, based on orthogonal basis/test functions, is developed for fluid flows on unstructured meshes. Orthogonality of basis functions is essential for enabling robust and efficient fully-implicit Newton-Krylov based time integration. The method is designed for generic partial differential equations, including transient, hyperbolic, parabolic or elliptic operators, which are attributed to many multiphysics problems. We demonstrate the method’s capabilities for solving compressible fluid-solid systems (in the low Mach number limit), with phase change (melting/solidification), as motivated by applications in Additive Manufacturing (AM). We focus on the method’s accuracy (in both space and time), as wellmore » as robustness and solvability of the system of linear equations involved in the linearization steps of Newton-based methods. The performance of the developed method is investigated for highly-stiff problems with melting/solidification, emphasizing the advantages from tight coupling of mass, momentum and energy conservation equations, as well as orthogonality of basis functions, which leads to better conditioning of the underlying (approximate) Jacobian matrices, and rapid convergence of the Krylov-based linear solver.« less

  20. On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations

    NASA Astrophysics Data System (ADS)

    Bassi, F.; Botti, L.; Colombo, A.; Di Pietro, D. A.; Tesini, P.

    2012-01-01

    In this work we show that the flexibility of the discontinuous Galerkin (dG) discretization can be fruitfully exploited to implement numerical solution strategies based on the use of elements with very general shapes. Thanks to the freedom in defining the mesh topology, we propose a new h-adaptive technique based on agglomeration coarsening of a fine mesh. The possibility to enhance the error distribution over the computational domain is investigated on a Poisson problem with the goal of obtaining a mesh independent discretization. The main building block of our dG method consists of defining discrete polynomial spaces directly on physical frame elements. For this purpose we orthonormalize with respect to the L2-product a set of monomials relocated in a specific element frame and we introduce an easy way to reduce the cost related to numerical integration on agglomerated meshes. To complete the dG formulation for second order problems, two extensions of the BR2 scheme to arbitrary polyhedral grids, including an estimate of the stabilization parameter ensuring the coercivity property, are here proposed.

  1. A hybridizable discontinuous Galerkin method for modeling fluid–structure interaction

    SciTech Connect

    Sheldon, Jason P.; Miller, Scott T.; Pitt, Jonathan S.

    2016-08-31

    This study presents a novel application of the hybridizable discontinuous Galerkin (HDG) finite element method to the multi-physics simulation of coupled fluid–structure interaction (FSI) problems. Recent applications of the HDG method have primarily been for single-physics problems including both solids and fluids, which are necessary building blocks for FSI modeling. Utilizing these established models, HDG formulations for linear elastostatics, a nonlinear elastodynamic model, and arbitrary Lagrangian–Eulerian Navier–Stokes are derived. The elasticity formulations are written in a Lagrangian reference frame, with the nonlinear formulation restricted to hyperelastic materials. With these individual solid and fluid formulations, the remaining challenge in FSI modeling is coupling together their disparate mathematics on the fluid–solid interface. This coupling is presented, along with the resultant HDG FSI formulation. Verification of the component models, through the method of manufactured solutions, is performed and each model is shown to converge at the expected rate. The individual components, along with the complete FSI model, are then compared to the benchmark problems proposed by Turek and Hron [1]. The solutions from the HDG formulation presented in this work trend towards the benchmark as the spatial polynomial order and the temporal order of integration are increased.

  2. A Spalart-Allmaras turbulence model implementation in a discontinuous Galerkin solver for incompressible flows

    NASA Astrophysics Data System (ADS)

    Crivellini, Andrea; D'Alessandro, Valerio; Bassi, Francesco

    2013-05-01

    In this paper the artificial compressibility flux Discontinuous Galerkin (DG) method for the solution of the incompressible Navier-Stokes equations has been extended to deal with the Reynolds-Averaged Navier-Stokes (RANS) equations coupled with the Spalart-Allmaras (SA) turbulence model. DG implementations of the RANS and SA equations for compressible flows have already been reported in the literature, including the description of limiting or stabilization techniques adopted in order to prevent the turbulent viscosity ν˜ from becoming negative. In this paper we introduce an SA model implementation that deals with negative ν˜ values by modifying the source and diffusion terms in the SA model equation only when the working variable or one of the model closure functions become negative. This results in an efficient high-order implementation where either stabilization terms or even additional equations are avoided. We remark that the proposed implementation is not DG specific and it is well suited for any numerical discretization of the RANS-SA governing equations. The reliability, robustness and accuracy of the proposed implementation have been assessed by computing several high Reynolds number turbulent test cases: the flow over a flat plate (Re=107), the flow past a backward-facing step (Re=37400) and the flow around a NACA 0012 airfoil at different angles of attack (α=0°, 10°, 15°) and Reynolds numbers (Re=2.88×106,6×106).

  3. Stereographic projection for three-dimensional global discontinuous Galerkin atmospheric modeling

    NASA Astrophysics Data System (ADS)

    Blaise, Sébastien; Lambrechts, Jonathan; Deleersnijder, Eric

    2015-09-01

    A method to solve the three-dimensional compressible Navier-Stokes equations on the sphere is suggested, based on a stereographic projection with a high-order mapping of the elements from the stereographic space to the sphere. The projection is slightly modified, in order to take into account the domain thickness without introducing any approximation about the aspect ratio (deep-atmosphere). In a discontinuous Galerkin framework, the elements alongside the equator are exactly represented using a nonpolynomial geometry, in order to avoid the numerical issues associated with the seam connecting the two hemispheres. This is an crucial point, necessary to avoid mass loss and spurious deviations of the velocity. The resulting model is validated on idealized three-dimensional atmospheric test cases on the sphere, demonstrating the good convergence properties of the scheme, its mass conservation, and its satisfactory behavior in terms of accuracy and low numerical dissipation. A simulation is performed on a variable resolution unstructured grid, producing accurate results despite a substantial reduction of the number of elements.

  4. Energy-conserving discontinuous Galerkin methods for the Vlasov-Maxwell system

    NASA Astrophysics Data System (ADS)

    Cheng, Yingda; Christlieb, Andrew J.; Zhong, Xinghui

    2014-12-01

    In this paper, we generalize the idea in our previous work for the Vlasov-Ampère (VA) system (Y. Cheng, A.J. Christlieb, and X. Zhong (2014) [10]) and develop energy-conserving discontinuous Galerkin (DG) methods for the Vlasov-Maxwell (VM) system. The VM system is a fundamental model in the simulation of collisionless magnetized plasmas. Compared to Y. Cheng, A.J. Christlieb, and X. Zhong (2014) [10], additional care needs to be taken for both the temporal and spatial discretizations to achieve similar type of conservation when the magnetic field is no longer negligible. Our proposed schemes conserve the total particle number and the total energy at the same time, therefore can obtain accurate and physically relevant numerical solutions. The main components of our methods include second order and above, explicit or implicit energy-conserving temporal discretizations, and DG methods for Vlasov and Maxwell's equations with carefully chosen numerical fluxes. Benchmark numerical tests such as the streaming Weibel instability are provided to validate the accuracy and conservation of the schemes.

  5. Discontinuous Galerkin methods for plasma physics in the scrape-off layer of tokamaks

    SciTech Connect

    Michoski, C.; Meyerson, D.; Isaac, T.; Waelbroeck, F.

    2014-10-01

    A new parallel discontinuous Galerkin solver, called ArcOn, is developed to describe the intermittent turbulent transport of filamentary blobs in the scrape-off layer (SOL) of fusion plasma. The model is comprised of an elliptic subsystem coupled to two convection-dominated reaction–diffusion–convection equations. Upwinding is used for a class of numerical fluxes developed to accommodate cross product driven convection, and the elliptic solver uses SIPG, NIPG, IIPG, Brezzi, and Bassi–Rebay fluxes to formulate the stiffness matrix. A novel entropy sensor is developed for this system, designed for a space–time varying artificial diffusion/viscosity regularization algorithm. Some numerical experiments are performed to show convergence order on manufactured solutions, regularization of blob/streamer dynamics in the SOL given unstable parameterizations, long-time stability of modon (or dipole drift vortex) solutions arising in simulations of drift-wave turbulence, and finally the formation of edge mode turbulence in the scrape-off layer under turbulent saturation conditions.

  6. The hybridized Discontinuous Galerkin method for Implicit Large-Eddy Simulation of transitional turbulent flows

    NASA Astrophysics Data System (ADS)

    Fernandez, P.; Nguyen, N. C.; Peraire, J.

    2017-05-01

    We present a high-order Implicit Large-Eddy Simulation (ILES) approach for transitional aerodynamic flows. The approach encompasses a hybridized Discontinuous Galerkin (DG) method for the discretization of the Navier-Stokes (NS) equations, and a parallel preconditioned Newton-GMRES solver for the resulting nonlinear system of equations. The combination of hybridized DG methods with an efficient solution procedure leads to a high-order accurate NS solver that is competitive to alternative approaches, such as finite volume and finite difference codes, in terms of computational cost. The proposed approach is applied to transitional flows over the NACA 65-(18)10 compressor cascade and the Eppler 387 wing at Reynolds numbers up to 460,000. Grid convergence studies are presented and the required resolution to capture transition at different Reynolds numbers is investigated. Numerical results show rapid convergence and excellent agreement with experimental data. In short, this work aims to demonstrate the potential of high-order ILES for simulating transitional aerodynamic flows. This is illustrated through numerical results and supported by theoretical considerations.

  7. Dynamic mesh adaptation for front evolution using discontinuous Galerkin based weighted condition number relaxation

    DOE PAGES

    Greene, Patrick T.; Schofield, Samuel P.; Nourgaliev, Robert

    2017-01-27

    A new mesh smoothing method designed to cluster cells near a dynamically evolving interface is presented. The method is based on weighted condition number mesh relaxation with the weight function computed from a level set representation of the interface. The weight function is expressed as a Taylor series based discontinuous Galerkin projection, which makes the computation of the derivatives of the weight function needed during the condition number optimization process a trivial matter. For cases when a level set is not available, a fast method for generating a low-order level set from discrete cell-centered fields, such as a volume fractionmore » or index function, is provided. Results show that the low-order level set works equally well as the actual level set for mesh smoothing. Meshes generated for a number of interface geometries are presented, including cases with multiple level sets. Lastly, dynamic cases with moving interfaces show the new method is capable of maintaining a desired resolution near the interface with an acceptable number of relaxation iterations per time step, which demonstrates the method's potential to be used as a mesh relaxer for arbitrary Lagrangian Eulerian (ALE) methods.« less

  8. Implicit high-order discontinuous Galerkin method with HWENO type limiters for steady viscous flow simulations

    NASA Astrophysics Data System (ADS)

    Jiang, Zhen-Hua; Yan, Chao; Yu, Jian

    2013-08-01

    Two types of implicit algorithms have been improved for high order discontinuous Galerkin (DG) method to solve compressible Navier-Stokes (NS) equations on triangular grids. A block lower-upper symmetric Gauss-Seidel (BLU-SGS) approach is implemented as a nonlinear iterative scheme. And a modified LU-SGS (LLU-SGS) approach is suggested to reduce the memory requirements while retain the good convergence performance of the original LU-SGS approach. Both implicit schemes have the significant advantage that only the diagonal block matrix is stored. The resulting implicit high-order DG methods are applied, in combination with Hermite weighted essentially non-oscillatory (HWENO) limiters, to solve viscous flow problems. Numerical results demonstrate that the present implicit methods are able to achieve significant efficiency improvements over explicit counterparts and for viscous flows with shocks, and the HWENO limiters can be used to achieve the desired essentially non-oscillatory shock transition and the designed high-order accuracy simultaneously.

  9. CosmosDG: An hp-adaptive Discontinuous Galerkin Code for Hyper-resolved Relativistic MHD

    NASA Astrophysics Data System (ADS)

    Anninos, Peter; Bryant, Colton; Fragile, P. Chris; Holgado, A. Miguel; Lau, Cheuk; Nemergut, Daniel

    2017-08-01

    We have extended Cosmos++, a multidimensional unstructured adaptive mesh code for solving the covariant Newtonian and general relativistic radiation magnetohydrodynamic (MHD) equations, to accommodate both discrete finite volume and arbitrarily high-order finite element structures. The new finite element implementation, called CosmosDG, is based on a discontinuous Galerkin (DG) formulation, using both entropy-based artificial viscosity and slope limiting procedures for the regularization of shocks. High-order multistage forward Euler and strong-stability preserving Runge-Kutta time integration options complement high-order spatial discretization. We have also added flexibility in the code infrastructure allowing for both adaptive mesh and adaptive basis order refinement to be performed separately or simultaneously in a local (cell-by-cell) manner. We discuss in this report the DG formulation and present tests demonstrating the robustness, accuracy, and convergence of our numerical methods applied to special and general relativistic MHD, although we note that an equivalent capability currently also exists in CosmosDG for Newtonian systems.

  10. A hybridizable discontinuous Galerkin method for modeling fluid-structure interaction

    NASA Astrophysics Data System (ADS)

    Sheldon, Jason P.; Miller, Scott T.; Pitt, Jonathan S.

    2016-12-01

    This work presents a novel application of the hybridizable discontinuous Galerkin (HDG) finite element method to the multi-physics simulation of coupled fluid-structure interaction (FSI) problems. Recent applications of the HDG method have primarily been for single-physics problems including both solids and fluids, which are necessary building blocks for FSI modeling. Utilizing these established models, HDG formulations for linear elastostatics, a nonlinear elastodynamic model, and arbitrary Lagrangian-Eulerian Navier-Stokes are derived. The elasticity formulations are written in a Lagrangian reference frame, with the nonlinear formulation restricted to hyperelastic materials. With these individual solid and fluid formulations, the remaining challenge in FSI modeling is coupling together their disparate mathematics on the fluid-solid interface. This coupling is presented, along with the resultant HDG FSI formulation. Verification of the component models, through the method of manufactured solutions, is performed and each model is shown to converge at the expected rate. The individual components, along with the complete FSI model, are then compared to the benchmark problems proposed by Turek and Hron [1]. The solutions from the HDG formulation presented in this work trend towards the benchmark as the spatial polynomial order and the temporal order of integration are increased.

  11. Energy Conserving Forms of Discontinuous Galerkin Algorithms, and Sparse Grid Methods

    NASA Astrophysics Data System (ADS)

    Hakim, Ammar; Hammett, Greg; Shi, Eric

    2016-10-01

    A hybrid discontinuous/continuous Galerkin scheme for gyrokinetic equations is presented. Discretizing the Poisson bracket form of the equations, along with a careful choice of basis functions allows conserving the total (particle+field) energy exactly, even with upwinding to reduce artificial oscillations. Straightforward use of tensor basis functions can get expensive in higher dimensions and high polynomial order. Savings might be possible by using basis sets that have fewer monomials and combining these with a version of sparse grid quadrature methods. For example, a tensor product of piecewise parabolic basis functions in 5D involves 243 basis functions per cell, but this drops to 21 basis functions if only second order monomials are needed. Enforcing continuity needed for energy conservation in configuration space might reduce the savings, but would still be a gain over Gaussian quadrature. Our version of sparse grid methods could use non-nested quadrature points as well as well as anisotropic basis. Energy conservation with use of reduced basis sets is discussed. Supported by the Max-Planck/Princeton Center for Plasma Physics, the SciDAC Center for the Study of Plasma Microturbulence, and DOE Contract DE-AC02-09CH11466.

  12. Development and evaluation of a hydrostatic dynamical core using the spectral element/discontinuous Galerkin methods

    NASA Astrophysics Data System (ADS)

    Choi, S.-J.; Giraldo, F. X.

    2014-06-01

    In this paper, we present a dynamical core for the atmospheric primitive hydrostatic equations using a unified formulation of spectral element (SE) and discontinuous Galerkin (DG) methods in the horizontal direction with a finite difference (FD) method in the radial direction. The CG and DG horizontal discretization employs high-order nodal basis functions associated with Lagrange polynomials based on Gauss-Lobatto-Legendre (GLL) quadrature points, which define the common machinery. The atmospheric primitive hydrostatic equations are solved on the cubed-sphere grid using the flux form governing equations in a three-dimensional (3-D) Cartesian space. By using Cartesian space, we can avoid the pole singularity problem due to spherical coordinates and this also allows us to use any quadrilateral-based grid naturally. In order to consider an easy way for coupling the dynamics with existing physics packages, we use a FD in the radial direction. The models are verified by conducting conventional benchmark test cases: the Rossby-Haurwitz wavenumber 4, Jablonowski-Williamson tests for balanced initial state and baroclinic instability, and Held-Suarez tests. The results from those tests demonstrate that the present dynamical core can produce numerical solutions of good quality comparable to other models.

  13. Bound-preserving discontinuous Galerkin methods for conservative phase space advection in curvilinear coordinates

    NASA Astrophysics Data System (ADS)

    Endeve, Eirik; Hauck, Cory D.; Xing, Yulong; Mezzacappa, Anthony

    2015-04-01

    We extend the positivity-preserving method of Zhang and Shu [49] to simulate the advection of neutral particles in phase space using curvilinear coordinates. The ability to utilize these coordinates is important for non-equilibrium transport problems in general relativity and also in science and engineering applications with specific geometries. The method achieves high-order accuracy using Discontinuous Galerkin (DG) discretization of phase space and strong stability-preserving, Runge-Kutta (SSP-RK) time integration. Special care is taken to ensure that the method preserves strict bounds for the phase space distribution function f; i.e., f ∈ [ 0 , 1 ]. The combination of suitable CFL conditions and the use of the high-order limiter proposed in [49] is sufficient to ensure positivity of the distribution function. However, to ensure that the distribution function satisfies the upper bound, the discretization must, in addition, preserve the divergence-free property of the phase space flow. Proofs that highlight the necessary conditions are presented for general curvilinear coordinates, and the details of these conditions are worked out for some commonly used coordinate systems (i.e., spherical polar spatial coordinates in spherical symmetry and cylindrical spatial coordinates in axial symmetry, both with spherical momentum coordinates). Results from numerical experiments - including one example in spherical symmetry adopting the Schwarzschild metric - demonstrate that the method achieves high-order accuracy and that the distribution function satisfies the maximum principle.

  14. Verification of higher-order discontinuous Galerkin method for hexahedral elements

    NASA Astrophysics Data System (ADS)

    Özdemir, Hüseyin; Hagmeijer, Rob; Hoeijmakers, Hendrik Willem Marie

    2005-09-01

    A high-order implementation of the Discontinuous Galerkin ( DG) method is presented for solving the three-dimensional Linearized Euler Equations on an unstructured hexahedral grid. The method is based on a quadrature free implementation and the high-order accuracy is obtained by employing higher-degree polynomials as basis functions. The present implementation is up to fourth-order accurate in space. For the time discretization a four-stage Runge-Kutta scheme is used which is fourth-order accurate. Non-reflecting boundary conditions are implemented at the boundaries of the computational domain.The method is verified for the case of the convection of a 1D compact acoustic disturbance. The numerical results show that the rate of convergence of the method is of order p+1 in the mesh size, with p the order of the basis functions. This observation is in agreement with analysis presented in the literature. To cite this article: H. Özdemir et al., C. R. Mecanique 333 (2005).

  15. Assessment of a Hybrid Continuous/Discontinuous Galerkin Finite Element Code for Geothermal Reservoir Simulations

    NASA Astrophysics Data System (ADS)

    Xia, Yidong; Podgorney, Robert; Huang, Hai

    2017-03-01

    FALCON (Fracturing And Liquid CONvection) is a hybrid continuous/discontinuous Galerkin finite element geothermal reservoir simulation code based on the MOOSE (Multiphysics Object-Oriented Simulation Environment) framework being developed and used for multiphysics applications. In the present work, a suite of verification and validation (V&V) test problems for FALCON was defined to meet the design requirements, and solved to the interests of enhanced geothermal system modeling and simulation. The intent for this test problem suite is to provide baseline comparison data that demonstrates the performance of FALCON solution methods. The test problems vary in complexity from a single mechanical or thermal process, to coupled thermo-hydro-mechanical processes in geological porous medium. Numerical results obtained by FALCON agreed well with either the available analytical solutions or experimental data, indicating the verified and validated implementation of these capabilities in FALCON. Whenever possible, some form of solution verification has been attempted to identify sensitivities in the solution methods, and suggest best practices when using the FALCON code.

  16. Dynamic Mesh Adaptation for Front Evolution Using Discontinuous Galerkin Based Weighted Condition Number Mesh Relaxation

    SciTech Connect

    Greene, Patrick T.; Schofield, Samuel P.; Nourgaliev, Robert

    2016-06-21

    A new mesh smoothing method designed to cluster mesh cells near a dynamically evolving interface is presented. The method is based on weighted condition number mesh relaxation with the weight function being computed from a level set representation of the interface. The weight function is expressed as a Taylor series based discontinuous Galerkin projection, which makes the computation of the derivatives of the weight function needed during the condition number optimization process a trivial matter. For cases when a level set is not available, a fast method for generating a low-order level set from discrete cell-centered elds, such as a volume fraction or index function, is provided. Results show that the low-order level set works equally well for the weight function as the actual level set. Meshes generated for a number of interface geometries are presented, including cases with multiple level sets. Dynamic cases for moving interfaces are presented to demonstrate the method's potential usefulness to arbitrary Lagrangian Eulerian (ALE) methods.

  17. Extending the dynamic slip-wall model to a compressible discontinuous-Galerkin method

    NASA Astrophysics Data System (ADS)

    Carton de Wiart, Corentin; Murman, Scott

    2016-11-01

    Standard equilibrium wall models suffer from both a strong dependence upon mesh resolution and the equilibrium turbulence assumption. Non-equilibrium wall models similarly have limitations for complex geometry due to the need for an auxiliary semi-structured mesh solver, and coupling between the LES and wall-model regions. Bose and Moin's dynamic slip-wall model offers a new modeling paradigm that does not rely upon assumptions about the local flow physics and uses a dynamic procedure so that the results are independent of resolution. Despite this, the model has not gained significant traction and few independent implementations have been tested. The current work implements the dynamic slip-wall model in an entropy-stable Discontinuous-Galerkin spectral-element solver with a dynamic variational multiscale sub-grid model. This involves both extending the model to a compressible formulation and to a different numerical method. The compressible model is outlined and tested on both attached and separated flows of aerodynamic interest.

  18. Stage-parallel fully implicit Runge-Kutta solvers for discontinuous Galerkin fluid simulations

    NASA Astrophysics Data System (ADS)

    Pazner, Will; Persson, Per-Olof

    2017-04-01

    In this paper, we develop new techniques for solving the large, coupled linear systems that arise from fully implicit Runge-Kutta methods. This method makes use of the iterative preconditioned GMRES algorithm for solving the linear systems, which has seen success for fluid flow problems and discontinuous Galerkin discretizations. By transforming the resulting linear system of equations, one can obtain a method which is much less computationally expensive than the untransformed formulation, and which compares competitively with other time-integration schemes, such as diagonally implicit Runge-Kutta (DIRK) methods. We develop and test several ILU-based preconditioners effective for these large systems. We additionally employ a parallel-in-time strategy to compute the Runge-Kutta stages simultaneously. Numerical experiments are performed on the Navier-Stokes equations using Euler vortex and 2D and 3D NACA airfoil test cases in serial and in parallel settings. The fully implicit Radau IIA Runge-Kutta methods compare favorably with equal-order DIRK methods in terms of accuracy, number of GMRES iterations, number of matrix-vector multiplications, and wall-clock time, for a wide range of time steps.

  19. High-order nodal discontinuous Galerkin particle-in-cell method on unstructured grids

    SciTech Connect

    Jacobs, G.B. . E-mail: gjacobs2@dam.brown.edu; Hesthaven, J.S. . E-mail: Jan.Hesthaven@brown.edu

    2006-05-01

    We present a high-order particle-in-cell (PIC) algorithm for the simulation of kinetic plasmas dynamics. The core of the algorithm utilizes an unstructured grid discontinuous Galerkin Maxwell field solver combining high-order accuracy with geometric flexibility. We introduce algorithms in the Lagrangian framework that preserve the favorable properties of the field solver in the PIC solver. Fast full-order interpolation and effective search algorithms are used for tracking individual particles on the general grid and smooth particle shape functions are introduced to ensure low noise in the charge and current density. A pre-computed levelset distance function is employed to represent the geometry and facilitates complex particle-boundary interaction. To enforce charge conservation we consider two different techniques, one based on projection and one on hyperbolic cleaning. Both are found to work well, although the latter is found be too expensive when used with explicit time integration. Examples of simple plasma phenomena, e.g., plasma waves, instabilities, and Landau damping are shown to agree well with theoretical predictions and/or results found by other computational methods. We also discuss generic well known problems such as numerical Cherenkov radiation and grid heating before presenting a few two-dimensional tests, showing the potential of the current method to handle fully relativistic plasma dynamics in complex geometries.

  20. 3D dynamic rupture with anelastic wave propagation using an hp-adaptive Discontinuous Galerkin method

    NASA Astrophysics Data System (ADS)

    Tago, J.; Cruz-Atienza, V. M.; Etienne, V.; Virieux, J.; Benjemaa, M.; Sanchez-Sesma, F. J.

    2010-12-01

    Simulating any realistic seismic scenario requires incorporating physical basis into the model. Considering both the dynamics of the rupture process and the anelastic attenuation of seismic waves is essential to this purpose and, therefore, we choose to extend the hp-adaptive Discontinuous Galerkin finite-element method to integrate these physical aspects. The 3D elastodynamic equations in an unstructured tetrahedral mesh are solved with a second-order time marching approach in a high-performance computing environment. The first extension incorporates the viscoelastic rheology so that the intrinsic attenuation of the medium is considered in terms of frequency dependent quality factors (Q). On the other hand, the extension related to dynamic rupture is integrated through explicit boundary conditions over the crack surface. For this visco-elastodynamic formulation, we introduce an original discrete scheme that preserves the optimal code performance of the elastodynamic equations. A set of relaxation mechanisms describes the behavior of a generalized Maxwell body. We approximate almost constant Q in a wide frequency range by selecting both suitable relaxation frequencies and anelastic coefficients characterizing these mechanisms. In order to do so, we solve an optimization problem which is critical to minimize the amount of relaxation mechanisms. Two strategies are explored: 1) a least squares method and 2) a genetic algorithm (GA). We found that the improvement provided by the heuristic GA method is negligible. Both optimization strategies yield Q values within the 5% of the target constant Q mechanism. Anelastic functions (i.e. memory variables) are introduced to efficiently evaluate the time convolution terms involved in the constitutive equations and thus to minimize the computational cost. The incorporation of anelastic functions implies new terms with ordinary differential equations in the mathematical formulation. We solve these equations using the same order

  1. Relaxation and Preconditioning for High Order Discontinuous Galerkin Methods with Applications to Aeroacoustics and High Speed Flows

    NASA Technical Reports Server (NTRS)

    Shu, Chi-Wang

    2004-01-01

    This project is about the investigation of the development of the discontinuous Galerkin finite element methods, for general geometry and triangulations, for solving convection dominated problems, with applications to aeroacoustics. Other related issues in high order WENO finite difference and finite volume methods have also been investigated. methods are two classes of high order, high resolution methods suitable for convection dominated simulations with possible discontinuous or sharp gradient solutions. In [18], we first review these two classes of methods, pointing out their similarities and differences in algorithm formulation, theoretical properties, implementation issues, applicability, and relative advantages. We then present some quantitative comparisons of the third order finite volume WENO methods and discontinuous Galerkin methods for a series of test problems to assess their relative merits in accuracy and CPU timing. In [3], we review the development of the Runge-Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge-Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier

  2. An entropy-residual shock detector for solving conservation laws using high-order discontinuous Galerkin methods

    NASA Astrophysics Data System (ADS)

    Lv, Yu; See, Yee Chee; Ihme, Matthias

    2016-10-01

    This manuscript is concerned with the detection of shock discontinuities in the solution of conservation laws for high-order discontinuous Galerkin methods. A shock detector based on the entropy residual is proposed to distinguish smooth and non-smooth parts of the solution. The numerical analysis shows that the proposed entropy residual converges if the true solution is smooth and sufficiently regularized in space and time. To precisely localize discontinuities of different natures, an approach is developed that dynamically sets the threshold on the detection function, such that the detection criterion retains its sensitivity to the characteristics of the local solution. The implementation is conducted in an entropy-bounded discontinuous Galerkin framework, and numerical tests confirm the convergence property of the entropy-residual formulation and the effectiveness of the thresholding procedure. This shock detector is combined with an artificial viscosity scheme for shock stabilization. Comparison with other detectors is performed to demonstrate the excellent performance of the entropy-residual based shock detector for a wide range of problems on regular and triangular grids.

  3. Development and application of discontinuous Galerkin method for the solution of two-dimensional Maxwell equations

    NASA Astrophysics Data System (ADS)

    Wong, See-Cheuk

    We inhabit an environment of electromagnetic (EM) waves. The waves within the EM spectrum---whether light, radio, or microwaves---all obey the same physical laws. A band in the spectrum is designated to the microwave frequencies (30MHz--300GHz), at which radar systems operate. The precise modeling of the scattered EM-ields about a target, as well as the numerical prediction of the radar return is the crux of the computational electromagnetics (CEM) problems. The signature or return from a target observed by radar is commonly provided in the form of radar cross section (RCS). Incidentally, the efforts in the reduction of such return forms the basis of stealth aircraft design. The object of this dissertation is to extend Discontinuous Galerkin (DG) method to solve numerically the Maxwell equations for scatterings from perfect electric conductor (PEC) objects. The governing equations are derived by writing the Maxwell equations in conservation-law form for scattered field quantities. The transverse magnetic (TM) and the transverse electric (TE) waveforms of the Maxwell equations are considered. A finite-element scheme is developed with proper representations for the electric and magnetic fluxes at a cell interface to account for variations in properties, in both space and time. A characteristic sub-path integration process, known as the "Riemann solver" is involved. An explicit Runge-Kutta Discontinuous Galerkin (RKDG) upwind scheme, which is fourth-order accurate in time and second-order in space, is employed to solve the TM and TE equations. Arbitrary cross-sectioned bodies are modeled, around which computational grids using random triangulation are generated. The RKDG method, in its development stage, was constructed and studied for solving hyperbolic conservation equations numerically. It was later extended to multidimensional nonlinear systems of conservation laws. The algorithms are described, including the formulations and treatments to the numerical fluxes

  4. Scalable direct Vlasov solver with discontinuous Galerkin method on unstructured mesh.

    SciTech Connect

    Xu, J.; Ostroumov, P. N.; Mustapha, B.; Nolen, J.

    2010-12-01

    This paper presents the development of parallel direct Vlasov solvers with discontinuous Galerkin (DG) method for beam and plasma simulations in four dimensions. Both physical and velocity spaces are in two dimesions (2P2V) with unstructured mesh. Contrary to the standard particle-in-cell (PIC) approach for kinetic space plasma simulations, i.e., solving Vlasov-Maxwell equations, direct method has been used in this paper. There are several benefits to solving a Vlasov equation directly, such as avoiding noise associated with a finite number of particles and the capability to capture fine structure in the plasma. The most challanging part of a direct Vlasov solver comes from higher dimensions, as the computational cost increases as N{sup 2d}, where d is the dimension of the physical space. Recently, due to the fast development of supercomputers, the possibility has become more realistic. Many efforts have been made to solve Vlasov equations in low dimensions before; now more interest has focused on higher dimensions. Different numerical methods have been tried so far, such as the finite difference method, Fourier Spectral method, finite volume method, and spectral element method. This paper is based on our previous efforts to use the DG method. The DG method has been proven to be very successful in solving Maxwell equations, and this paper is our first effort in applying the DG method to Vlasov equations. DG has shown several advantages, such as local mass matrix, strong stability, and easy parallelization. These are particularly suitable for Vlasov equations. Domain decomposition in high dimensions has been used for parallelization; these include a highly scalable parallel two-dimensional Poisson solver. Benchmark results have been shown and simulation results will be reported.

  5. Fast discontinuous Galerkin lattice-Boltzmann simulations on GPUs via maximal kernel fusion

    NASA Astrophysics Data System (ADS)

    Mazzeo, Marco D.

    2013-03-01

    A GPU implementation of the discontinuous Galerkin lattice-Boltzmann method with square spectral elements, and highly optimised for speed and precision of calculations is presented. An extensive analysis of the numerous variants of the fluid solver unveils that best performance is obtained by maximising CUDA kernel fusion and by arranging the resulting kernel tasks so as to trigger memory coherent and scattered loads in a specific manner, albeit at the cost of introducing cross-thread load unbalancing. Surprisingly, any attempt to vanish this, to maximise thread occupancy and to adopt conventional work tiling or distinct custom kernels highly tuned via ad hoc data and computation layouts invariably deteriorate performance. As such, this work sheds light into the possibility to hide fetch latencies of workloads involving heterogeneous loads in a way that is more effective than what is achieved with frequently suggested techniques. When simulating the lid-driven cavity on a NVIDIA GeForce GTX 480 via a 5-stage 4th-order Runge-Kutta (RK) scheme, the first four digits of the obtained centreline velocity values, or more, converge to those of the state-of-the-art literature data at a simulation speed of 7.0G primitive variable updates per second during the collision stage and 4.4G ones during each RK step of the advection by employing double-precision arithmetic (DPA) and a computational grid of 642 4×4-point elements only. The new programming engine leads to about 2× performance w.r.t. the best programming guidelines in the field. The new fluid solver on the above GPU is also 20-30 times faster than a highly optimised version running on a single core of a Intel Xeon X5650 2.66 GHz.

  6. A GPU Accelerated Discontinuous Galerkin Conservative Level Set Method for Simulating Atomization

    NASA Astrophysics Data System (ADS)

    Jibben, Zechariah J.

    This dissertation describes a process for interface capturing via an arbitrary-order, nearly quadrature free, discontinuous Galerkin (DG) scheme for the conservative level set method (Olsson et al., 2005, 2008). The DG numerical method is utilized to solve both advection and reinitialization, and executed on a refined level set grid (Herrmann, 2008) for effective use of processing power. Computation is executed in parallel utilizing both CPU and GPU architectures to make the method feasible at high order. Finally, a sparse data structure is implemented to take full advantage of parallelism on the GPU, where performance relies on well-managed memory operations. With solution variables projected into a kth order polynomial basis, a k + 1 order convergence rate is found for both advection and reinitialization tests using the method of manufactured solutions. Other standard test cases, such as Zalesak's disk and deformation of columns and spheres in periodic vortices are also performed, showing several orders of magnitude improvement over traditional WENO level set methods. These tests also show the impact of reinitialization, which often increases shape and volume errors as a result of level set scalar trapping by normal vectors calculated from the local level set field. Accelerating advection via GPU hardware is found to provide a 30x speedup factor comparing a 2.0GHz Intel Xeon E5-2620 CPU in serial vs. a Nvidia Tesla K20 GPU, with speedup factors increasing with polynomial degree until shared memory is filled. A similar algorithm is implemented for reinitialization, which relies on heavier use of shared and global memory and as a result fills them more quickly and produces smaller speedups of 18x.

  7. Hybridizable discontinuous Galerkin projection methods for Navier-Stokes and Boussinesq equations

    NASA Astrophysics Data System (ADS)

    Ueckermann, M. P.; Lermusiaux, P. F. J.

    2016-02-01

    Schemes for the incompressible Navier-Stokes and Boussinesq equations are formulated and derived combining the novel Hybridizable Discontinuous Galerkin (HDG) method, a projection method, and Implicit-Explicit Runge-Kutta (IMEX-RK) time-integration schemes. We employ an incremental pressure correction and develop the corresponding HDG finite element discretization including consistent edge-space fluxes for the velocity predictor and pressure correction. We then derive the proper forms of the element-local and HDG edge-space final corrections for both velocity and pressure, including the HDG rotational correction. We also find and explain a consistency relation between the HDG stability parameters of the pressure correction and velocity predictor. We discuss and illustrate the effects of the time-splitting error. We then detail how to incorporate the HDG projection method time-split within standard IMEX-RK time-stepping schemes. Our high-order HDG projection schemes are implemented for arbitrary, mixed-element unstructured grids, with both straight-sided and curved meshes. In particular, we provide a quadrature-free integration method for a nodal basis that is consistent with the HDG method. To prevent numerical oscillations, we develop a selective nodal limiting approach. Its applications show that it can stabilize high-order schemes while retaining high-order accuracy in regions where the solution is sufficiently smooth. We perform spatial and temporal convergence studies to evaluate the properties of our integration and selective limiting schemes and to verify that our solvers are properly formulated and implemented. To complete these studies and to illustrate a range of properties for our new schemes, we employ an unsteady tracer advection benchmark, a manufactured solution for the steady diffusion and Stokes equations, and a standard lock-exchange Boussinesq problem.

  8. Discontinuous Galerkin methodology for Large-Eddy Simulations of wind turbine airfoils

    NASA Astrophysics Data System (ADS)

    Frére, A.; Sørensen, N. N.; Hillewaert, K.; Winckelmans, G.

    2016-09-01

    This paper aims at evaluating the potential of the Discontinuous Galerkin (DG) methodology for Large-Eddy Simulation (LES) of wind turbine airfoils. The DG method has shown high accuracy, excellent scalability and capacity to handle unstructured meshes. It is however not used in the wind energy sector yet. The present study aims at evaluating this methodology on an application which is relevant for that sector and focuses on blade section aerodynamics characterization. To be pertinent for large wind turbines, the simulations would need to be at low Mach numbers (M ≤ 0.3) where compressible approaches are often limited and at large Reynolds numbers (Re ≥ 106) where wall-resolved LES is still unaffordable. At these high Re, a wall-modeled LES (WMLES) approach is thus required. In order to first validate the LES methodology, before the WMLES approach, this study presents airfoil flow simulations at low and high Reynolds numbers and compares the results to state-of-the-art models used in industry, namely the panel method (XFOIL with boundary layer modeling) and Reynolds Averaged Navier-Stokes (RANS). At low Reynolds number (Re = 6 x 104), involving laminar boundary layer separation and transition in the detached shear layer, the Eppler 387 airfoil is studied at two angles of attack. The LES results agree slightly better with the experimental chordwise pressure distribution than both XFOIL and RANS results. At high Reynolds number (Re = 1.64 x 106), the NACA4412 airfoil is studied close to stall condition. In this case, although the wall model approach used for the WMLES is very basic and not supposed to handle separation nor adverse pressure gradients, all three methods provide equivalent accuracy on averaged quantities. The present work is hence considered as a strong step forward in the use of LES at high Reynolds numbers.

  9. Preconditioning methods for discontinuous Galerkin solutions of the Navier-Stokes equations

    NASA Astrophysics Data System (ADS)

    Diosady, Laslo T.; Darmofal, David L.

    2009-06-01

    A Newton-Krylov method is developed for the solution of the steady compressible Navier-Stokes equations using a discontinuous Galerkin (DG) discretization on unstructured meshes. Steady-state solutions are obtained using a Newton-Krylov approach where the linear system at each iteration is solved using a restarted GMRES algorithm. Several different preconditioners are examined to achieve fast convergence of the GMRES algorithm. An element Line-Jacobi preconditioner is presented which solves a block-tridiagonal system along lines of maximum coupling in the flow. An incomplete block-LU factorization (Block-ILU(0)) is also presented as a preconditioner, where the factorization is performed using a reordering of elements based upon the lines of maximum coupling. This reordering is shown to be superior to standard reordering techniques (Nested Dissection, One-way Dissection, Quotient Minimum Degree, Reverse Cuthill-Mckee) especially for viscous test cases. The Block-ILU(0) factorization is performed in-place and an algorithm is presented for the application of the linearization which reduces both the memory and CPU time over the traditional dual matrix storage format. Additionally, a linear p-multigrid preconditioner is also considered, where Block-Jacobi, Line-Jacobi and Block-ILU(0) are used as smoothers. The linear multigrid preconditioner is shown to significantly improve convergence in term of number of iterations and CPU time compared to a single-level Block-Jacobi or Line-Jacobi preconditioner. Similarly the linear multigrid preconditioner with Block-ILU smoothing is shown to reduce the number of linear iterations to achieve convergence over a single-level Block-ILU(0) preconditioner, though no appreciable improvement in CPU time is shown.

  10. A Parallel Implicit Reconstructed Discontinuous Galerkin Method for Compressible Flows on Hybrid Grids

    NASA Astrophysics Data System (ADS)

    Xia, Yidong

    The objective this work is to develop a parallel, implicit reconstructed discontinuous Galerkin (RDG) method using Taylor basis for the solution of the compressible Navier-Stokes equations on 3D hybrid grids. This third-order accurate RDG method is based on a hierarchical weighed essentially non- oscillatory reconstruction scheme, termed as HWENO(P1P 2) to indicate that a quadratic polynomial solution is obtained from the underlying linear polynomial DG solution via a hierarchical WENO reconstruction. The HWENO(P1P2) is designed not only to enhance the accuracy of the underlying DG(P1) method but also to ensure non-linear stability of the RDG method. In this reconstruction scheme, a quadratic polynomial (P2) solution is first reconstructed using a least-squares approach from the underlying linear (P1) discontinuous Galerkin solution. The final quadratic solution is then obtained using a Hermite WENO reconstruction, which is necessary to ensure the linear stability of the RDG method on 3D unstructured grids. The first derivatives of the quadratic polynomial solution are then reconstructed using a WENO reconstruction in order to eliminate spurious oscillations in the vicinity of strong discontinuities, thus ensuring the non-linear stability of the RDG method. The parallelization in the RDG method is based on a message passing interface (MPI) programming paradigm, where the METIS library is used for the partitioning of a mesh into subdomain meshes of approximately the same size. Both multi-stage explicit Runge-Kutta and simple implicit backward Euler methods are implemented for time advancement in the RDG method. In the implicit method, three approaches: analytical differentiation, divided differencing (DD), and automatic differentiation (AD) are developed and implemented to obtain the resulting flux Jacobian matrices. The automatic differentiation is a set of techniques based on the mechanical application of the chain rule to obtain derivatives of a function given as

  11. Chebyshev polynomial filtered subspace iteration in the discontinuous Galerkin method for large-scale electronic structure calculations.

    PubMed

    Banerjee, Amartya S; Lin, Lin; Hu, Wei; Yang, Chao; Pask, John E

    2016-10-21

    The Discontinuous Galerkin (DG) electronic structure method employs an adaptive local basis (ALB) set to solve the Kohn-Sham equations of density functional theory in a discontinuous Galerkin framework. The adaptive local basis is generated on-the-fly to capture the local material physics and can systematically attain chemical accuracy with only a few tens of degrees of freedom per atom. A central issue for large-scale calculations, however, is the computation of the electron density (and subsequently, ground state properties) from the discretized Hamiltonian in an efficient and scalable manner. We show in this work how Chebyshev polynomial filtered subspace iteration (CheFSI) can be used to address this issue and push the envelope in large-scale materials simulations in a discontinuous Galerkin framework. We describe how the subspace filtering steps can be performed in an efficient and scalable manner using a two-dimensional parallelization scheme, thanks to the orthogonality of the DG basis set and block-sparse structure of the DG Hamiltonian matrix. The on-the-fly nature of the ALB functions requires additional care in carrying out the subspace iterations. We demonstrate the parallel scalability of the DG-CheFSI approach in calculations of large-scale two-dimensional graphene sheets and bulk three-dimensional lithium-ion electrolyte systems. Employing 55 296 computational cores, the time per self-consistent field iteration for a sample of the bulk 3D electrolyte containing 8586 atoms is 90 s, and the time for a graphene sheet containing 11 520 atoms is 75 s.

  12. Chebyshev polynomial filtered subspace iteration in the discontinuous Galerkin method for large-scale electronic structure calculations

    NASA Astrophysics Data System (ADS)

    Banerjee, Amartya S.; Lin, Lin; Hu, Wei; Yang, Chao; Pask, John E.

    2016-10-01

    The Discontinuous Galerkin (DG) electronic structure method employs an adaptive local basis (ALB) set to solve the Kohn-Sham equations of density functional theory in a discontinuous Galerkin framework. The adaptive local basis is generated on-the-fly to capture the local material physics and can systematically attain chemical accuracy with only a few tens of degrees of freedom per atom. A central issue for large-scale calculations, however, is the computation of the electron density (and subsequently, ground state properties) from the discretized Hamiltonian in an efficient and scalable manner. We show in this work how Chebyshev polynomial filtered subspace iteration (CheFSI) can be used to address this issue and push the envelope in large-scale materials simulations in a discontinuous Galerkin framework. We describe how the subspace filtering steps can be performed in an efficient and scalable manner using a two-dimensional parallelization scheme, thanks to the orthogonality of the DG basis set and block-sparse structure of the DG Hamiltonian matrix. The on-the-fly nature of the ALB functions requires additional care in carrying out the subspace iterations. We demonstrate the parallel scalability of the DG-CheFSI approach in calculations of large-scale two-dimensional graphene sheets and bulk three-dimensional lithium-ion electrolyte systems. Employing 55 296 computational cores, the time per self-consistent field iteration for a sample of the bulk 3D electrolyte containing 8586 atoms is 90 s, and the time for a graphene sheet containing 11 520 atoms is 75 s.

  13. Mass Conservation of the Unified Continuous and Discontinuous Element-Based Galerkin Methods on Dynamically Adaptive Grids with Application to Atmospheric Simulations

    DTIC Science & Technology

    2015-09-01

    that matter. [12] (with a later follow-up in [13]) addressed the conservation issue of the conforming continuous Galerkin method on the cubed -sphere...atmospheric dynamical core on the cubed -sphere grid, in: J. Phys. Conf. Ser., Vol. 78, IOP Publishing, 2007, p. 012074. [13] M. A. Taylor, A. Fournier, A...Discontinuous Element-Based Galerkin Methods on Dynamically Adaptive Grids with Application to Atmospheric Simulations 5a. CONTRACT NUMBER 5b. GRANT NUMBER

  14. Discontinuous Galerkin method for the spherically reduced Baumgarte-Shapiro-Shibata-Nakamura system with second-order operators

    SciTech Connect

    Field, Scott E.; Hesthaven, Jan S.; Lau, Stephen R.; Mroue, Abdul H.

    2010-11-15

    We present a high-order accurate discontinuous Galerkin method for evolving the spherically reduced Baumgarte-Shapiro-Shibata-Nakamura (BSSN) system expressed in terms of second-order spatial operators. Our multidomain method achieves global spectral accuracy and longtime stability on short computational domains. We discuss in detail both our scheme for the BSSN system and its implementation. After a theoretical and computational verification of the proposed scheme, we conclude with a brief discussion of issues likely to arise when one considers the full BSSN system.

  15. Optimized Schwarz algorithms for solving time-harmonic Maxwell's equations discretized by a hybridizable discontinuous Galerkin method

    NASA Astrophysics Data System (ADS)

    He, Yu-Xuan; Li, Liang; Lanteri, Stéphane; Huang, Ting-Zhu

    2016-03-01

    This work is concerned with the development of numerical methods for the simulation of time-harmonic electromagnetic wave propagation problems. A hybridizable discontinuous Galerkin (HDG) method is adopted for the discretization of the two-dimensional time-harmonic Maxwell's equations on a triangular mesh. A distinguishing feature of the present work is that this discretization method is employed at the subdomain level in the framework of a Schwarz-type domain decomposition algorithm (DDM). We show that HDG method naturally couples with a Schwarz method relying on optimized transmission conditions. The presented numerical results show the effectiveness of the optimized DDM-HDG method.

  16. Discontinuous Galerkin method for the problem of linear elasticity with applications to the fluid-structure interaction

    NASA Astrophysics Data System (ADS)

    Hadrava, M.; Feistauer, M.; Horáček, J.; Kosík, A.

    2013-10-01

    The paper is concerned with the numerical solution of static and dynamic elasticity problems. The purpose of this subject is the computation of the so-called ALE mapping (representing the mesh deformation) in the solution of flow in time-dependent domains and the computation of the time-dependent deformation of an elastic body. These two problems represent important ingredients in the fluid-structure interaction (FSI). They are discretized by the discontinuous Galerkin method (DGM). Here we describe the method and present some test problems. The developed method is applied to the FSI problem treated in [2].

  17. A discontinuous Galerkin method on refined meshes for the two-dimensional time-harmonic Maxwell equations in composite materials

    NASA Astrophysics Data System (ADS)

    Lohrengel, Stephanie; Nicaise, Serge

    2007-09-01

    In this paper, a discontinuous Galerkin method for the two-dimensional time-harmonic Maxwell equations in composite materials is presented. The divergence constraint is taken into account by a regularized variational formulation and the tangential and normal jumps of the discrete solution at the element interfaces are penalized. Due to an appropriate mesh refinement near exterior and interior corners, the singular behaviour of the electromagnetic field is taken into account. Optimal error estimates in a discrete energy norm and in the L2-norm are proved in the case where the exact solution is singular.

  18. An improved predictor/multi-corrector algorithm for a time-discontinuous Galerkin finite element method in structural dynamics

    NASA Astrophysics Data System (ADS)

    Chien, C.-C.; Wu, T.-Y.

    This work presents an improved predictor/multi-corrector algorithm for linear structural dynamics problems, based on the time-discontinuous Galerkin finite element method. The improved algorithm employs the Gauss-Seidel method to calculate iteratively the solutions that exist in the phase of the predictor/multi-corrector of the numerical implementation. Stability analyses of iterative algorithms reveal that such an improved scheme retains the unconditionally stable behavior with greater efficiency than another iterative algorithm. Also, numerical examples are presented, demonstrating that the proposed method is more stable and accurate than several commonly used algorithms in structural dynamic applications.

  19. Quantum hydrodynamics with trajectories: The nonlinear conservation form mixed/discontinuous Galerkin method with applications in chemistry

    SciTech Connect

    Michoski, C. Evans, J.A.; Schmitz, P.G.; Vasseur, A.

    2009-12-10

    We present a solution to the conservation form (Eulerian form) of the quantum hydrodynamic equations which arise in chemical dynamics by implementing a mixed/discontinuous Galerkin (MDG) finite element numerical scheme. We show that this methodology is stable, showing good accuracy and a remarkable scale invariance in its solution space. In addition the MDG method is robust, adapting well to various initial-boundary value problems of particular significance in a range of physical and chemical applications. We further show explicitly how to recover the Lagrangian frame (or pathline) solutions.

  20. Numerical Solution of Poroelastic Wave Equation Using Nodal Discontinuous Galerkin Finite Element Method

    NASA Astrophysics Data System (ADS)

    Shukla, K.; Wang, Y.; Jaiswal, P.

    2014-12-01

    In a porous medium the seismic energy not only propagates through matrix but also through pore-fluids. The differential movement between sediment grains of the matrix and interstitial fluid generates a diffusive wave which is commonly referred to as the slow P-wave. A combined system of equation which includes both elastic and diffusive phases is known as the poroelasticity. Analyzing seismic data through poroelastic modeling results in accurate interpretation of amplitude and separation of wave modes, leading to more accurate estimation of geomehanical properties of rocks. Despite its obvious multi-scale application, from sedimentary reservoir characterization to deep-earth fractured crust, poroelasticity remains under-developed primarily due to the complex nature of its constituent equations. We present a detail formulation of poroleastic wave equations for isotropic media by combining the Biot's and Newtonian mechanics. System of poroelastic wave equation constitutes for eight time dependent hyperbolic PDEs in 2D whereas in case of 3D number goes up to thirteen. Eigen decomposition of Jacobian of these systems confirms the presence of an additional slow-P wave phase with velocity lower than shear wave, posing stability issues on numerical scheme. To circumvent the issue, we derived a numerical scheme using nodal discontinuous Galerkin approach by adopting the triangular meshes in 2D which is extended to tetrahedral for 3D problems. In our nodal DG approach the basis function over a triangular element is interpolated using Legendre-Gauss-Lobatto (LGL) function leading to a more accurate local solutions than in the case of simple DG. We have tested the numerical scheme for poroelastic media in 1D and 2D case, and solution obtained for the systems offers high accuracy in results over other methods such as finite difference , finite volume and pseudo-spectral. The nodal nature of our approach makes it easy to convert the application into a multi-threaded algorithm

  1. 3D Discontinuous Galerkin elastic seismic wave modeling based upon a grid injection method

    NASA Astrophysics Data System (ADS)

    Monteiller, V.

    2015-12-01

    Full waveform inversion (FWI) is a seismic imaging method that estimates thesub-surface physical properties with a spatial resolution of the order of thewavelength. FWI is generally recast as the iterative optimization of anobjective function that measures the distance between modeled and recordeddata. In the framework of local descent methods, FWI requires to perform atleast two seismic modelings per source and per FWI iteration.Due to the resulting computational burden, applications of elastic FWI have been usuallyrestricted to 2D geometries. Despite the continuous growth of high-performancecomputing facilities, application of 3D elastic FWI to real-scale problemsremain computationally too expensive. To perform elastic seismic modeling with a reasonable amount of time, weconsider a reduced computational domain embedded in a larger background modelin which seismic sources are located. Our aim is to compute repeatedly thefull wavefield in the targeted domain after model alteration, once theincident wavefield has been computed once for all in the background model. Toachieve this goal, we use a grid injection method referred to as the Total-Field/Scattered-Field (TF/SF) technique in theelectromagnetic community. We implemented the Total-Field/Scattered-Field approach in theDiscontinuous Galerkin Finite Element method (DG-FEM) that is used to performmodeling in the local domain. We show how to interface the DG-FEM with any modeling engine (analytical solution, finite difference or finite elements methods) that is suitable for the background simulation. One advantage of the Total-Field/Scattered-Field approach is related to thefact that the scattered wavefield instead of the full wavefield enter thePMLs, hence making more efficient the absorption of the outgoing waves at theouter edges of the computational domain. The domain reduction in which theDG-FEM is applied allows us to use modest computational resources opening theway for high-resolution imaging by full

  2. High-Fidelity Lagrangian Coherent Structures Analysis and DNS with Discontinuous-Galerkin Methods

    NASA Astrophysics Data System (ADS)

    Nelson, Daniel Alan Wendell

    High-fidelity numerical tools based on high-order Discontinuous-Galerkin (DG) methods and Lagrangian Coherent Structure (LCS) theory are developed and validated for the study of separated, vortex-dominated flows over complex geometry. The numerical framework couples prediction of separated turbulent flows using DG with time-dependent analysis of the flow through LCS and is intended for the development of separation control strategies for aerodynamic surfaces. The compressible viscous flow over a NACA 65-(1)412 airfoil is solved with a DG based Navier-Stokes solver in two and three dimensions. A method is presented in which high-order polynomial element edges adjacent to curved boundaries are matched to boundaries defined by non-smooth splines. Artificial surface roughness introduced by the piecewise-linear boundary approximation of straight-sided meshes results in the simulation of incorrect physics, including wake instabilities and spurious time-dependent modes. Spectral accuracy in the boundary approximation is not achieved for non-analytic boundary functions, particularly in high curvature regions. An algorithm is developed for the high-order computation of Finite-Time Lyapunov Exponent (FTLE) fields simultaneously and efficiently with two and three dimensional DG-based flow solvers. Fluid tracers are initialized at Gauss-Lobatto quadrature nodes within an element and form the high-order basis for a flow map at later time. Gradients of the flow map and FTLE are evaluated with DG operators. Multiple flow maps are determined from a single particle trace by remapping the flow map to the quadrature nodes on deformed mesh elements. For large integration times, excessive subdomain deformation deteriorates the interpolating conditioning. The conditioning provides information on the fluid deformation and identifies subdomains that contain LCS. An exponential filter smooths the flow map in highly deformed areas. The algorithm is tested on several benchmarks and is shown

  3. A hybridizable discontinuous Galerkin method combined to a Schwarz algorithm for the solution of 3d time-harmonic Maxwell's equation

    NASA Astrophysics Data System (ADS)

    Li, Liang; Lanteri, Stéphane; Perrussel, Ronan

    2014-01-01

    A Schwarz-type domain decomposition method is presented for the solution of the system of 3d time-harmonic Maxwell's equations. We introduce a hybridizable discontinuous Galerkin (HDG) scheme for the discretization of the problem based on a tetrahedrization of the computational domain. The discrete system of the HDG method on each subdomain is solved by an optimized sparse direct (LU factorization) solver. The solution of the interface system in the domain decomposition framework is accelerated by a Krylov subspace method. The formulation and the implementation of the resulting DD-HDG (Domain Decomposed-Hybridizable Discontinuous Galerkin) method are detailed. Numerical results show that the resulting DD-HDG solution strategy has an optimal convergence rate and can save both CPU time and memory cost compared to a classical upwind flux-based DD-DG (Domain Decomposed-Discontinuous Galerkin) approach.

  4. A 3D discontinuous Galerkin finite-element method for teleseismic modelling.

    NASA Astrophysics Data System (ADS)

    monteiller, vadim; Beller, Stephen; Nolet, Guust; Operto, Stephane; Virieux, Jean

    2014-05-01

    The massive development of dense seismic arrays and the rapide increase in computing capacity allow today to consider application of full waveform inversion of teleseismic data for high-resolution lithospheric imaging. We present an hybrid numerical method that allows for the modelling of short period telesismic waves in 3D lithospheric target with the discontinuous Galerkin finite elements method, opennig the possibility to perform waveform inversion of seismograms recorded by dense regional broadband arrays. In order to reduce the computational cost of the forward-problem, we developed a method that relies on multi-core parallel computing and computational-domain reduction. We defined two nested levels for parallelism based on MPI library, which are managed by two MPI communicators. Firstly, we use a domain partitionning strategy, assigning one subdomain to one cpu and, secondly we distribute telesismic sources on different copies of the partitioned domain. However, despite the supercomputer ability, the forward-problem remains expensive for telesismic configuration especially when 3D numerical methods are considered. In order to perform the forward problem in a reasonable amount of time, we reduce the computational domain in which full waveform modelling is performed. We defined a 3D regional domain located below the seismological network that is embeded in a background homogeneous or axisymetric model, in which the seismic wavefield can be computed efficiently. The background wavefield is used to compute the full wavefield in the 3D regional domain using the so-called total-field/scattered-field technique (Alterman & Karal (1968),Taflove & Hagness (2005)), which relies on the decomposition of the wavefield into a background and a scattered wavefields. The computational domain is subdivided intro three subdomains: an outer domain formed by the perfectly-mathed absorbing layers, an intermediate zone in which only the outgoing wavefield scattered by the

  5. Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations

    NASA Astrophysics Data System (ADS)

    Gassner, Gregor J.; Winters, Andrew R.; Kopriva, David A.

    2016-12-01

    Fisher and Carpenter (2013) [12] found a remarkable equivalence of general diagonal norm high-order summation-by-parts operators to a subcell based high-order finite volume formulation. This equivalence enables the construction of provably entropy stable schemes by a specific choice of the subcell finite volume flux. We show that besides the construction of entropy stable high-order schemes, a careful choice of subcell finite volume fluxes generates split formulations of quadratic or cubic terms. Thus, by changing the subcell finite volume flux to a specific choice, we are able to generate, in a systematic way, all common split forms of the compressible Euler advection terms, such as the Ducros splitting and the Kennedy and Gruber splitting. Although these split forms are not entropy stable, we present a systematic way to prove which of those split forms are at least kinetic energy preserving. With this, we construct a unified high-order split form DG framework. We investigate with three dimensional numerical simulations of the inviscid Taylor-Green vortex and show that the new split forms enhance the robustness of high-order simulations in comparison to the standard scheme when solving turbulent vortex dominated flows. In fact, we show that for certain test cases, the novel split form discontinuous Galerkin schemes are more robust than the discontinuous Galerkin scheme with over-integration.

  6. Numerical Evaluation of P-Multigrid Method for the Solution of Discontinuous Galerkin Discretizations of Diffusive Equations

    NASA Technical Reports Server (NTRS)

    Atkins, H. L.; Helenbrook, B. T.

    2005-01-01

    This paper describes numerical experiments with P-multigrid to corroborate analysis, validate the present implementation, and to examine issues that arise in the implementations of the various combinations of relaxation schemes, discretizations and P-multigrid methods. The two approaches to implement P-multigrid presented here are equivalent for most high-order discretization methods such as spectral element, SUPG, and discontinuous Galerkin applied to advection; however it is discovered that the approach that mimics the common geometric multigrid implementation is less robust, and frequently unstable when applied to discontinuous Galerkin discretizations of di usion. Gauss-Seidel relaxation converges 40% faster than block Jacobi, as predicted by analysis; however, the implementation of Gauss-Seidel is considerably more expensive that one would expect because gradients in most neighboring elements must be updated. A compromise quasi Gauss-Seidel relaxation method that evaluates the gradient in each element twice per iteration converges at rates similar to those predicted for true Gauss-Seidel.

  7. A discontinuous Galerkin numerical model for the simulation of multiphase gas-particle flows in explosive volcanic eruptions

    NASA Astrophysics Data System (ADS)

    Carcano, Susanna; Bonaventura, Luca

    2014-05-01

    in the energy equations. As an alternative to existing finite volume methods, a p-adaptive discontinuous Galerkin space discretization is proposed for the multiphase conservation equations, following the method of lines. An operator splitting approach is adopted, coupled with explicit Runge-Kutta methods for advective terms and semi-implicit time averaging methods for interphase exchange terms. Discontinuous Galerkin methods can be interpreted as an extension of finite volume methods to arbitrary order of accuracy. As finite volume methods, discontinuous Galerkin methods provide discrete conservation laws that reproduce at the discrete level the fundamental physical balances characterizing the continuous problem. Moreover, both the methods represent a good choice for the approximation of problems whose solution presents discontinuities, i.e. explosive volcanic phenomena. However, with discontinuous Galerkin methods high order accuracy can be obtained without extending the computational stencil, thus allowing for a good scalability on parallel architectures. Limiting techniques are introduced on the advective terms of the system, in order to prevent the formation of spurious oscillations and avoid unphysical negative values in the numerical solution. An automatic criterion is introduced to adapt the local number of degrees of freedom and to improve the accuracy locally. The employed technique is simple and relies on the use of orthogonal hierarchical basis functions. The p-adaptivity algorithm allows to reduce the computational cost while maintaining the accuracy of the numerical approximation. The numerical model is applied and tested to several relevant test cases, with special focus on pyroclastic flows arising in volcanic eruptions, in order to assess its accuracy and stability properties. Moreover we analyse the efficiency of the p-adaptivity approach.

  8. A Dynamic Eddy Viscosity Model for the Shallow Water Equations Solved by Spectral Element and Discontinuous Galerkin Methods

    NASA Astrophysics Data System (ADS)

    Marras, Simone; Suckale, Jenny; Giraldo, Francis X.; Constantinescu, Emil

    2016-04-01

    We present the solution of the viscous shallow water equations where viscosity is built as a residual-based subgrid scale model originally designed for large eddy simulation of compressible [1] and stratified flows [2]. The necessity of viscosity for a shallow water model not only finds motivation from mathematical analysis [3], but is supported by physical reasoning as can be seen by an analysis of the energetics of the solution. We simulated the flow of an idealized wave as it hits a set of obstacles. The kinetic energy spectrum of this flow shows that, although the inviscid Galerkin solutions -by spectral elements and discontinuous Galerkin [4]- preserve numerical stability in spite of the spurious oscillations in the proximity of the wave fronts, the slope of the energy cascade deviates from the theoretically expected values. We show that only a sufficiently small amount of dynamically adaptive viscosity removes the unwanted high-frequency modes while preserving the overall sharpness of the solution. In addition, it yields a physically plausible energy decay. This work is motivated by a larger interest in the application of a shallow water model to the solution of tsunami triggered coastal flows. In particular, coastal flows in regions around the world where coastal parks made of mitigation hills of different sizes and configurations are considered as a means to deviate the power of the incoming wave. References [1] M. Nazarov and J. Hoffman (2013) "Residual-based artificial viscosity for simulation of turbulent compressible flow using adaptive finite element methods" Int. J. Numer. Methods Fluids, 71:339-357 [2] S. Marras, M. Nazarov, F. X. Giraldo (2015) "Stabilized high-order Galerkin methods based on a parameter-free dynamic SGS model for LES" J. Comput. Phys. 301:77-101 [3] J. F. Gerbeau and B. Perthame (2001) "Derivation of the viscous Saint-Venant system for laminar shallow water; numerical validation" Discrete Contin. Dyn. Syst. Ser. B, 1:89?102 [4] F

  9. A high-order discontinuous Galerkin method for fluid–structure interaction with efficient implicit–explicit time stepping

    SciTech Connect

    Froehle, Bradley Persson, Per-Olof

    2014-09-01

    We present a high-order accurate scheme for coupled fluid–structure interaction problems. The fluid is discretized using a discontinuous Galerkin method on unstructured tetrahedral meshes, and the structure uses a high-order volumetric continuous Galerkin finite element method. Standard radial basis functions are used for the mesh deformation. The time integration is performed using a partitioned approach based on implicit–explicit Runge–Kutta methods. The resulting scheme fully decouples the implicit solution procedures for the fluid and the solid parts, which we perform using two separate efficient parallel solvers. We demonstrate up to fifth order accuracy in time on a non-trivial test problem, on which we also show that additional subiterations are not required. We solve a benchmark problem of a cantilever beam in a shedding flow, and show good agreement with other results in the literature. Finally, we solve for the flow around a thin membrane at a high angle of attack in both 2D and 3D, and compare with the results obtained with a rigid plate.

  10. A triangular discontinuous Galerkin method for non-Newtonian implicit constitutive models of rarefied and microscale gases

    NASA Astrophysics Data System (ADS)

    Le, N. T. P.; Xiao, H.; Myong, R. S.

    2014-09-01

    The discontinuous Galerkin (DG) method has been popular as a numerical technique for solving the conservation laws of gas dynamics. In the present study, we develop an explicit modal DG scheme for multi-dimensional conservation laws on unstructured triangular meshes in conjunction with non-Newtonian implicit nonlinear coupled constitutive relations (NCCR). Special attention is given to how to treat the complex non-Newtonian type constitutive relations arising from the high degree of thermal nonequilibrium in multi-dimensional gas flows within the Galerkin framework. The Langmuir velocity slip and temperature jump conditions are also implemented into the two-dimensional DG scheme for high Knudsen number flows. As a canonical scalar case, Newtonian and non-Newtonian convection-diffusion Burgers equations are studied to develop the basic building blocks for the scheme. In order to verify and validate the scheme, we applied the scheme to a stiff problem of the shock wave structure for all Mach numbers and to the two-dimensional hypersonic rarefied and low-speed microscale gas flows past a circular cylinder. The computational results show that the NCCR model yields the solutions in better agreement with the direct simulation Monte Carlo (DSMC) data than the Newtonian linear Navier-Stokes-Fourier (NSF) results in all cases of the problem studied.

  11. Third order maximum-principle-satisfying direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangular meshes

    SciTech Connect

    Chen, Zheng; Huang, Hongying; Yan, Jue

    2015-12-21

    We develop 3rd order maximum-principle-satisfying direct discontinuous Galerkin methods [8], [9], [19] and [21] for convection diffusion equations on unstructured triangular mesh. We carefully calculate the normal derivative numerical flux across element edges and prove that, with proper choice of parameter pair (β01) in the numerical flux formula, the quadratic polynomial solution satisfies strict maximum principle. The polynomial solution is bounded within the given range and third order accuracy is maintained. There is no geometric restriction on the meshes and obtuse triangles are allowed in the partition. As a result, a sequence of numerical examples are carried out to demonstrate the accuracy and capability of the maximum-principle-satisfying limiter.

  12. Third order maximum-principle-satisfying direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangular meshes

    DOE PAGES

    Chen, Zheng; Huang, Hongying; Yan, Jue

    2015-12-21

    We develop 3rd order maximum-principle-satisfying direct discontinuous Galerkin methods [8], [9], [19] and [21] for convection diffusion equations on unstructured triangular mesh. We carefully calculate the normal derivative numerical flux across element edges and prove that, with proper choice of parameter pair (β0,β1) in the numerical flux formula, the quadratic polynomial solution satisfies strict maximum principle. The polynomial solution is bounded within the given range and third order accuracy is maintained. There is no geometric restriction on the meshes and obtuse triangles are allowed in the partition. As a result, a sequence of numerical examples are carried out to demonstratemore » the accuracy and capability of the maximum-principle-satisfying limiter.« less

  13. Pseudo-time stepping methods for space time discontinuous Galerkin discretizations of the compressible Navier Stokes equations

    NASA Astrophysics Data System (ADS)

    Klaij, C. M.; van der Vegt, J. J. W.; van der Ven, H.

    2006-12-01

    The space-time discontinuous Galerkin discretization of the compressible Navier-Stokes equations results in a non-linear system of algebraic equations, which we solve with pseudo-time stepping methods. We show that explicit Runge-Kutta methods developed for the Euler equations suffer from a severe stability constraint linked to the viscous part of the equations and propose an alternative to relieve this constraint while preserving locality. To evaluate its effectiveness, we compare with an implicit-explicit Runge-Kutta method which does not suffer from the viscous stability constraint. We analyze the stability of the methods and illustrate their performance by computing the flow around a 2D airfoil and a 3D delta wing at low and moderate Reynolds numbers.

  14. A set of parallel, implicit methods for a reconstructed discontinuous Galerkin method for compressible flows on 3D hybrid grids

    DOE PAGES

    Xia, Yidong; Luo, Hong; Frisbey, Megan; ...

    2014-07-01

    A set of implicit methods are proposed for a third-order hierarchical WENO reconstructed discontinuous Galerkin method for compressible flows on 3D hybrid grids. An attractive feature in these methods are the application of the Jacobian matrix based on the P1 element approximation, resulting in a huge reduction of memory requirement compared with DG (P2). Also, three approaches -- analytical derivation, divided differencing, and automatic differentiation (AD) are presented to construct the Jacobian matrix respectively, where the AD approach shows the best robustness. A variety of compressible flow problems are computed to demonstrate the fast convergence property of the implemented flowmore » solver. Furthermore, an SPMD (single program, multiple data) programming paradigm based on MPI is proposed to achieve parallelism. The numerical results on complex geometries indicate that this low-storage implicit method can provide a viable and attractive DG solution for complicated flows of practical importance.« less

  15. Sparse Grid Methods in Discontinuous Galerkin Algorithms, and Initial Continuum Gyrokinetic Simulations of NSTX-like SOL Turbulence

    NASA Astrophysics Data System (ADS)

    Hakim, Ammar; Hammett, Greg; Shi, Eric

    2016-10-01

    We have developed a hybrid discontinuous/continuous Galerkin scheme for gyrokinetic equations. Our scheme solves the equations in the Poisson bracket formulation and, with a careful choice of basis functions, conserves energy exactly. We use a sparse grid representation to reduce cost. We have developed a novel form of sheath boundary conditions, going beyond logical-sheath BCs, that allows current flow into the boundaries, yet retains overall charge continuity as well as conserves energy. First applications to a simplified model of the NSTX-like scrape-off-layer (SOL) are presented. We treat the SOL in simplified geometry, retaining magnetic curvature effects, and study the turbulent spreading of particle and heat flux. Extensions to include magnetic fluctuations and collisions are discussed. Supported by the Max-Planck/Princeton Center for Plasma Physics, the SciDAC Center for the Study of Plasma Microturbulence, and DOE Contract DE-AC02-09CH11466.

  16. Application of the discontinuous Galerkin method to study the source mechanisms for Love waves in ambient noise

    NASA Astrophysics Data System (ADS)

    Wenk, S.; Hadziioannou, C.; Igel, H.

    2012-12-01

    To produce detailed images of the crust using noise correlation studies and to understand the ocean - solid Earth interaction processes we investigate the behavior and distribution of ambient noise sources, especially focused on the relative contribution of Love waves to the ambient noise field. Therefore, we apply the discontinuous Galerkin (DG) method, which makes use of unstructured tetrahedral meshes combined with a time integration scheme solving the arbitrary high-order derivative (ADER) Riemann problem. The ADER-DG method is high-order accurate in space and time, beneficial for reliable simulations of high-frequency wavefields over long propagation distances. Due to the ease with which tetrahedral grids can be adapted to complex geometries, undulating topography of the Earth's surface and interior interfaces can be readily implemented in the computational domain.

  17. A new efficient 3D Discontinuous Galerkin Time Domain (DGTD) method for large and multiscale electromagnetic simulations

    NASA Astrophysics Data System (ADS)

    Tobón, Luis E.; Ren, Qiang; Liu, Qing Huo

    2015-02-01

    A new Discontinuous Galerkin Time Domain (DGTD) method for solving the 3D time dependent Maxwell's equations via the electric field intensity E and magnetic flux density B fields is proposed for the first time. It uses curl-conforming and divergence-conforming basis functions for E and B, respectively, with the same order of interpolation. In this way, higher accuracy is achieved at lower memory consumption than the conventional approach based on the field variables E and H. The centered flux and Riemann solver are both used to treat interfaces with non-conforming meshes, and both explicit Runge-Kutta method and implicit Crank-Nicholson method are implemented for time integration. Numerical examples for realistic cases will be presented to verify that the proposed method is a non-spurious and efficient DGTD scheme.

  18. Simulation of near-field plasmonic interactions with a local approximation order discontinuous Galerkin time-domain method

    NASA Astrophysics Data System (ADS)

    Viquerat, Jonathan; Lanteri, Stéphane

    2016-01-01

    During the last ten years, the discontinuous Galerkin time-domain (DGTD) method has progressively emerged as a viable alternative to well established finite-difference time-domain (FDTD) and finite-element time-domain (FETD) methods for the numerical simulation of electromagnetic wave propagation problems in the time-domain. The method is now actively studied in various application contexts including those requiring to model light/matter interactions on the nanoscale. Several recent works have demonstrated the viability of the DGDT method for nanophotonics. In this paper we further demonstrate the capabilities of the method for the simulation of near-field plasmonic interactions by considering more particularly the possibility of combining the use of a locally refined conforming tetrahedral mesh with a local adaptation of the approximation order.

  19. DNS of Flow in a Low-Pressure Turbine Cascade Using a Discontinuous-Galerkin Spectral-Element Method

    NASA Technical Reports Server (NTRS)

    Garai, Anirban; Diosady, Laslo Tibor; Murman, Scott; Madavan, Nateri

    2015-01-01

    A new computational capability under development for accurate and efficient high-fidelity direct numerical simulation (DNS) and large eddy simulation (LES) of turbomachinery is described. This capability is based on an entropy-stable Discontinuous-Galerkin spectral-element approach that extends to arbitrarily high orders of spatial and temporal accuracy and is implemented in a computationally efficient manner on a modern high performance computer architecture. A validation study using this method to perform DNS of flow in a low-pressure turbine airfoil cascade are presented. Preliminary results indicate that the method captures the main features of the flow. Discrepancies between the predicted results and the experiments are likely due to the effects of freestream turbulence not being included in the simulation and will be addressed in the final paper.

  20. Stability of a Leap-Frog Discontinuous Galerkin Method for Time-Domain Maxwell's Equations in Anisotropic Materials

    NASA Astrophysics Data System (ADS)

    Araújo, Adérito; Barbeiro, Sílvia; Ghalati, Maryam Khaksar

    2017-05-01

    In this work we discuss the numerical discretization of the time-dependent Maxwell's equations using a fully explicit leap-frog type discontinuous Galerkin method. We present a sufficient condition for the stability, for cases of typical boundary conditions, either perfect electric, perfect magnetic or first order Silver-M\\"uller. The bounds of the stability region point out the influence of not only the mesh size but also the dependence on the choice of the numerical flux and the degree of the polynomials used in the construction of the finite element space, making possible to balance accuracy and computational efficiency. In the model we consider heterogeneous anisotropic permittivity tensors which arise naturally in many applications of interest. Numerical results supporting the analysis are provided.

  1. A set of parallel, implicit methods for a reconstructed discontinuous Galerkin method for compressible flows on 3D hybrid grids

    SciTech Connect

    Xia, Yidong; Luo, Hong; Frisbey, Megan; Nourgaliev, Robert

    2014-07-01

    A set of implicit methods are proposed for a third-order hierarchical WENO reconstructed discontinuous Galerkin method for compressible flows on 3D hybrid grids. An attractive feature in these methods are the application of the Jacobian matrix based on the P1 element approximation, resulting in a huge reduction of memory requirement compared with DG (P2). Also, three approaches -- analytical derivation, divided differencing, and automatic differentiation (AD) are presented to construct the Jacobian matrix respectively, where the AD approach shows the best robustness. A variety of compressible flow problems are computed to demonstrate the fast convergence property of the implemented flow solver. Furthermore, an SPMD (single program, multiple data) programming paradigm based on MPI is proposed to achieve parallelism. The numerical results on complex geometries indicate that this low-storage implicit method can provide a viable and attractive DG solution for complicated flows of practical importance.

  2. Simulation of underresolved turbulent flows by adaptive filtering using the high order discontinuous Galerkin spectral element method

    NASA Astrophysics Data System (ADS)

    Flad, David; Beck, Andrea; Munz, Claus-Dieter

    2016-05-01

    Scale-resolving simulations of turbulent flows in complex domains demand accurate and efficient numerical schemes, as well as geometrical flexibility. For underresolved situations, the avoidance of aliasing errors is a strong demand for stability. For continuous and discontinuous Galerkin schemes, an effective way to prevent aliasing errors is to increase the quadrature precision of the projection operator to account for the non-linearity of the operands (polynomial dealiasing, overintegration). But this increases the computational costs extensively. In this work, we present a novel spatially and temporally adaptive dealiasing strategy by projection filtering. We show this to be more efficient for underresolved turbulence than the classical overintegration strategy. For this novel approach, we discuss the implementation strategy and the indicator details, show its accuracy and efficiency for a decaying homogeneous isotropic turbulence and the transitional Taylor-Green vortex and compare it to the original overintegration approach and a state of the art variational multi-scale eddy viscosity formulation.

  3. High-order local maximum principle preserving (MPP) discontinuous Galerkin finite element method for the transport equation

    NASA Astrophysics Data System (ADS)

    Anderson, R.; Dobrev, V.; Kolev, Tz.; Kuzmin, D.; Quezada de Luna, M.; Rieben, R.; Tomov, V.

    2017-04-01

    In this work we present a FCT-like Maximum-Principle Preserving (MPP) method to solve the transport equation. We use high-order polynomial spaces; in particular, we consider up to 5th order spaces in two and three dimensions and 23rd order spaces in one dimension. The method combines the concepts of positive basis functions for discontinuous Galerkin finite element spatial discretization, locally defined solution bounds, element-based flux correction, and non-linear local mass redistribution. We consider a simple 1D problem with non-smooth initial data to explain and understand the behavior of different parts of the method. Convergence tests in space indicate that high-order accuracy is achieved. Numerical results from several benchmarks in two and three dimensions are also reported.

  4. Development of a Perfectly Matched Layer Technique for a Discontinuous-Galerkin Spectral-Element Method

    NASA Technical Reports Server (NTRS)

    Garai, Anirban; Murman, Scott M.; Madavan, Nateri K.

    2016-01-01

    used involves modeling the pressure fluctuations as acoustic waves propagating in the far-field relative to a single noise-source inside the buffer region. This approach treats vorticity-induced pressure fluctuations the same as acoustic waves. Another popular approach, often referred to as the "sponge layer," attempts to dampen the flow perturbations by introducing artificial dissipation in the buffer region. Although the artificial dissipation removes all perturbations inside the sponge layer, incoming waves are still reflected from the interface boundary between the computational domain and the sponge layer. The effect of these refkections can be somewhat mitigated by appropriately selecting the artificial dissipation strength and the extent of the sponge layer. One of the most promising variants on the buffer region approach is the Perfectly Matched Layer (PML) technique. The PML technique mitigates spurious reflections from boundaries and interfaces by dampening the perturbation modes inside the buffer region such that their eigenfunctions remain unchanged. The technique was first developed by Berenger for application to problems involving electromagnetic wave propagation. It was later extended to the linearized Euler, Euler and Navier-Stokes equations by Hu and his coauthors. The PML technique ensures the no-reflection property for all waves, irrespective of incidence angle, wavelength, and propagation direction. Although the technique requires the solution of a set of auxiliary equations, the computational overhead is easily justified since it allows smaller domain sizes and can provide better accuracy, stability, and convergence of the numerical solution. In this paper, the PML technique is developed in the context of a high-order spectral-element Discontinuous Galerkin (DG) method. The technique is compared to other approaches to treating the in flow and out flow boundary, such as those based on using characteristic boundary conditions and sponge layers. The

  5. Towards Next Generation Ocean Models: Novel Discontinuous Galerkin Schemes for 2D Unsteady Biogeochemical Models

    DTIC Science & Technology

    2009-09-01

    interest ∂Ω The boundary of the domain of interest φ A nodal basis function ψ A modal basis function C Convection (or stiffness) matrix Ckji = ∫ K θi...x) · ∇θj(x)dK F The functional form of the flux K A single element in the triangulation ∂ K The boundary of a single element in the triangulation M...C| K , that is w is chosen as a constant, C, over the triangle K . With the Galerkin method, w is chosen to be the same as the basis function, θ, used

  6. An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry

    NASA Astrophysics Data System (ADS)

    Wintermeyer, Niklas; Winters, Andrew R.; Gassner, Gregor J.; Kopriva, David A.

    2017-07-01

    We design an arbitrary high-order accurate nodal discontinuous Galerkin spectral element approximation for the non-linear two dimensional shallow water equations with non-constant, possibly discontinuous, bathymetry on unstructured, possibly curved, quadrilateral meshes. The scheme is derived from an equivalent flux differencing formulation of the split form of the equations. We prove that this discretization exactly preserves the local mass and momentum. Furthermore, combined with a special numerical interface flux function, the method exactly preserves the mathematical entropy, which is the total energy for the shallow water equations. By adding a specific form of interface dissipation to the baseline entropy conserving scheme we create a provably entropy stable scheme. That is, the numerical scheme discretely satisfies the second law of thermodynamics. Finally, with a particular discretization of the bathymetry source term we prove that the numerical approximation is well-balanced. We provide numerical examples that verify the theoretical findings and furthermore provide an application of the scheme for a partial break of a curved dam test problem.

  7. A new limiting procedure for discontinuous Galerkin methods applied to compressible multiphase flows with shocks and interfaces

    NASA Astrophysics Data System (ADS)

    Henry de Frahan, Marc T.; Varadan, Sreenivas; Johnsen, Eric

    2015-01-01

    Although the Discontinuous Galerkin (DG) method has seen widespread use for compressible flow problems in a single fluid with constant material properties, it has yet to be implemented in a consistent fashion for compressible multiphase flows with shocks and interfaces. Specifically, it is challenging to design a scheme that meets the following requirements: conservation, high-order accuracy in smooth regions and non-oscillatory behavior at discontinuities (in particular, material interfaces). Following the interface-capturing approach of Abgrall [1], we model flows of multiple fluid components or phases using a single equation of state with variable material properties; discontinuities in these properties correspond to interfaces. To represent compressible phenomena in solids, liquids, and gases, we present our analysis for equations of state belonging to the Mie-Grüneisen family. Within the DG framework, we propose a conservative, high-order accurate, and non-oscillatory limiting procedure, verified with simple multifluid and multiphase problems. We show analytically that two key elements are required to prevent spurious pressure oscillations at interfaces and maintain conservation: (i) the transport equation(s) describing the material properties must be solved in a non-conservative weak form, and (ii) the suitable variables must be limited (density, momentum, pressure, and appropriate properties entering the equation of state), coupled with a consistent reconstruction of the energy. Further, we introduce a physics-based discontinuity sensor to apply limiting in a solution-adaptive fashion. We verify this approach with one- and two-dimensional problems with shocks and interfaces, including high pressure and density ratios, for fluids obeying different equations of state to illustrate the robustness and versatility of the method. The algorithm is implemented on parallel graphics processing units (GPU) to achieve high speedup.

  8. A New Runge-Kutta Discontinuous Galerkin Method with Conservation Constraint to Improve CFL Condition for Solving Conservation Laws.

    PubMed

    Xu, Zhiliang; Chen, Xu-Yan; Liu, Yingjie

    2014-12-01

    We present a new formulation of the Runge-Kutta discontinuous Galerkin (RKDG) method [9, 8, 7, 6] for solving conservation Laws with increased CFL numbers. The new formulation requires the computed RKDG solution in a cell to satisfy additional conservation constraint in adjacent cells and does not increase the complexity or change the compactness of the RKDG method. Numerical computations for solving one-dimensional and two-dimensional scalar and systems of nonlinear hyperbolic conservation laws are performed with approximate solutions represented by piecewise quadratic and cubic polynomials, respectively. The hierarchical reconstruction [17, 33] is applied as a limiter to eliminate spurious oscillations in discontinuous solutions. From both numerical experiments and the analytic estimate of the CFL number of the newly formulated method, we find that: 1) this new formulation improves the CFL number over the original RKDG formulation by at least three times or more and thus reduces the overall computational cost; and 2) the new formulation essentially does not compromise the resolution of the numerical solutions of shock wave problems compared with ones computed by the RKDG method.

  9. Mixed-hybrid and vertex-discontinuous-Galerkin finite element modeling of multiphase compositional flow on 3D unstructured grids

    NASA Astrophysics Data System (ADS)

    Moortgat, Joachim; Firoozabadi, Abbas

    2016-06-01

    Problems of interest in hydrogeology and hydrocarbon resources involve complex heterogeneous geological formations. Such domains are most accurately represented in reservoir simulations by unstructured computational grids. Finite element methods accurately describe flow on unstructured meshes with complex geometries, and their flexible formulation allows implementation on different grid types. In this work, we consider for the first time the challenging problem of fully compositional three-phase flow in 3D unstructured grids, discretized by any combination of tetrahedra, prisms, and hexahedra. We employ a mass conserving mixed hybrid finite element (MHFE) method to solve for the pressure and flux fields. The transport equations are approximated with a higher-order vertex-based discontinuous Galerkin (DG) discretization. We show that this approach outperforms a face-based implementation of the same polynomial order. These methods are well suited for heterogeneous and fractured reservoirs, because they provide globally continuous pressure and flux fields, while allowing for sharp discontinuities in compositions and saturations. The higher-order accuracy improves the modeling of strongly non-linear flow, such as gravitational and viscous fingering. We review the literature on unstructured reservoir simulation models, and present many examples that consider gravity depletion, water flooding, and gas injection in oil saturated reservoirs. We study convergence rates, mesh sensitivity, and demonstrate the wide applicability of our chosen finite element methods for challenging multiphase flow problems in geometrically complex subsurface media.

  10. Physical-constraint-preserving Central Discontinuous Galerkin Methods for Special Relativistic Hydrodynamics with a General Equation of State

    NASA Astrophysics Data System (ADS)

    Wu, Kailiang; Tang, Huazhong

    2017-01-01

    The ideal gas equation of state (EOS) with a constant adiabatic index is a poor approximation for most relativistic astrophysical flows, although it is commonly used in relativistic hydrodynamics (RHD). This paper develops high-order accurate, physical-constraints-preserving (PCP), central, discontinuous Galerkin (DG) methods for the one- and two-dimensional special RHD equations with a general EOS. It is built on our theoretical analysis of the admissible states for RHD and the PCP limiting procedure that enforce the admissibility of central DG solutions. The convexity, scaling invariance, orthogonal invariance, and Lax-Friedrichs splitting property of the admissible state set are first proved with the aid of its equivalent form. Then, the high-order central DG methods with the PCP limiting procedure and strong stability-preserving time discretization are proved, to preserve the positivity of the density, pressure, specific internal energy, and the bound of the fluid velocity, maintain high-order accuracy, and be L1-stable. The accuracy, robustness, and effectiveness of the proposed methods are demonstrated by several 1D and 2D numerical examples involving large Lorentz factor, strong discontinuities, or low density/pressure, etc.

  11. Finite volume - space-time discontinuous Galerkin method for the numerical simulation of compressible turbulent flow in time dependent domains

    NASA Astrophysics Data System (ADS)

    Česenek, Jan

    The article is concerned with the numerical simulation of the compressible turbulent flow in time dependent domains. The mathematical model of flow is represented by the system of non-stationary Reynolds- Averaged Navier-Stokes (RANS) equations. The motion of the domain occupied by the fluid is taken into account with the aid of the ALE (Arbitrary Lagrangian-Eulerian) formulation of the RANS equations. This RANS system is equipped with two-equation k - ω turbulence model. These two systems of equations are solved separately. Discretization of the RANS system is carried out by the space-time discontinuous Galerkin method which is based on piecewise polynomial discontinuous approximation of the sought solution in space and in time. Discretization of the two-equation k - ω turbulence model is carried out by the implicit finite volume method, which is based on piecewise constant approximation of the sought solution. We present some numerical experiments to demonstrate the applicability of the method using own-developed code.

  12. High-order accurate solution of acoustic-elastic interface problems on adapted meshes using a discontinuous Galerkin method

    NASA Astrophysics Data System (ADS)

    Wilcox, L. C.; Burstedde, C.; Ghattas, O.; Stadler, G.

    2009-12-01

    Our goal is to develop scalable methods for global full-waveform seismic inversion. The first step we have taken towards this goal is the creation of an accurate solver for the numerical simulation of wave propagation in media with fluid-solid interfaces. We have chosen a high-order discontinuous Galerkin (DG) method to effectively eliminate numerical dispersion, enabling simulation over many wave periods. Our numerical method uses a strain-velocity formulation that enables the solution of acoustic and elastic wave equations within the same framework. Careful attention has been directed at the formulation of a numerical flux that preserves high-order accuracy in the presence of material discontinuities and at fluid-solid interfaces. We use adaptive mesh refinement (AMR) to resolve local variations in wave speeds with appropriate element sizes. To study the numerical accuracy and convergence of the proposed method we compare with reference solutions of classical interface problems, including Raleigh waves, Lamb waves, Stoneley waves, Scholte waves, and Love waves. We report strong and weak parallel scaling results for generation of the mesh and solution of the wave equations on adaptively resolved global Earth models.

  13. A New Runge-Kutta Discontinuous Galerkin Method with Conservation Constraint to Improve CFL Condition for Solving Conservation Laws

    PubMed Central

    Xu, Zhiliang; Chen, Xu-Yan; Liu, Yingjie

    2014-01-01

    We present a new formulation of the Runge-Kutta discontinuous Galerkin (RKDG) method [9, 8, 7, 6] for solving conservation Laws with increased CFL numbers. The new formulation requires the computed RKDG solution in a cell to satisfy additional conservation constraint in adjacent cells and does not increase the complexity or change the compactness of the RKDG method. Numerical computations for solving one-dimensional and two-dimensional scalar and systems of nonlinear hyperbolic conservation laws are performed with approximate solutions represented by piecewise quadratic and cubic polynomials, respectively. The hierarchical reconstruction [17, 33] is applied as a limiter to eliminate spurious oscillations in discontinuous solutions. From both numerical experiments and the analytic estimate of the CFL number of the newly formulated method, we find that: 1) this new formulation improves the CFL number over the original RKDG formulation by at least three times or more and thus reduces the overall computational cost; and 2) the new formulation essentially does not compromise the resolution of the numerical solutions of shock wave problems compared with ones computed by the RKDG method. PMID:25414520

  14. High-Order Discontinuous Galerkin Level Set Method for Interface Tracking and Re-Distancing on Unstructured Meshes

    NASA Astrophysics Data System (ADS)

    Greene, Patrick; Nourgaliev, Robert; Schofield, Sam

    2015-11-01

    A new sharp high-order interface tracking method for multi-material flow problems on unstructured meshes is presented. The method combines the marker-tracking algorithm with a discontinuous Galerkin (DG) level set method to implicitly track interfaces. DG projection is used to provide a mapping from the Lagrangian marker field to the Eulerian level set field. For the level set re-distancing, we developed a novel marching method that takes advantage of the unique features of the DG representation of the level set. The method efficiently marches outward from the zero level set with values in the new cells being computed solely from cell neighbors. Results are presented for a number of different interface geometries including ones with sharp corners and multiple hierarchical level sets. The method can robustly handle the level set discontinuities without explicit utilization of solution limiters. Results show that the expected high order (3rd and higher) of convergence for the DG representation of the level set is obtained for smooth solutions on unstructured meshes. High-order re-distancing on irregular meshes is a must for applications were the interfacial curvature is important for underlying physics, such as surface tension, wetting and detonation shock dynamics. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. Information management release number LLNL-ABS-675636.

  15. A third-order implicit discontinuous Galerkin method based on a Hermite WENO reconstruction for time-accurate solution of the compressible Navier-Stokes equations

    DOE PAGES

    Xia, Yidong; Liu, Xiaodong; Luo, Hong; ...

    2015-06-01

    Here, a space and time third-order discontinuous Galerkin method based on a Hermite weighted essentially non-oscillatory reconstruction is presented for the unsteady compressible Euler and Navier–Stokes equations. At each time step, a lower-upper symmetric Gauss–Seidel preconditioned generalized minimal residual solver is used to solve the systems of linear equations arising from an explicit first stage, single diagonal coefficient, diagonally implicit Runge–Kutta time integration scheme. The performance of the developed method is assessed through a variety of unsteady flow problems. Numerical results indicate that this method is able to deliver the designed third-order accuracy of convergence in both space and time,more » while requiring remarkably less storage than the standard third-order discontinous Galerkin methods, and less computing time than the lower-order discontinous Galerkin methods to achieve the same level of temporal accuracy for computing unsteady flow problems.« less

  16. A third-order implicit discontinuous Galerkin method based on a Hermite WENO reconstruction for time-accurate solution of the compressible Navier-Stokes equations

    SciTech Connect

    Xia, Yidong; Liu, Xiaodong; Luo, Hong; Nourgaliev, Robert

    2015-06-01

    Here, a space and time third-order discontinuous Galerkin method based on a Hermite weighted essentially non-oscillatory reconstruction is presented for the unsteady compressible Euler and Navier–Stokes equations. At each time step, a lower-upper symmetric Gauss–Seidel preconditioned generalized minimal residual solver is used to solve the systems of linear equations arising from an explicit first stage, single diagonal coefficient, diagonally implicit Runge–Kutta time integration scheme. The performance of the developed method is assessed through a variety of unsteady flow problems. Numerical results indicate that this method is able to deliver the designed third-order accuracy of convergence in both space and time, while requiring remarkably less storage than the standard third-order discontinous Galerkin methods, and less computing time than the lower-order discontinous Galerkin methods to achieve the same level of temporal accuracy for computing unsteady flow problems.

  17. High-Order Hybridized Discontinuous Galerkin (HDG) Method for Wave Propagation Simulation in Complex Geophysical Media - Elastic, Acoustic and Hydro-Acoustic - an Unifying Framework to Couple Continuous Spectral Element and Discontinuous Galerkin Methods.

    NASA Astrophysics Data System (ADS)

    Sébastien, T.; Vilotte, J. P.; Guillot, L.; Mariotti, C.

    2014-12-01

    Today seismological observation systems combine broadband seismic receivers, hydrophones and micro-barometers antenna that provide complementary observations of source-radiated waves in heterogeneous and complex geophysical media. Exploiting these observations requires accurate and multi-physics - elastic, hydro-acoustic, infrasonic - wave simulation methods. A popular approach is the Spectral Element Method (SEM) (Chaljub et al, 2006) which is high-order accurate (low dispersion error), very flexible to parallelization and computationally attractive due to efficient sum factorization technique and diagonal mass matrix. However SEMs suffer from lack of flexibility in handling complex geometry and multi-physics wave propagation. High-order Discontinuous Galerkin Methods (DGMs), i.e. Dumbser et al (2006), Etienne et al. (2010), Wilcox et al (2010), are recent alternatives that can handle complex geometry, space-and-time adaptativity, and allow efficient multi-physics wave coupling at interfaces. However, DGMs are more memory demanding and less computationally attractive than SEMs, especially when explicit time stepping is used. We propose a new class of higher-order Hybridized Discontinuous Galerkin Spectral Elements (HDGSEM) methods for spatial discretization of wave equations, following the unifying framework for hybridization of Cockburn et al (2009) and Nguyen et al (2011), which allows for a single implementation of conforming and non-conforming SEMs. When used with energy conserving explicit time integration schemes, HDGSEM is flexible to handle complex geometry, computationally attractive and has significantly less degrees of freedom than classical DGMs, i.e., the only coupled unknowns are the single-valued numerical traces of the velocity field on the element's faces. The formulation can be extended to model fractional energy loss at interfaces between elastic, acoustic and hydro-acoustic media. Accuracy and performance of the HDGSEM are illustrated and

  18. Multilevel Preconditioners for Discontinuous Galerkin Approximations of Elliptic Problems with Jump Coefficients

    DTIC Science & Technology

    2010-12-01

    discontinuous coefficients on geometrically nonconforming substructures. Technical Report Serie A 634, Instituto de Matematica Pura e Aplicada, Brazil, 2009...Instituto de Matematica Pura e Aplicada, Brazil, 2010. submitted. [41] M. Dryja, M. V. Sarkis, and O. B. Widlund. Multilevel Schwarz methods for

  19. Application of the multigrid approach for solving 3D Navier-Stokes equations on hexahedral grids using the discontinuous Galerkin method

    NASA Astrophysics Data System (ADS)

    Wolkov, A. V.

    2010-03-01

    The Galerkin method with discontinuous basis functions is adapted for solving the Euler and Navier-Stokes equations on unstructured hexahedral grids. A hybrid multigrid algorithm involving the finite element and grid stages is used as an iterative solution method. Numerical results of calculating the sphere inviscid flow, viscous flow in a bent pipe, and turbulent flow past a wing are presented. The numerical results and the computational cost are compared with those obtained using the finite volume method.

  20. Superconvergence of Discontinuous Galerkin Solutions for a Nonlinear Scalar Hyperbolic Problem

    DTIC Science & Technology

    2006-01-01

    inflow element we prove that the p -degree discontinuous finite element solution con- verges at Radau points with an O(/f+2 ) rate. We further show that...hp+2) superconvergence rate at the roots of Radau polynomial of degree p + 1. They used this result to construct asymptotically correct a posteriori...the true local error is spanned by two ( p + 1)-degree Radau polynomials in the x- and y-directions, respectively. They further showed that a p -degree

  1. An operator-based local discontinuous Galerkin method compatible with the BSSN formulation of the Einstein equations

    NASA Astrophysics Data System (ADS)

    Miller, Jonah M.; Schnetter, Erik

    2017-01-01

    Discontinuous Galerkin finite element (DGFE) methods offer a mathematically beautiful, computationally efficient, and efficiently parallelizable way to solve partial differential equations (PDEs). These properties make them highly desirable for numerical calculations in relativistic astrophysics and many other fields. The BSSN formulation of the Einstein equations has repeatedly demonstrated its robustness. The formulation is not only stable but allows for puncture-type evolutions of black hole systems. To-date no one has been able to solve the full (3  +  1)-dimensional BSSN equations using DGFE methods. This is partly because DGFE discretization often occurs at the level of the equations, not the derivative operator, and partly because DGFE methods are traditionally formulated for manifestly flux-conservative systems. By discretizing the derivative operator, we generalize a particular flavor of DGFE methods, Local DG methods, to solve arbitrary second-order hyperbolic equations. Because we discretize at the level of the derivative operator, our method can be interpreted as either a DGFE method or as a finite differences stencil with non-constant coefficients.

  2. Accelerating the discontinuous Galerkin method for seismic wave propagation simulations using multiple GPUs with CUDA and MPI

    NASA Astrophysics Data System (ADS)

    Mu, Dawei; Chen, Po; Wang, Liqiang

    2013-12-01

    We have successfully ported an arbitrary high-order discontinuous Galerkin method for solving the three-dimensional isotropic elastic wave equation on unstructured tetrahedral meshes to multiple Graphic Processing Units (GPUs) using the Compute Unified Device Architecture (CUDA) of NVIDIA and Message Passing Interface (MPI) and obtained a speedup factor of about 28.3 for the single-precision version of our codes and a speedup factor of about 14.9 for the double-precision version. The GPU used in the comparisons is NVIDIA Tesla C2070 Fermi, and the CPU used is Intel Xeon W5660. To effectively overlap inter-process communication with computation, we separate the elements on each subdomain into inner and outer elements and complete the computation on outer elements and fill the MPI buffer first. While the MPI messages travel across the network, the GPU performs computation on inner elements, and all other calculations that do not use information of outer elements from neighboring subdomains. A significant portion of the speedup also comes from a customized matrix-matrix multiplication kernel, which is used extensively throughout our program. Preliminary performance analysis on our parallel GPU codes shows favorable strong and weak scalabilities.

  3. Coupling of a compressible vortex particle-mesh method with a near-body compressible discontinuous Galerkin solver

    NASA Astrophysics Data System (ADS)

    Parmentier, Philippe; Winckelmans, Gregoire; Chatelain, Philippe; Hillewaert, Koen

    2015-11-01

    A hybrid approach, coupling a compressible vortex particle-mesh method (CVPM, also with efficient Poisson solver) and a high order compressible discontinuous Galerkin Eulerian solver, is being developed in order to efficiently simulate flows past bodies; also in the transonic regime. The Eulerian solver is dedicated to capturing the anisotropic flow structures in the near-wall region whereas the CVPM solver is exploited away from the body and in the wake. An overlapping domain decomposition approach is used. The Eulerian solver, which captures the near-body region, also corrects the CVPM solution in that region at every time step. The CVPM solver, which captures the region away from the body and the wake, also provides the outer boundary conditions to the Eulerian solver. Because of the coupling, a boundary element method is also required for consistency. The approach is assessed on typical 2D benchmark cases. Supported by the Fund for Research Training in Industry and Agriculture (F.R.I.A.).

  4. Analysis of adaptive mesh refinement for IMEX discontinuous Galerkin solutions of the compressible Euler equations with application to atmospheric simulations

    NASA Astrophysics Data System (ADS)

    Kopera, Michal A.; Giraldo, Francis X.

    2014-10-01

    The resolutions of interests in atmospheric simulations require prohibitively large computational resources. Adaptive mesh refinement (AMR) tries to mitigate this problem by putting high resolution in crucial areas of the domain. We investigate the performance of a tree-based AMR algorithm for the high order discontinuous Galerkin method on quadrilateral grids with non-conforming elements. We perform a detailed analysis of the cost of AMR by comparing this to uniform reference simulations of two standard atmospheric test cases: density current and rising thermal bubble. The analysis shows up to 15 times speed-up of the AMR simulations with the cost of mesh adaptation below 1% of the total runtime. We pay particular attention to the implicit-explicit (IMEX) time integration methods and show that the ARK2 method is more robust with respect to dynamically adapting meshes than BDF2. Preliminary analysis of preconditioning reveals that it can be an important factor in the AMR overhead. The compiler optimizations provide significant runtime reduction and positively affect the effectiveness of AMR allowing for speed-ups greater than it would follow from the simple performance model.

  5. Convergence and superconvergence of staggered discontinuous Galerkin methods for the three-dimensional Maxwell’s equations on Cartesian grids

    NASA Astrophysics Data System (ADS)

    Chung, Eric T.; Ciarlet, Patrick; Yu, Tang Fei

    2013-02-01

    In this paper, a new type of staggered discontinuous Galerkin methods for the three dimensional Maxwell’s equations is developed and analyzed. The spatial discretization is based on staggered Cartesian grids so that many good properties are obtained. First of all, our method has the advantages that the numerical solution preserves the electromagnetic energy and automatically fulfills a discrete version of the Gauss law. Moreover, the mass matrices are diagonal, thus time marching is explicit and is very efficient. Our method is high order accurate and the optimal order of convergence is rigorously proved. It is also very easy to implement due to its Cartesian structure and can be regarded as a generalization of the classical Yee’s scheme as well as the quadrilateral edge finite elements. Furthermore, a superconvergence result, that is the convergence rate is one order higher at interpolation nodes, is proved. Numerical results are shown to confirm our theoretical statements, and applications to problems in unbounded domains with the use of PML are presented. A comparison of our staggered method and non-staggered method is carried out and shows that our method has better accuracy and efficiency.

  6. A nodal discontinuous Galerkin method for site effects assessment in viscoelastic media—verification and validation in the Nice basin

    NASA Astrophysics Data System (ADS)

    Peyrusse, Fabien; Glinsky, Nathalie; Gélis, Céline; Lanteri, Stéphane

    2014-10-01

    We present a discontinuous Galerkin method for site effects assessment. The P-SV seismic wave propagation is studied in 2-D space heterogeneous media. The first-order velocity-stress system is obtained by assuming that the medium is linear, isotropic and viscoelastic, thus considering intrinsic attenuation. The associated stress-strain relation in the time domain being a convolution, which is numerically intractable, we consider the rheology of a generalized Maxwell body replacing the convolution by a set of differential equations. This results in a velocity-stress system which contains additional equations for the anelastic functions expressing the strain history of the material. Our numerical method, suitable for complex triangular unstructured meshes, is based on centred numerical fluxes and a leap-frog time-discretization. The method is validated through numerical simulations including comparisons with a finite-difference scheme. We study the influence of the geological structures of the Nice basin on the surface ground motion through the comparison of 1-D and 2-D soil response in homogeneous and heterogeneous soil. At last, we compare numerical results with real recordings data. The computed multiple-sediment basin response allows to reproduce the shape of the recorded amplification in the basin. This highlights the importance of knowing the lithological structures of a basin, layers properties and interface geometry.

  7. OpenACC acceleration of an unstructured CFD solver based on a reconstructed discontinuous Galerkin method for compressible flows

    SciTech Connect

    Xia, Yidong; Lou, Jialin; Luo, Hong; Edwards, Jack; Mueller, Frank

    2015-02-09

    Here, an OpenACC directive-based graphics processing unit (GPU) parallel scheme is presented for solving the compressible Navier–Stokes equations on 3D hybrid unstructured grids with a third-order reconstructed discontinuous Galerkin method. The developed scheme requires the minimum code intrusion and algorithm alteration for upgrading a legacy solver with the GPU computing capability at very little extra effort in programming, which leads to a unified and portable code development strategy. A face coloring algorithm is adopted to eliminate the memory contention because of the threading of internal and boundary face integrals. A number of flow problems are presented to verify the implementation of the developed scheme. Timing measurements were obtained by running the resulting GPU code on one Nvidia Tesla K20c GPU card (Nvidia Corporation, Santa Clara, CA, USA) and compared with those obtained by running the equivalent Message Passing Interface (MPI) parallel CPU code on a compute node (consisting of two AMD Opteron 6128 eight-core CPUs (Advanced Micro Devices, Inc., Sunnyvale, CA, USA)). Speedup factors of up to 24× and 1.6× for the GPU code were achieved with respect to one and 16 CPU cores, respectively. The numerical results indicate that this OpenACC-based parallel scheme is an effective and extensible approach to port unstructured high-order CFD solvers to GPU computing.

  8. A conservative discontinuous Galerkin scheme with O(N2) operations in computing Boltzmann collision weight matrix

    NASA Astrophysics Data System (ADS)

    Gamba, Irene M.; Zhang, Chenglong

    2014-12-01

    In the present work, we propose a deterministic numerical solver for the homogeneous Boltzmann equation based on Discontinuous Galerkin (DG) methods. The weak form of the collision operator is approximated by a quadratic form in linear algebra setting. We employ the property of "shifting symmetry" in the weight matrix to reduce the computing complexity from theoretical O(N3) down to O(N2) , with N the total number of freedom for d-dimensional velocity space. In addition, the sparsity is also explored to further reduce the storage complexity. To apply lower order polynomials and resolve loss of conserved quantities, we invoke the conservation routine at every time step to enforce the conservation of desired moments (mass, momentum and/or energy), with only linear complexity. Due to the locality of the DG schemes, the whole computing process is well parallelized using hybrid OpenMP and MPI. The current work only considers integrable angular cross-sections under elastic and/or inelastic interaction laws. Numerical results on 2-D and 3-D problems are shown.

  9. A fully-coupled discontinuous Galerkin spectral element method for two-phase flow in petroleum reservoirs

    NASA Astrophysics Data System (ADS)

    Taneja, Ankur; Higdon, Jonathan

    2016-11-01

    A spectral element method (SEM) is presented to simulate two-phase fluid flow (oil and water phase) in petroleum reservoirs. Petroleum reservoirs are porous media with heterogeneous geologic features, and the flow of two immiscible phases involves sharp, moving interfaces. The governing equations of motion are time-dependent, non-linear PDEs with strong hyperbolic nature. A fully-coupled numerical scheme using discontinuous Galerkin (DG) method with nodal spectral element basis functions for spatial discretization, and an implicit Runge-Kutta type time-stepping is developed to solve the PDEs in a robust, stable manner. Isoparameteric mapping is used to generate grids for reservoir and well geometry. We present the performance capabilities of the DG scheme with high-order basis functions to accurately resolve sharp fluid interfaces and a variety of heterogeneous geologic features. High-order convergence of SEM is demonstrated. Numerical results are presented for reservoir flows with various injection-production patterns. Typical reservoir heterogeneities like low-permeable regions, impermeable shale barriers, etc. are included in the numerical tests. Comparisons with commonly used finite volume methods and linear and quadratic finite element methods are presented. ExxonMobil Upstream Research Co.

  10. Simulation study of planar and nonplanar super rogue waves in an electronegative plasma: Local discontinuous Galerkin method

    NASA Astrophysics Data System (ADS)

    El-Tantawy, S. A.; Aboelenen, Tarek

    2017-05-01

    Planar and nonplanar (cylindrical and spherical) ion-acoustic super rogue waves in an unmagnetized electronegative plasma are investigated, both analytically (for planar geometry) and numerically (for planar and nonplanar geometries). Using a reductive perturbation technique, the basic set of fluid equations is reduced to a nonplanar/modified nonlinear Schrödinger equation (NLSE), which describes a slow modulation of the nonlinear wave amplitude. The local modulational instability of the ion-acoustic structures governed by the planar and nonplanar NLSE is reported. Furthermore, the existence region of rogue waves is strictly defined. The parameters used in our calculations are from the lab observation data. The local discontinuous Galerkin (LDG) method is used to find rogue wave solutions of the planar and nonplanar NLSE and to prove L2 stability of this method. Also, it is found that the numerical simulations and the exact (analytical) solutions of the planar NLSE match remarkably well and numerical examples show that the convergence orders of the proposed LDG method are N + 1 when polynomials of degree N are used. Moreover, it is noted that the spherical rogue waves travel faster than their cylindrical counterpart. Also, the numerical solution showed that the spherical and cylindrical amplitudes of the localized pulses decrease with the increase in the time | τ |.

  11. Effect of boundary representation on viscous, separated flows in a discontinuous-Galerkin Navier-Stokes solver

    NASA Astrophysics Data System (ADS)

    Nelson, Daniel A.; Jacobs, Gustaaf B.; Kopriva, David A.

    2016-08-01

    The effect of curved-boundary representation on the physics of the separated flow over a NACA 65(1)-412 airfoil is thoroughly investigated. A method is presented to approximate curved boundaries with a high-order discontinuous-Galerkin spectral element method for the solution of the Navier-Stokes equations. Multiblock quadrilateral element meshes are constructed with the grid generation software GridPro. The boundary of a NACA 65(1)-412 airfoil, defined by a cubic natural spline, is piecewise-approximated by isoparametric polynomial interpolants that represent the edges of boundary-fitted elements. Direct numerical simulation of the airfoil is performed on a coarse mesh and fine mesh with polynomial orders ranging from four to twelve. The accuracy of the curve fitting is investigated by comparing the flows computed on curved-sided meshes with those given by straight-sided meshes. Straight-sided meshes yield irregular wakes, whereas curved-sided meshes produce a regular Karman street wake. Straight-sided meshes also produce lower lift and higher viscous drag as compared with curved-sided meshes. When the mesh is refined by reducing the sizes of the elements, the lift decrease and viscous drag increase are less pronounced. The differences in the aerodynamic performance between the straight-sided meshes and the curved-sided meshes are concluded to be the result of artificial surface roughness introduced by the piecewise-linear boundary approximation provided by the straight-sided meshes.

  12. OpenACC acceleration of an unstructured CFD solver based on a reconstructed discontinuous Galerkin method for compressible flows

    DOE PAGES

    Xia, Yidong; Lou, Jialin; Luo, Hong; ...

    2015-02-09

    Here, an OpenACC directive-based graphics processing unit (GPU) parallel scheme is presented for solving the compressible Navier–Stokes equations on 3D hybrid unstructured grids with a third-order reconstructed discontinuous Galerkin method. The developed scheme requires the minimum code intrusion and algorithm alteration for upgrading a legacy solver with the GPU computing capability at very little extra effort in programming, which leads to a unified and portable code development strategy. A face coloring algorithm is adopted to eliminate the memory contention because of the threading of internal and boundary face integrals. A number of flow problems are presented to verify the implementationmore » of the developed scheme. Timing measurements were obtained by running the resulting GPU code on one Nvidia Tesla K20c GPU card (Nvidia Corporation, Santa Clara, CA, USA) and compared with those obtained by running the equivalent Message Passing Interface (MPI) parallel CPU code on a compute node (consisting of two AMD Opteron 6128 eight-core CPUs (Advanced Micro Devices, Inc., Sunnyvale, CA, USA)). Speedup factors of up to 24× and 1.6× for the GPU code were achieved with respect to one and 16 CPU cores, respectively. The numerical results indicate that this OpenACC-based parallel scheme is an effective and extensible approach to port unstructured high-order CFD solvers to GPU computing.« less

  13. Local Discontinuous Galerkin (LDG) Method for Advection of Active Compositional Fields with Discontinuous Boundaries: Demonstration and Comparison with Other Methods in the Mantle Convection Code ASPECT

    NASA Astrophysics Data System (ADS)

    He, Y.; Billen, M. I.; Puckett, E. G.

    2015-12-01

    Flow in the Earth's mantle is driven by thermo-chemical convection in which the properties and geochemical signatures of rocks vary depending on their origin and composition. For example, tectonic plates are composed of compositionally-distinct layers of crust, residual lithosphere and fertile mantle, while in the lower-most mantle there are large compositionally distinct "piles" with thinner lenses of different material. Therefore, tracking of active or passive fields with distinct compositional, geochemical or rheologic properties is important for incorporating physical realism into mantle convection simulations, and for investigating the long term mixing properties of the mantle. The difficulty in numerically advecting fields arises because they are non-diffusive and have sharp boundaries, and therefore require different methods than usually used for temperature. Previous methods for tracking fields include the marker-chain, tracer particle, and field-correction (e.g., the Lenardic Filter) methods: each of these has different advantages or disadvantages, trading off computational speed with accuracy in tracking feature boundaries. Here we present a method for modeling active fields in mantle dynamics simulations using a new solver implemented in the deal.II package that underlies the ASPECT software. The new solver for the advection-diffusion equation uses a Local Discontinuous Galerkin (LDG) algorithm, which combines features of both finite element and finite volume methods, and is particularly suitable for problems with a dominant first-order term and discontinuities. Furthermore, we have applied a post-processing technique to insure that the solution satisfies a global maximum/minimum. One potential drawback for the LDG method is that the total number of degrees of freedom is larger than the finite element method. To demonstrate the capabilities of this new method we present results for two benchmarks used previously: a falling cube with distinct buoyancy and

  14. Local Discontinuous Galerkin (LDG) Method for Advection of Active Compositional Fields with Discontinuous Boundaries: Demonstration and Comparison with Other Methods in the Mantle Convection Code ASPECT

    NASA Astrophysics Data System (ADS)

    Hsu, S. K.; Armada, L. T.; Yeh, Y. C.; Bacolcol, T. C.; Dimalanta, C. B.; Doo, W. B.; Liang, C. W.

    2014-12-01

    Flow in the Earth's mantle is driven by thermo-chemical convection in which the properties and geochemical signatures of rocks vary depending on their origin and composition. For example, tectonic plates are composed of compositionally-distinct layers of crust, residual lithosphere and fertile mantle, while in the lower-most mantle there are large compositionally distinct "piles" with thinner lenses of different material. Therefore, tracking of active or passive fields with distinct compositional, geochemical or rheologic properties is important for incorporating physical realism into mantle convection simulations, and for investigating the long term mixing properties of the mantle. The difficulty in numerically advecting fields arises because they are non-diffusive and have sharp boundaries, and therefore require different methods than usually used for temperature. Previous methods for tracking fields include the marker-chain, tracer particle, and field-correction (e.g., the Lenardic Filter) methods: each of these has different advantages or disadvantages, trading off computational speed with accuracy in tracking feature boundaries. Here we present a method for modeling active fields in mantle dynamics simulations using a new solver implemented in the deal.II package that underlies the ASPECT software. The new solver for the advection-diffusion equation uses a Local Discontinuous Galerkin (LDG) algorithm, which combines features of both finite element and finite volume methods, and is particularly suitable for problems with a dominant first-order term and discontinuities. Furthermore, we have applied a post-processing technique to insure that the solution satisfies a global maximum/minimum. One potential drawback for the LDG method is that the total number of degrees of freedom is larger than the finite element method. To demonstrate the capabilities of this new method we present results for two benchmarks used previously: a falling cube with distinct buoyancy and

  15. Discontinuous Galerkin discretization of the Reynolds-averaged Navier-Stokes equations with the shear-stress transport model

    NASA Astrophysics Data System (ADS)

    Schoenawa, Stefan; Hartmann, Ralf

    2014-04-01

    In this article we consider the development of Discontinuous Galerkin (DG) methods for the numerical approximation of the Reynolds-averaged Navier-Stokes (RANS) equations with the shear-stress transport (SST) model by Menter. This turbulence model is based on a blending of the Wilcox k-ω model used near the wall and the k-ɛ model used in the rest of the domain where the blending functions depend on the distance to the nearest wall. For the computation of the distance of each quadrature point in the domain to the nearest of the curved, piecewise polynomial wall boundaries, we propose a stabilized continuous finite element (FE) discretization of the eikonal equation. Furthermore, we propose a new wall boundary condition for the dissipation rate ω based on the projection of the analytic near-wall behavior of ω onto the discrete ansatz space of the DG discretization. Finally, we introduce an artificial viscosity to the discretization of the turbulence kinetic energy (k-)equation to suppress oscillations of k near the underresolved boundary layer edge. The wall distance computation based on the continuous FE discretization of the eikonal equation is demonstrated for an internal and three external/aerodynamic flow geometries including a three-element high-lift configuration. The DG discretization of the RANS equations with the SST model is demonstrated for turbulent flows past a flat plate and the RAE2822 airfoil (Cases 9 and 10). The results are compared to the underlying k-ω model and experimental data.

  16. A New Ice-sheet / Ocean Interaction Model for Greenland Fjords using High-Order Discontinuous Galerkin Methods

    NASA Astrophysics Data System (ADS)

    Kopera, M. A.; Maslowski, W.; Giraldo, F.

    2015-12-01

    One of the key outstanding challenges in modeling of climate change and sea-level rise is the ice-sheet/ocean interaction in narrow, elongated and geometrically complicated fjords around Greenland. To address this challenge we propose a new approach, a separate fjord model using discontinuous Galerkin (DG) methods, or FDG. The goal of this project is to build a separate, high-resolution module for use in Earth System Models (ESMs) to realistically represent the fjord bathymetry, coastlines, exchanges with the outside ocean, circulation and fine-scale processes occurring within the fjord and interactions at the ice shelf interface. FDG is currently at the first stage of development. The DG method provides FDG with high-order accuracy as well as geometrical flexibility, including the capacity to handle non-conforming adaptive mesh refinement to resolve the processes occurring near the ice-sheet/ocean interface without introducing prohibitive computational costs. Another benefit of this method is its excellent performance on multi- and many-core architectures, which allows for utilizing modern high performance computing systems for high-resolution simulations. The non-hydrostatic model of the incompressible Navier-Stokes equation will account for the stationary ice-shelf with sub-shelf ocean interaction, basal melting and subglacial meltwater influx and with boundary conditions at the surface to account for floating sea ice. The boundary conditions will be provided to FDG via a flux coupler to emulate the integration with an ESM. Initially, FDG will be tested for the Sermilik Fjord settings, using real bathymetry, boundary and initial conditions, and evaluated against available observations and other model results for this fjord. The overarching goal of the project is to be able to resolve the ice-sheet/ocean interactions around the entire coast of Greenland and two-way coupling with regional and global climate models such as the Regional Arctic System Model (RASM

  17. Implicit finite volume and discontinuous Galerkin methods for multicomponent flow in unstructured 3D fractured porous media

    NASA Astrophysics Data System (ADS)

    Moortgat, Joachim; Amooie, Mohammad Amin; Soltanian, Mohamad Reza

    2016-10-01

    We present a new implicit higher-order finite element (FE) approach to efficiently model compressible multicomponent fluid flow on unstructured grids and in fractured porous subsurface formations. The scheme is sequential implicit: pressures and fluxes are updated with an implicit Mixed Hybrid Finite Element (MHFE) method, and the transport of each species is approximated with an implicit second-order Discontinuous Galerkin (DG) FE method. Discrete fractures are incorporated with a cross-flow equilibrium approach. This is the first investigation of all-implicit higher-order MHFE-DG for unstructured triangular, quadrilateral (2D), and hexahedral (3D) grids and discrete fractures. A lowest-order implicit finite volume (FV) transport update is also developed for the same grid types. The implicit methods are compared to an Implicit-Pressure-Explicit-Composition (IMPEC) scheme. For fractured domains, the unconditionally stable implicit transport update is shown to increase computational efficiency by orders of magnitude as compared to IMPEC, which has a time-step constraint proportional to the pore volume of discrete fracture grid cells. However, when lowest-order Euler time-discretizations are used, numerical errors increase linearly with the larger implicit time-steps, resulting in high numerical dispersion. Second-order Crank-Nicolson implicit MHFE-DG and MHFE-FV are therefore presented as well. Convergence analyses show twice the convergence rate for the DG methods as compared to FV, resulting in two to three orders of magnitude higher computational efficiency. Numerical experiments demonstrate the efficiency and robustness in modeling compressible multicomponent flow on irregular and fractured 2D and 3D grids, even in the presence of fingering instabilities.

  18. NEXD: A Software Package for High Order Simulation of Seismic Waves using the Nodal Discontinuous Galerkin Method

    NASA Astrophysics Data System (ADS)

    Boxberg, Marc S.; Lamert, Andre; Möller, Thomas; Lambrecht, Lasse; Friederich, Wolfgang

    2017-04-01

    Numerical simulations are a key tool to improve the knowledge of the interior of the earth. For example, global simulations of seismic waves excited by earthquakes are essential to infer the velocity structure within the earth. Numerical investigations on local scales can be helpful to find and characterize oil and gas reservoirs. Moreover, simulations help to understand wave propagation in boreholes and other complex geological structures. Even on laboratory scales, numerical simulations of seismic waves can help to increase knowledge about the behaviour of materials, e.g., to understand the mechanisms of attenuation or crack propagation in rocks. To deal with highly complex heterogeneous models, the Nodal Discontinuous Galerkin Method (NDG) is used to calculate synthetic seismograms. The main advantage of this method is the ability to mesh complex geometries by using triangular or tetrahedral elements together with a high order spatial approximation of the wave field. The presented simulation tool NEXD has the capability of simulating elastic, anelastic, and poroelastic wave fields for seismic experiments for one-, two- and three-dimensional settings. In addition, fractures can be modelled using linear slip interfaces. NEXD also provides adjoint kernel capabilities to invert for seismic wave velocities. External models provided by, e.g., Trelis can be used for parallelized computations. For absorbing boundary conditions, Perfectly Matched Layers (PML) can be used. Examples are presented to validate the method and to show the capability of the software for complex models such as the simulation of a tunnel reconaissance experiment. The software is available on GitHub: https://github.com/seismology-RUB

  19. Semi-implicit discontinuous Galerkin methods for the incompressible Navier-Stokes equations on adaptive staggered Cartesian grids

    NASA Astrophysics Data System (ADS)

    Fambri, Francesco; Dumbser, Michael

    2017-09-01

    In this paper a new high order semi-implicit discontinuous Galerkin method (SI-DG) is presented for the solution of the incompressible Navier-Stokes equations on staggered space-time adaptive Cartesian grids (AMR) in two and three space-dimensions. The pressure is written in the form of piecewise polynomials on the main grid, which is dynamically adapted within a cell-by-cell AMR framework. According to the time dependent main grid, different face-based spatially staggered dual grids are defined for the piece-wise polynomials of the respective velocity components. Arbitrary high order of accuracy is achieved in space, while a very simple semi-implicit time discretization is obtained via an explicit discretization of the nonlinear convective terms, and an implicit discretization of the pressure gradient in the momentum equation and of the divergence of the velocity field in the continuity equation. The real advantages of the staggered grid arise in the solution of the Schur complement associated with the saddle point problem of the discretized incompressible Navier-Stokes equations, i.e. after substituting the discrete momentum equations into the discrete continuity equation. This leads to a linear system for only one unknown, the scalar pressure. Indeed, the resulting linear pressure system is shown to be symmetric and positive-definite. The new space-time adaptive staggered DG scheme has been thoroughly verified for a large set of non-trivial test problems in two and three space dimensions, for which analytical, numerical or experimental reference solutions exist. To the knowledge of the authors, this is the first staggered semi-implicit DG scheme for the incompressible Navier-Stokes equations on space-time adaptive meshes in two and three space dimensions.

  20. NEXD: A Software Package for High Order Simulation of Seismic Waves using the Nodal Discontinuous Galerkin Method

    NASA Astrophysics Data System (ADS)

    Schumacher, F.; Lambrecht, L.; Friederich, W.

    2015-12-01

    In geophysics numerical simulations are a key tool to understand the processes of earth. For example, global simulations of seismic waves excited by earthquakes are essential to infer the velocity structure within the earth. Furthermore, numerical investigations can be helpful on local scales in order to find and characterize oil and gas reservoirs. Moreover, simulations enable a better understanding of wave propagation in borehole and tunnel seismic applications. Even on microscopic scales, numerical simulations of elastic waves can help to increase knowledge about the behaviour of materials, e.g. to understand the mechanism of crack propagation in rocks. To deal with highly complex heterogeneous models, here the Nodal Discontinuous Galerkin Method (NDG) is used to calculate synthetic seismograms. The advantage of this method is that complex mesh geometries can be computed by using triangular or tetrahedral elements for domain discretization together with a high order spatial approximation of the wave field. The simulation tool NEXD is presented which has the capability of simulating elastic and anelastic wave fields for seismic experiments for one-, two- and three- dimensional settings. The implementation of poroelasticity and simulation of slip interfaces are currently in progress and are working for the one dimensional part. External models provided by e.g. Trelis/Cubit can be used for parallelized computations on triangular or tetrahedral meshes. For absorbing boundary conditions either a fluxes based approach or a Nearly Perfectly Matched Layer (NPML) can be used. Examples are presented to validate the method and to show the capability of the software for complex models such as the simulation of a tunnel seismic experiment.

  1. Three-dimensional dynamic rupture simulation with a high-order discontinuous Galerkin method on unstructured tetrahedral meshes

    NASA Astrophysics Data System (ADS)

    Pelties, Christian; de la Puente, Josep; Ampuero, Jean-Paul; Brietzke, Gilbert B.; Käser, Martin

    2012-02-01

    Accurate and efficient numerical methods to simulate dynamic earthquake rupture and wave propagation in complex media and complex fault geometries are needed to address fundamental questions in earthquake dynamics, to integrate seismic and geodetic data into emerging approaches for dynamic source inversion, and to generate realistic physics-based earthquake scenarios for hazard assessment. Modeling of spontaneous earthquake rupture and seismic wave propagation by a high-order discontinuous Galerkin (DG) method combined with an arbitrarily high-order derivatives (ADER) time integration method was introduced in two dimensions by de la Puente et al. (2009). The ADER-DG method enables high accuracy in space and time and discretization by unstructured meshes. Here we extend this method to three-dimensional dynamic rupture problems. The high geometrical flexibility provided by the usage of tetrahedral elements and the lack of spurious mesh reflections in the ADER-DG method allows the refinement of the mesh close to the fault to model the rupture dynamics adequately while concentrating computational resources only where needed. Moreover, ADER-DG does not generate spurious high-frequency perturbations on the fault and hence does not require artificial Kelvin-Voigt damping. We verify our three-dimensional implementation by comparing results of the SCEC TPV3 test problem with two well-established numerical methods, finite differences, and spectral boundary integral. Furthermore, a convergence study is presented to demonstrate the systematic consistency of the method. To illustrate the capabilities of the high-order accurate ADER-DG scheme on unstructured meshes, we simulate an earthquake scenario, inspired by the 1992 Landers earthquake, that includes curved faults, fault branches, and surface topography.

  2. High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms

    SciTech Connect

    Xing Yulong . E-mail: xing@dam.brown.edu; Shu Chiwang . E-mail: shu@dam.brown.edu

    2006-05-20

    Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source term. In our earlier work [J. Comput. Phys. 208 (2005) 206-227; J. Sci. Comput., accepted], we designed a well-balanced finite difference weighted essentially non-oscillatory (WENO) scheme, which at the same time maintains genuine high order accuracy for general solutions, to a class of hyperbolic systems with separable source terms including the shallow water equations, the elastic wave equation, the hyperbolic model for a chemosensitive movement, the nozzle flow and a two phase flow model. In this paper, we generalize high order finite volume WENO schemes and Runge-Kutta discontinuous Galerkin (RKDG) finite element methods to the same class of hyperbolic systems to maintain a well-balanced property. Finite volume and discontinuous Galerkin finite element schemes are more flexible than finite difference schemes to treat complicated geometry and adaptivity. However, because of a different computational framework, the maintenance of the well-balanced property requires different technical approaches. After the description of our well-balanced high order finite volume WENO and RKDG schemes, we perform extensive one and two dimensional simulations to verify the properties of these schemes such as the exact preservation of the balance laws for certain steady state solutions, the non-oscillatory property for general solutions with discontinuities, and the genuine high order accuracy in smooth regions.

  3. Arbitrary-Lagrangian-Eulerian Discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes

    NASA Astrophysics Data System (ADS)

    Boscheri, Walter; Dumbser, Michael

    2017-10-01

    We present a new family of high order accurate fully discrete one-step Discontinuous Galerkin (DG) finite element schemes on moving unstructured meshes for the solution of nonlinear hyperbolic PDE in multiple space dimensions, which may also include parabolic terms in order to model dissipative transport processes, like molecular viscosity or heat conduction. High order piecewise polynomials of degree N are adopted to represent the discrete solution at each time level and within each spatial control volume of the computational grid, while high order of accuracy in time is achieved by the ADER approach, making use of an element-local space-time Galerkin finite element predictor. A novel nodal solver algorithm based on the HLL flux is derived to compute the velocity for each nodal degree of freedom that describes the current mesh geometry. In our algorithm the spatial mesh configuration can be defined in two different ways: either by an isoparametric approach that generates curved control volumes, or by a piecewise linear decomposition of each spatial control volume into simplex sub-elements. Each technique generates a corresponding number of geometrical degrees of freedom needed to describe the current mesh configuration and which must be considered by the nodal solver for determining the grid velocity. The connection of the old mesh configuration at time tn with the new one at time t n + 1 provides the space-time control volumes on which the governing equations have to be integrated in order to obtain the time evolution of the discrete solution. Our numerical method belongs to the category of so-called direct Arbitrary-Lagrangian-Eulerian (ALE) schemes, where a space-time conservation formulation of the governing PDE system is considered and which already takes into account the new grid geometry (including a possible rezoning step) directly during the computation of the numerical fluxes. We emphasize that our method is a moving mesh method, as opposed to total

  4. Discontinuous Galerkin particle-in-cell simulation of longitudinal plasma wave damping and comparison to the Landau approximation and the exact solution of the dispersion relation

    SciTech Connect

    Foust, F. R.; Bell, T. F.; Spasojevic, M.; Inan, U. S.

    2011-06-15

    We present results showing the measured Landau damping rate using a high-order discontinuous Galerkin particle-in-cell (DG-PIC) [G. B. Jacobs and J. S. Hesthaven, J. Comput. Phys. 214, 96 (2006)] method. We show that typical damping rates measured in particle-in-cell (PIC) simulations can differ significantly from the linearized Landau damping coefficient and propose a simple numerical method to solve the plasma dispersion function exactly for moderate to high damping rates. Simulation results show a high degree of agreement between the high-order PIC results and this calculated theoretical damping rate.

  5. Second order discontinuous Galerkin scheme for compound natural channels with movable bed. Applications for the computation of rating curves

    NASA Astrophysics Data System (ADS)

    Minatti, Lorenzo; De Cicco, Pina Nicoletta; Solari, Luca

    2016-07-01

    A new higher order 1D numerical scheme for the propagation of flood waves in compound channels with a movable bed is presented. The model equations are solved by means of an ADER Discontinuous Galerkin explicit scheme which can, in principle, reach any order of space-time accuracy. The higher order nature of the scheme allows the numerical coupling between flux and source terms appearing in the governing equations and, importantly, to handle moderately stiff and stiff source terms. Stiff source terms arise in the case of abrupt changes of river geometry such as in the case of hydraulic structures like bridges and weirs. Hydraulic interpretation of these conditions with 1D numerical modelling requires particular attention; for instance, a 1st order scheme might either lead to inaccurate solutions or impossibility to simulate these complex conditions. Validation is carried out with several test cases with the aim to check the scheme capability to deal with abrupt geometric changes and to capture the direction and celerity of propagation of bed and water surface disturbances. Validation is done also in a real case by using stage-discharge field measurements in the Ombrone river (Tuscany). The proposed scheme is further employed for the computation of flow rating curves in cross-sections just upstream of an abrupt narrowing, considering both fixed and movable bed conditions and different ratios of contraction for cross-section width. This problem is of particular relevance as, in common engineering practice, rating curves are derived from stage-measuring gauges installed on bridges with flow conditions that are likely to be influenced by local width narrowing. Results show that a higher order scheme is needed in order to deal with stiff source terms and reproduce realistic flow rating curves, unless a strong refinement of the computational grid is performed. This capability appears to be crucial for the computation of rating curves on coarse grids as it allows the

  6. A Parallel Icosahedral, Higher Order Discontinuous Galerkin, Global Shallow Water Model: Global Ocean Tides and Aquaplanet Benchmarks

    NASA Astrophysics Data System (ADS)

    Salehipour, H.; Stuhne, G.; Peltier, W. R.

    2012-12-01

    The development of models of the ocean tides with higher resolution near the coastlines and courser mesh offshore, has been required due to the significant impacts of coastline configuration and bathymetry (associated with sea level rise) on the amplitude and phase of tidal constituents, not only under present conditions but also in the deep past [Griffiths and Peltier GRL 2008, Griffiths and Peltier AMS 2009, Hill et al. JGR 2011]. A global tidal model with enhanced resolution at the poles has been developed by Griffiths and Peltier [2008, 2009], which, although capable of highly resolving polar ocean tides , is based upon a standard structured Arakawa C grid and hence is not capable of resolving coastlines locally. Furthermore the use of a nested modelling approach, although it may enable local spatial refinement [Hill et al. 2011], nevertheless suffers from its inherent dependence on the availability of a global tidal model with necessarily low spatial resolution to provide the open boundary conditions required for the local high resolution model. On the other hand, an unstructured triangulation of the global domain provides a standalone framework that may be employed to study highly resolved regions without relying on secondary models. The first step in the development of the structure we are employing was described in Stuhne and Peltier [Ocean Modeling, 2009]. In further extending this modelling structure we are employing a new discontinuous Galerkin (DG) discretization of the governing equations in order to provide very high order of accuracy while also ensuring that momentum transport is locally conserved [Giraldo et al. JCP 2002]. After validating the 2D shallow water model with several test suites appropriate to aquaplanets [Williamson et al. JCP 1992, Galewsky et al. Tellus 2004, Nair and Lauritzen JCP 2010], the governing equations are extended to include the influence of internal tide drag in the deep ocean as well as the drag in shallow marginal seas

  7. New central and central discontinuous Galerkin schemes on overlapping cells of unstructured grids for solving ideal magnetohydrodynamic equations with globally divergence-free magnetic field

    NASA Astrophysics Data System (ADS)

    Xu, Zhiliang; Liu, Yingjie

    2016-12-01

    New schemes are developed on triangular grids for solving ideal magnetohydrodynamic equations while preserving globally divergence-free magnetic field. These schemes incorporate the constrained transport (CT) scheme of Evans and Hawley [34] with central schemes and central discontinuous Galerkin methods on overlapping cells which have no need for solving Riemann problems across cell edges where there are discontinuities of the numerical solution. These schemes are formally second-order accurate with major development on the reconstruction of globally divergence-free magnetic field on polygonal dual mesh. Moreover, the computational cost is reduced by solving the complete set of governing equations on the primal grid while only solving the magnetic induction equation on the polygonal dual mesh. Various numerical experiments are provided to validate the new schemes.

  8. A limiter-based well-balanced discontinuous Galerkin method for shallow-water flows with wetting and drying: One-dimensional case

    NASA Astrophysics Data System (ADS)

    Vater, Stefan; Beisiegel, Nicole; Behrens, Jörn

    2015-11-01

    An important part in the numerical simulation of tsunami and storm surge events is the accurate modeling of flooding and the appearance of dry areas when the water recedes. This paper proposes a new algorithm to model inundation events with piecewise linear Runge-Kutta discontinuous Galerkin approximations applied to the shallow water equations. This study is restricted to the one-dimensional case and shows a detailed analysis and the corresponding numerical treatment of the inundation problem. The main feature is a velocity based "limiting" of the momentum distribution in each cell, which prevents instabilities in case of wetting or drying situations. Additional limiting of the fluid depth ensures its positivity while preserving local mass conservation. A special flux modification in cells located at the wet/dry interface leads to a well-balanced method, which maintains the steady state at rest. The discontinuous Galerkin scheme is formulated in a nodal form using a Lagrange basis. The proposed wetting and drying treatment is verified with several numerical simulations. These test cases demonstrate the well-balancing property of the method and its stability in case of rapid transition of the wet/dry interface. We also verify the conservation of mass and investigate the convergence characteristics of the scheme.

  9. One-sided Post-processing for the Discontinuous Galerkin Method Using ENO Type Stencil Choosing and the Local Edge Detection Method

    SciTech Connect

    Archibald, Richard K; Gelb, Anne; Gottlieb, Sigal; Ryan, Jennifer

    2006-01-01

    In a previous paper by Ryan and Shu [Ryan, J. K., and Shu, C.-W. (2003). Methods Appl. Anal. 10(2), 295-307], a one-sided post-processing technique for the discontinuous Galerkin method was introduced for reconstructing solutions near computational boundaries and discontinuities in the boundaries, as well as for changes in mesh size. This technique requires prior knowledge of the discontinuity location in order to determine whether to use centered, partially one-sided, or one-sided post-processing. We now present two alternative stencil choosing schemes to automate the choice of post-processing stencil. The first is an ENO type stencil choosing procedure, which is designed to choose centered post-processing in smooth regions and one-sided or partially one-sided post-processing near a discontinuity, and the second method is based on the edge detection method designed by Archibald, Gelb, and Yoon [Archibald, R., Gelb, A., and Yoon, J. (2005). SIAM J. Numeric. Anal. 43, 259-279; Archibald, R., Gelb, A., and Yoon, J. (2006). Appl. Numeric. Math. (submitted)]. We compare these stencil choosing techniques and analyze their respective strengths and weaknesses. Finally, the automated stencil choices are applied in conjunction with the appropriate post-processing procedures and it is determine that the resulting numerical solutions are of the correct order.

  10. A fully-coupled upwind discontinuous Galerkin method for incompressible porous media flows: High-order computations of viscous fingering instabilities in complex geometry

    NASA Astrophysics Data System (ADS)

    Scovazzi, G.; Huang, H.; Collis, S. S.; Yin, J.

    2013-11-01

    We present a new approach to the simulation of viscous fingering instabilities in incompressible, miscible displacement flows in porous media. In the past, high resolution computational simulations of viscous fingering instabilities have always been performed using high-order finite difference or Fourier-spectral methods which do not posses the flexibility to compute very complex subsurface geometries. Our approach, instead, by means of a fully-coupled nonlinear implementation of the discontinuous Galerkin method, possesses a fundamental differentiating feature, in that it maintains high-order accuracy on fully unstructured meshes. In addition, the proposed method shows very low sensitivity to mesh orientation, in contrast with classical finite volume approximation used in porous media flow simulations. The robustness and accuracy of the method are demonstrated in a number of challenging computational problems.

  11. Eigensolution Analysis of the Discontinuous Galerkin Method with Non-uniform Grids, Part I: One Space Dimension

    DTIC Science & Technology

    2001-12-01

    analytically that the numerical dispersion relation is accurate to order 2p + 2where p is the highest order of the basis polynomials. The results of...non-existent. For the physically accurate mode, it is shown analytically that the numerical dispersion relation is accurate to order 2p+ 2 where p is...Galerkin scheme employing basis polynomials up to order p , the rate of convergence is hp+1=2 in general and hp+1 in some special cases ([13,12,17,19

  12. A staggered space-time discontinuous Galerkin method for the three-dimensional incompressible Navier-Stokes equations on unstructured tetrahedral meshes

    NASA Astrophysics Data System (ADS)

    Tavelli, Maurizio; Dumbser, Michael

    2016-08-01

    In this paper we propose a novel arbitrary high order accurate semi-implicit space-time discontinuous Galerkin method for the solution of the three-dimensional incompressible Navier-Stokes equations on staggered unstructured curved tetrahedral meshes. As is typical for space-time DG schemes, the discrete solution is represented in terms of space-time basis functions. This allows to achieve very high order of accuracy also in time, which is not easy to obtain for the incompressible Navier-Stokes equations. Similarly to staggered finite difference schemes, in our approach the discrete pressure is defined on the primary tetrahedral grid, while the discrete velocity is defined on a face-based staggered dual grid. While staggered meshes are state of the art in classical finite difference schemes for the incompressible Navier-Stokes equations, their use in high order DG schemes is still quite rare. A very simple and efficient Picard iteration is used in order to derive a space-time pressure correction algorithm that achieves also high order of accuracy in time and that avoids the direct solution of global nonlinear systems. Formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse five-point block system for the scalar pressure, which is conveniently solved with a matrix-free GMRES algorithm. From numerical experiments we find that the linear system seems to be reasonably well conditioned, since all simulations shown in this paper could be run without the use of any preconditioner, even up to very high polynomial degrees. For a piecewise constant polynomial approximation in time and if pressure boundary conditions are specified at least in one point, the resulting system is, in addition, symmetric and positive definite. This allows us to use even faster iterative solvers, like the conjugate gradient method. The flexibility and accuracy of high order space-time DG methods on curved

  13. A comparative study of Rosenbrock-type and implicit Runge-Kutta time integration for discontinuous Galerkin method for unsteady 3D compressible Navier-Stokes equations

    DOE PAGES

    Liu, Xiaodong; Xia, Yidong; Luo, Hong; ...

    2016-10-05

    A comparative study of two classes of third-order implicit time integration schemes is presented for a third-order hierarchical WENO reconstructed discontinuous Galerkin (rDG) method to solve the 3D unsteady compressible Navier-Stokes equations: — 1) the explicit first stage, single diagonally implicit Runge-Kutta (ESDIRK3) scheme, and 2) the Rosenbrock-Wanner (ROW) schemes based on the differential algebraic equations (DAEs) of Index-2. Compared with the ESDIRK3 scheme, a remarkable feature of the ROW schemes is that, they only require one approximate Jacobian matrix calculation every time step, thus considerably reducing the overall computational cost. A variety of test cases, ranging from inviscid flowsmore » to DNS of turbulent flows, are presented to assess the performance of these schemes. Here, numerical experiments demonstrate that the third-order ROW scheme for the DAEs of index-2 can not only achieve the designed formal order of temporal convergence accuracy in a benchmark test, but also require significantly less computing time than its ESDIRK3 counterpart to converge to the same level of discretization errors in all of the flow simulations in this study, indicating that the ROW methods provide an attractive alternative for the higher-order time-accurate integration of the unsteady compressible Navier-Stokes equations.« less

  14. A non-linear discontinuous Petrov-Galerkin method for removing oscillations in the solution of the time-dependent transport equation

    SciTech Connect

    Merton, S. R.; Smedley-Stevenson, R. P.; Pain, C. C.

    2012-07-01

    This paper describes a Non-Linear Discontinuous Petrov-Galerkin method and its application to the one-speed Boltzmann Transport Equation (BTE) for space-time problems. The purpose of the method is to remove unwanted oscillations in the transport solution which occur in the vicinity of sharp flux gradients, while improving computational efficiency and numerical accuracy. This is achieved by applying artificial dissipation in the solution gradient direction, internal to an element using a novel finite element (FE) Riemann approach. The added dissipation is calculated at each node of the finite element mesh based on local behaviour of the transport solution on both the spatial and temporal axes of the problem. Thus a different dissipation is used in different elements. The magnitude of dissipation that is used is obtained from a gradient-informed scaling of the advection velocities in the stabilisation term. This makes the method in its most general form non-linear. The method is implemented within a very general finite element Riemann framework. This makes it completely independent of choice of angular basis function allowing one to use different descriptions of the angular variation. Results show the non-linear scheme performs consistently well in demanding time-dependent multi-dimensional neutron transport problems. (authors)

  15. Accelerating the discontinuous Galerkin method for seismic wave propagation simulations using the graphic processing unit (GPU)—single-GPU implementation

    NASA Astrophysics Data System (ADS)

    Mu, Dawei; Chen, Po; Wang, Liqiang

    2013-02-01

    We have successfully ported an arbitrary high-order discontinuous Galerkin (ADER-DG) method for solving the three-dimensional elastic seismic wave equation on unstructured tetrahedral meshes to an Nvidia Tesla C2075 GPU using the Nvidia CUDA programming model. On average our implementation obtained a speedup factor of about 24.3 for the single-precision version of our GPU code and a speedup factor of about 12.8 for the double-precision version of our GPU code when compared with the double precision serial CPU code running on one Intel Xeon W5880 core. When compared with the parallel CPU code running on two, four and eight cores, the speedup factor of our single-precision GPU code is around 12.9, 6.8 and 3.6, respectively. In this article, we give a brief summary of the ADER-DG method, a short introduction to the CUDA programming model and a description of our CUDA implementation and optimization of the ADER-DG method on the GPU. To our knowledge, this is the first study that explores the potential of accelerating the ADER-DG method for seismic wave-propagation simulations using a GPU.

  16. A novel discontinuous Galerkin time-domain method for ground-penetrating radar simulation with applications to the ASSESS-GPR test site

    NASA Astrophysics Data System (ADS)

    Fahlke, J.; Buchner, J.; Ippisch, O.; Roth, K.; Bastian, P.

    2012-04-01

    The simulation of ground-penetrating radar (GPR) measurements requires the solution of Maxwell's equations. While finite-differences time-domain (FDTD) solvers are faster on structured grids, finite-element time-domain (FETD) and discontinuous Galerkin time-domain (DGTD) allow to resolve complicated structures and avoid staircase approximations. Soil horizon boundaries can be resolved exactly by the finite element mesh. In this contribution 3D simulations are compared with measurements from the ASSESS-GPR test site which is an artificial GPR testbed with a well known geometry and ground-truth on volumetric water content provided by 32 TDR probes. For the simulations a DGTD method is used in a dual-field formulation and compared to a standard FETD method with conforming edge-based finite elements. The software for the simulation has been developed using the Distributed and Unified Numerics Environment (DUNE) and its PDELab discretization module. The programs have been parallelized using MPI to make computations on the size of 108 unknowns feasible.

  17. On the eddy-resolving capability of high-order discontinuous Galerkin approaches to implicit LES / under-resolved DNS of Euler turbulence

    NASA Astrophysics Data System (ADS)

    Moura, R. C.; Mengaldo, G.; Peiró, J.; Sherwin, S. J.

    2017-02-01

    We present estimates of spectral resolution power for under-resolved turbulent Euler flows obtained with high-order discontinuous Galerkin (DG) methods. The '1% rule' based on linear dispersion-diffusion analysis introduced by Moura et al. (2015) [10] is here adapted for 3D energy spectra and validated through the inviscid Taylor-Green vortex problem. The 1% rule estimates the wavenumber beyond which numerical diffusion induces an artificial dissipation range on measured energy spectra. As the original rule relies on standard upwinding, different Riemann solvers are tested. Very good agreement is found for solvers which treat the different physical waves in a consistent manner. Relatively good agreement is still found for simpler solvers. The latter however displayed spurious features attributed to the inconsistent treatment of different physical waves. It is argued that, in the limit of vanishing viscosity, such features might have a significant impact on robustness and solution quality. The estimates proposed are regarded as useful guidelines for no-model DG-based simulations of free turbulence at very high Reynolds numbers.

  18. Jacobian-Free Newton-Krylov Discontinuous Galerkin (JFNK-DG) Method and Its Physics-Based Preconditioning for All-Speed Flows

    NASA Astrophysics Data System (ADS)

    Park, Hyeongkae; Nourgaliev, Robert; Knoll, Dana

    2007-11-01

    The Discontinuous Galerkin (DG) method for compressible fluid flows is incorporated into the Jacobian-Free Newton-Krylov (JFNK) framework. Advantages of combining the DG with the JFNK are two-fold: a) enabling robust and efficient high-order-accurate modeling of all-speed flows on unstructured grids, opening the possibility for high-fidelity simulation of nuclear-power-industry-relevant flows; and b) ability to tightly, robustly and high-order-accurately couple with other relevant physics (neutronics, thermal-structural response of solids, etc.). In the present study, we focus on the physics-based preconditioning (PBP) of the Krylov method (GMRES), used as the linear solver in our implicit higher-order-accurate Runge-Kutta (ESDIRK) time discretization scheme; exploiting the compactness of the spatial discretization of the DG family. In particular, we utilize the Implicit Continuous-fluid Eulerian (ICE) method and investigate its efficacy as the PBP within the JFNK-DG method. Using the eigenvalue analysis, it is found that the ICE collapses the complex components of the all eigenvalues of the Jacobian matrix (associated with pressure waves) onto the real axis, and thereby enabling at least an order of magnitude faster simulations in nearly-incompressible/weakly-compressible regimes with a significant storage saving.

  19. A comparative study of Rosenbrock-type and implicit Runge-Kutta time integration for discontinuous Galerkin method for unsteady 3D compressible Navier-Stokes equations

    SciTech Connect

    Liu, Xiaodong; Xia, Yidong; Luo, Hong; Xuan, Lijun

    2016-10-05

    A comparative study of two classes of third-order implicit time integration schemes is presented for a third-order hierarchical WENO reconstructed discontinuous Galerkin (rDG) method to solve the 3D unsteady compressible Navier-Stokes equations: — 1) the explicit first stage, single diagonally implicit Runge-Kutta (ESDIRK3) scheme, and 2) the Rosenbrock-Wanner (ROW) schemes based on the differential algebraic equations (DAEs) of Index-2. Compared with the ESDIRK3 scheme, a remarkable feature of the ROW schemes is that, they only require one approximate Jacobian matrix calculation every time step, thus considerably reducing the overall computational cost. A variety of test cases, ranging from inviscid flows to DNS of turbulent flows, are presented to assess the performance of these schemes. Here, numerical experiments demonstrate that the third-order ROW scheme for the DAEs of index-2 can not only achieve the designed formal order of temporal convergence accuracy in a benchmark test, but also require significantly less computing time than its ESDIRK3 counterpart to converge to the same level of discretization errors in all of the flow simulations in this study, indicating that the ROW methods provide an attractive alternative for the higher-order time-accurate integration of the unsteady compressible Navier-Stokes equations.

  20. A discontinuous Galerkin method with a bound preserving limiter for the advection of non-diffusive fields in solid Earth geodynamics

    NASA Astrophysics Data System (ADS)

    He, Ying; Puckett, Elbridge Gerry; Billen, Magali I.

    2017-02-01

    Mineral composition has a strong effect on the properties of rocks and is an essentially non-diffusive property in the context of large-scale mantle convection. Due to the non-diffusive nature and the origin of compositionally distinct regions in the Earth the boundaries between distinct regions can be nearly discontinuous. While there are different methods for tracking rock composition in numerical simulations of mantle convection, one must consider trade-offs between computational cost, accuracy or ease of implementation when choosing an appropriate method. Existing methods can be computationally expensive, cause over-/undershoots, smear sharp boundaries, or are not easily adapted to tracking multiple compositional fields. Here we present a Discontinuous Galerkin method with a bound preserving limiter (abbreviated as DG-BP) using a second order Runge-Kutta, strong stability-preserving time discretization method for the advection of non-diffusive fields. First, we show that the method is bound-preserving for a point-wise divergence free flow (e.g., a prescribed circular flow in a box). However, using standard adaptive mesh refinement (AMR) there is an over-shoot error (2%) because the cell average is not preserved during mesh coarsening. The effectiveness of the algorithm for convection-dominated flows is demonstrated using the falling box problem. We find that the DG-BP method maintains sharper compositional boundaries (3-5 elements) as compared to an artificial entropy-viscosity method (6-15 elements), although the over-/undershoot errors are similar. When used with AMR the DG-BP method results in fewer degrees of freedom due to smaller regions of mesh refinement in the neighborhood of the discontinuity. However, using Taylor-Hood elements and a uniform mesh there is an over-/undershoot error on the order of 0.0001%, but this error increases to 0.01-0.10% when using AMR. Therefore, for research problems in which a continuous field method is desired the DG

  1. Experiences on p-Version Time-Discontinuous Galerkin's Method for Nonlinear Heat Transfer Analysis and Sensitivity Analysis

    NASA Technical Reports Server (NTRS)

    Hou, Gene

    2004-01-01

    The focus of this research is on the development of analysis and sensitivity analysis equations for nonlinear, transient heat transfer problems modeled by p-version, time discontinuous finite element approximation. The resulting matrix equation of the state equation is simply in the form ofA(x)x = c, representing a single step, time marching scheme. The Newton-Raphson's method is used to solve the nonlinear equation. Examples are first provided to demonstrate the accuracy characteristics of the resultant finite element approximation. A direct differentiation approach is then used to compute the thermal sensitivities of a nonlinear heat transfer problem. The report shows that only minimal coding effort is required to enhance the analysis code with the sensitivity analysis capability.

  2. A posteriori error estimates for continuous/discontinuous Galerkin approximations of the Kirchhoff-Love buckling problem

    NASA Astrophysics Data System (ADS)

    Hansbo, Peter; Larson, Mats G.

    2015-11-01

    Second order buckling theory involves a one-way coupled coupled problem where the stress tensor from a plane stress problem appears in an eigenvalue problem for the fourth order Kirchhoff plate. In this paper we present an a posteriori error estimate for the critical buckling load and mode corresponding to the smallest eigenvalue and associated eigenvector. A particular feature of the analysis is that we take the effect of approximate computation of the stress tensor and also provide an error indicator for the plane stress problem. The Kirchhoff plate is discretized using a continuous/discontinuous finite element method based on standard continuous piecewise polynomial finite element spaces. The same finite element spaces can be used to solve the plane stress problem.

  3. High-order Discontinuous Element-based Schemes for the Inviscid Shallow Water Equations: Spectral Multidomain Penalty and Discontinuous Galerkin Methods

    DTIC Science & Technology

    2011-07-19

    Master’s thesis, Naval Postgraduate School, 2009. [35] P. K. Kundu , I. M. Cohen, Fluid Mechanics , Academic Press, San Diego, 2004. [36] J. D. Anderson...earth’s rotation, ambient stratification and the constraining effects of lateral and vertical boundaries, flow processes in geophysical fluids commonly...topography inter- actions and other mechanisms , can be highly localized in space and time and span a broad range of scales within their region of occurrence

  4. Linear dispersion-diffusion analysis and its application to under-resolved turbulence simulations using discontinuous Galerkin spectral/hp methods

    NASA Astrophysics Data System (ADS)

    Moura, R. C.; Sherwin, S. J.; Peiró, J.

    2015-10-01

    We investigate the potential of linear dispersion-diffusion analysis in providing direct guidelines for turbulence simulations through the under-resolved DNS (sometimes called implicit LES) approach via spectral/hp methods. The discontinuous Galerkin (DG) formulation is assessed in particular as a representative of these methods. We revisit the eigensolutions technique as applied to linear advection and suggest a new perspective to the role of multiple numerical modes, peculiar to spectral/hp methods. From this new perspective, "secondary" eigenmodes are seen to replicate the propagation behaviour of a "primary" mode, so that DG's propagation characteristics can be obtained directly from the dispersion-diffusion curves of the primary mode. Numerical dissipation is then appraised from these primary eigencurves and its effect over poorly-resolved scales is quantified. Within this scenario, a simple criterion is proposed to estimate DG's effective resolution in terms of the largest wavenumber it can accurately resolve in a given hp approximation space, also allowing us to present points per wavelength estimates typically used in spectral and finite difference methods. Although strictly valid for linear advection, the devised criterion is tested against (1D) Burgers turbulence and found to predict with good accuracy the beginning of the dissipation range on the energy spectra of under-resolved simulations. The analysis of these test cases through the proposed methodology clarifies why and how the DG formulation can be used for under-resolved turbulence simulations without explicit subgrid-scale modelling. In particular, when dealing with communication limited hardware which forces one to consider the performance for a fixed number of degrees of freedom, the use of higher polynomial orders along with moderately coarser meshes is shown to be the best way to translate available degrees of freedom into resolution power.

  5. A simple robust and accurate a posteriori sub-cell finite volume limiter for the discontinuous Galerkin method on unstructured meshes

    NASA Astrophysics Data System (ADS)

    Dumbser, Michael; Loubère, Raphaël

    2016-08-01

    In this paper we propose a simple, robust and accurate nonlinear a posteriori stabilization of the Discontinuous Galerkin (DG) finite element method for the solution of nonlinear hyperbolic PDE systems on unstructured triangular and tetrahedral meshes in two and three space dimensions. This novel a posteriori limiter, which has been recently proposed for the simple Cartesian grid case in [62], is able to resolve discontinuities at a sub-grid scale and is substantially extended here to general unstructured simplex meshes in 2D and 3D. It can be summarized as follows: At the beginning of each time step, an approximation of the local minimum and maximum of the discrete solution is computed for each cell, taking into account also the vertex neighbors of an element. Then, an unlimited discontinuous Galerkin scheme of approximation degree N is run for one time step to produce a so-called candidate solution. Subsequently, an a posteriori detection step checks the unlimited candidate solution at time t n + 1 for positivity, absence of floating point errors and whether the discrete solution has remained within or at least very close to the bounds given by the local minimum and maximum computed in the first step. Elements that do not satisfy all the previously mentioned detection criteria are flagged as troubled cells. For these troubled cells, the candidate solution is discarded as inappropriate and consequently needs to be recomputed. Within these troubled cells the old discrete solution at the previous time tn is scattered onto small sub-cells (Ns = 2 N + 1 sub-cells per element edge), in order to obtain a set of sub-cell averages at time tn. Then, a more robust second order TVD finite volume scheme is applied to update the sub-cell averages within the troubled DG cells from time tn to time t n + 1. The new sub-grid data at time t n + 1 are finally gathered back into a valid cell-centered DG polynomial of degree N by using a classical conservative and higher order

  6. Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid-structure interaction, and free surface flow: Part I

    NASA Astrophysics Data System (ADS)

    Saye, Robert

    2017-09-01

    In this two-part paper, a high-order accurate implicit mesh discontinuous Galerkin (dG) framework is developed for fluid interface dynamics, facilitating precise computation of interfacial fluid flow in evolving geometries. The framework uses implicitly defined meshes-wherein a reference quadtree or octree grid is combined with an implicit representation of evolving interfaces and moving domain boundaries-and allows physically prescribed interfacial jump conditions to be imposed or captured with high-order accuracy. Part one discusses the design of the framework, including: (i) high-order quadrature for implicitly defined elements and faces; (ii) high-order accurate discretisation of scalar and vector-valued elliptic partial differential equations with interfacial jumps in ellipticity coefficient, leading to optimal-order accuracy in the maximum norm and discrete linear systems that are symmetric positive (semi)definite; (iii) the design of incompressible fluid flow projection operators, which except for the influence of small penalty parameters, are discretely idempotent; and (iv) the design of geometric multigrid methods for elliptic interface problems on implicitly defined meshes and their use as preconditioners for the conjugate gradient method. Also discussed is a variety of aspects relating to moving interfaces, including: (v) dG discretisations of the level set method on implicitly defined meshes; (vi) transferring state between evolving implicit meshes; (vii) preserving mesh topology to accurately compute temporal derivatives; (viii) high-order accurate reinitialisation of level set functions; and (ix) the integration of adaptive mesh refinement. In part two, several applications of the implicit mesh dG framework in two and three dimensions are presented, including examples of single phase flow in nontrivial geometry, surface tension-driven two phase flow with phase-dependent fluid density and viscosity, rigid body fluid-structure interaction, and free

  7. Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid-structure interaction, and free surface flow: Part II

    NASA Astrophysics Data System (ADS)

    Saye, Robert

    2017-09-01

    In this two-part paper, a high-order accurate implicit mesh discontinuous Galerkin (dG) framework is developed for fluid interface dynamics, facilitating precise computation of interfacial fluid flow in evolving geometries. The framework uses implicitly defined meshes-wherein a reference quadtree or octree grid is combined with an implicit representation of evolving interfaces and moving domain boundaries-and allows physically prescribed interfacial jump conditions to be imposed or captured with high-order accuracy. Part one discusses the design of the framework, including: (i) high-order quadrature for implicitly defined elements and faces; (ii) high-order accurate discretisation of scalar and vector-valued elliptic partial differential equations with interfacial jumps in ellipticity coefficient, leading to optimal-order accuracy in the maximum norm and discrete linear systems that are symmetric positive (semi)definite; (iii) the design of incompressible fluid flow projection operators, which except for the influence of small penalty parameters, are discretely idempotent; and (iv) the design of geometric multigrid methods for elliptic interface problems on implicitly defined meshes and their use as preconditioners for the conjugate gradient method. Also discussed is a variety of aspects relating to moving interfaces, including: (v) dG discretisations of the level set method on implicitly defined meshes; (vi) transferring state between evolving implicit meshes; (vii) preserving mesh topology to accurately compute temporal derivatives; (viii) high-order accurate reinitialisation of level set functions; and (ix) the integration of adaptive mesh refinement. In part two, several applications of the implicit mesh dG framework in two and three dimensions are presented, including examples of single phase flow in nontrivial geometry, surface tension-driven two phase flow with phase-dependent fluid density and viscosity, rigid body fluid-structure interaction, and free

  8. Extending the Riemann-Solver-Free High-Order Space-Time Discontinuous Galerkin Cell Vertex Scheme (DG-CVS) to Solve Compressible Magnetohydrodynamics EquationsEquations

    DTIC Science & Technology

    2016-06-08

    non-physical overshoots and undershoots near discontinuities (left column in Fig. 8). These overshoots/undershoots can be suppressed (right column in...Solutions of Brio-Wu Case 2. B x = 0.75. T = 0.2. Left column : unlimited solutions. Right column : limited solutions. 12 DISTRIBUTION A: Distribution...292. [25] Gardiner, T. A. and Stone , J. M., “An unsplit Godunov method for ideal MHD via constrained transport in three dimensions,” J. Comput. Phys

  9. 3-D finite-difference, finite-element, discontinuous-Galerkin and spectral-element schemes analysed for their accuracy with respect to P-wave to S-wave speed ratio

    NASA Astrophysics Data System (ADS)

    Moczo, Peter; Kristek, Jozef; Galis, Martin; Chaljub, Emmanuel; Etienne, Vincent

    2011-12-01

    We analyse 13 3-D numerical time-domain explicit schemes for modelling seismic wave propagation and earthquake motion for their behaviour with a varying P-wave to S-wave speed ratio (VP/VS). The second-order schemes include three finite-difference, three finite-element and one discontinuous-Galerkin schemes. The fourth-order schemes include three finite-difference and two spectral-element schemes. All schemes are second-order in time. We assume a uniform cubic grid/mesh and present all schemes in a unified form. We assume plane S-wave propagation in an unbounded homogeneous isotropic elastic medium. We define relative local errors of the schemes in amplitude and the vector difference in one time step and normalize them for a unit time. We also define the equivalent spatial sampling ratio as a ratio at which the maximum relative error is equal to the reference maximum error. We present results of the extensive numerical analysis. We theoretically (i) show how a numerical scheme sees the P and S waves if the VP/VS ratio increases, (ii) show the structure of the errors in amplitude and the vector difference and (iii) compare the schemes in terms of the truncation errors of the discrete approximations to the second mixed and non-mixed spatial derivatives. We find that four of the tested schemes have errors in amplitude almost independent on the VP/VS ratio. The homogeneity of the approximations to the second mixed and non-mixed spatial derivatives in terms of the coefficients of the leading terms of their truncation errors as well as the absolute values of the coefficients are key factors for the behaviour of the schemes with increasing VP/VS ratio. The dependence of the errors in the vector difference on the VP/VS ratio should be accounted for by a proper (sufficiently dense) spatial sampling.

  10. Discontinuous Galerkin Dynamical Core in HOMME

    SciTech Connect

    Nair, R. D.; Tufo, Henry

    2012-08-14

    Atmospheric numerical modeling has been going through radical changes over the past decade. One major reason for this trend is due to the recent paradigm change in scientific computing , triggered by the arrival of petascale computing resources with core counts in the tens of thousands to hundreds of thousands range. Modern atmospheric modelers must adapt grid systems and numerical algorithms to facilitate an unprecedented levels of scalability on these modern highly parallel computer architectures. The numerical algorithms which can address these challenges should have the local properties such as high on-processor floating-point operation count to bytes moved and minimum parallel communication overhead.

  11. A pressure-based semi-implicit space-time discontinuous Galerkin method on staggered unstructured meshes for the solution of the compressible Navier-Stokes equations at all Mach numbers

    NASA Astrophysics Data System (ADS)

    Tavelli, Maurizio; Dumbser, Michael

    2017-07-01

    We propose a new arbitrary high order accurate semi-implicit space-time discontinuous Galerkin (DG) method for the solution of the two and three dimensional compressible Euler and Navier-Stokes equations on staggered unstructured curved meshes. The method is pressure-based and semi-implicit and is able to deal with all Mach number flows. The new DG scheme extends the seminal ideas outlined in [1], where a second order semi-implicit finite volume method for the solution of the compressible Navier-Stokes equations with a general equation of state was introduced on staggered Cartesian grids. Regarding the high order extension we follow [2], where a staggered space-time DG scheme for the incompressible Navier-Stokes equations was presented. In our scheme, the discrete pressure is defined on the primal grid, while the discrete velocity field and the density are defined on a face-based staggered dual grid. Then, the mass conservation equation, as well as the nonlinear convective terms in the momentum equation and the transport of kinetic energy in the energy equation are discretized explicitly, while the pressure terms appearing in the momentum and energy equation are discretized implicitly. Formal substitution of the discrete momentum equation into the total energy conservation equation yields a linear system for only one unknown, namely the scalar pressure. Here the equation of state is assumed linear with respect to the pressure. The enthalpy and the kinetic energy are taken explicitly and are then updated using a simple Picard procedure. Thanks to the use of a staggered grid, the final pressure system is a very sparse block five-point system for three dimensional problems and it is a block four-point system in the two dimensional case. Furthermore, for high order in space and piecewise constant polynomials in time, the system is observed to be symmetric and positive definite. This allows to use fast linear solvers such as the conjugate gradient (CG) method. In

  12. A stable interface element scheme for the p-adaptive lifting collocation penalty formulation

    NASA Astrophysics Data System (ADS)

    Cagnone, J. S.; Nadarajah, S. K.

    2012-02-01

    This paper presents a procedure for adaptive polynomial refinement in the context of the lifting collocation penalty (LCP) formulation. The LCP scheme is a high-order unstructured discretization method unifying the discontinuous Galerkin, spectral volume, and spectral difference schemes in single differential formulation. Due to the differential nature of the scheme, the treatment of inter-cell fluxes for spatially varying polynomial approximations is not straightforward. Specially designed elements are proposed to tackle non-conforming polynomial approximations. These elements are constructed such that a conforming interface between polynomial approximations of different degrees is recovered. The stability and conservation properties of the scheme are analyzed and various inviscid compressible flow calculations are performed to demonstrate the potential of the proposed approach.

  13. Collocation and Galerkin Time-Stepping Methods

    NASA Technical Reports Server (NTRS)

    Huynh, H. T.

    2011-01-01

    We study the numerical solutions of ordinary differential equations by one-step methods where the solution at tn is known and that at t(sub n+1) is to be calculated. The approaches employed are collocation, continuous Galerkin (CG) and discontinuous Galerkin (DG). Relations among these three approaches are established. A quadrature formula using s evaluation points is employed for the Galerkin formulations. We show that with such a quadrature, the CG method is identical to the collocation method using quadrature points as collocation points. Furthermore, if the quadrature formula is the right Radau one (including t(sub n+1)), then the DG and CG methods also become identical, and they reduce to the Radau IIA collocation method. In addition, we present a generalization of DG that yields a method identical to CG and collocation with arbitrary collocation points. Thus, the collocation, CG, and generalized DG methods are equivalent, and the latter two methods can be formulated using the differential instead of integral equation. Finally, all schemes discussed can be cast as s-stage implicit Runge-Kutta methods.

  14. A partially penalty immersed Crouzeix-Raviart finite element method for interface problems.

    PubMed

    An, Na; Yu, Xijun; Chen, Huanzhen; Huang, Chaobao; Liu, Zhongyan

    2017-01-01

    The elliptic equations with discontinuous coefficients are often used to describe the problems of the multiple materials or fluids with different densities or conductivities or diffusivities. In this paper we develop a partially penalty immersed finite element (PIFE) method on triangular grids for anisotropic flow models, in which the diffusion coefficient is a piecewise definite-positive matrix. The standard linear Crouzeix-Raviart type finite element space is used on non-interface elements and the piecewise linear Crouzeix-Raviart type immersed finite element (IFE) space is constructed on interface elements. The piecewise linear functions satisfying the interface jump conditions are uniquely determined by the integral averages on the edges as degrees of freedom. The PIFE scheme is given based on the symmetric, nonsymmetric or incomplete interior penalty discontinuous Galerkin formulation. The solvability of the method is proved and the optimal error estimates in the energy norm are obtained. Numerical experiments are presented to confirm our theoretical analysis and show that the newly developed PIFE method has optimal-order convergence in the [Formula: see text] norm as well. In addition, numerical examples also indicate that this method is valid for both the isotropic and the anisotropic elliptic interface problems.

  15. Meshless Local Petrov-Galerkin Method for Bending Problems

    NASA Technical Reports Server (NTRS)

    Phillips, Dawn R.; Raju, Ivatury S.

    2002-01-01

    Recent literature shows extensive research work on meshless or element-free methods as alternatives to the versatile Finite Element Method. One such meshless method is the Meshless Local Petrov-Galerkin (MLPG) method. In this report, the method is developed for bending of beams - C1 problems. A generalized moving least squares (GMLS) interpolation is used to construct the trial functions, and spline and power weight functions are used as the test functions. The method is applied to problems for which exact solutions are available to evaluate its effectiveness. The accuracy of the method is demonstrated for problems with load discontinuities and continuous beam problems. A Petrov-Galerkin implementation of the method is shown to greatly reduce computational time and effort and is thus preferable over the previously developed Galerkin approach. The MLPG method for beam problems yields very accurate deflections and slopes and continuous moment and shear forces without the need for elaborate post-processing techniques.

  16. Time-stepping stability of continuous and discontinuous finite-element methods for 3-D wave propagation

    NASA Astrophysics Data System (ADS)

    Mulder, W. A.; Zhebel, E.; Minisini, S.

    2014-02-01

    We analyse the time-stepping stability for the 3-D acoustic wave equation, discretized on tetrahedral meshes. Two types of methods are considered: mass-lumped continuous finite elements and the symmetric interior-penalty discontinuous Galerkin method. Combining the spatial discretization with the leap-frog time-stepping scheme, which is second-order accurate and conditionally stable, leads to a fully explicit scheme. We provide estimates of its stability limit for simple cases, namely, the reference element with Neumann boundary conditions, its distorted version of arbitrary shape, the unit cube that can be partitioned into six tetrahedra with periodic boundary conditions and its distortions. The Courant-Friedrichs-Lewy stability limit contains an element diameter for which we considered different options. The one based on the sum of the eigenvalues of the spatial operator for the first-degree mass-lumped element gives the best results. It resembles the diameter of the inscribed sphere but is slightly easier to compute. The stability estimates show that the mass-lumped continuous and the discontinuous Galerkin finite elements of degree 2 have comparable stability conditions, whereas the mass-lumped elements of degree one and three allow for larger time steps.

  17. Software for the parallel adaptive solution of conservation laws by discontinous Galerkin methods.

    SciTech Connect

    Flaherty, J. E.; Loy, R. M.; Shephard, M. S.; Teresco, J. D.

    1999-08-17

    The authors develop software tools for the solution of conservation laws using parallel adaptive discontinuous Galerkin methods. In particular, the Rensselaer Partition Model (RPM) provides parallel mesh structures within an adaptive framework to solve the Euler equations of compressible flow by a discontinuous Galerkin method (LOCO). Results are presented for a Rayleigh-Taylor flow instability for computations performed on 128 processors of an IBM SP computer. In addition to managing the distributed data and maintaining a load balance, RPM provides information about the parallel environment that can be used to tailor partitions to a specific computational environment.

  18. MIB Galerkin method for elliptic interface problems

    PubMed Central

    Xia, Kelin; Zhan, Meng; Wei, Guo-Wei

    2014-01-01

    Summary Material interfaces are omnipresent in the real-world structures and devices. Mathematical modeling of material interfaces often leads to elliptic partial differential equations (PDEs) with discontinuous coefficients and singular sources, which are commonly called elliptic interface problems. The development of high-order numerical schemes for elliptic interface problems has become a well defined field in applied and computational mathematics and attracted much attention in the past decades. Despite of significant advances, challenges remain in the construction of high-order schemes for nonsmooth interfaces, i.e., interfaces with geometric singularities, such as tips, cusps and sharp edges. The challenge of geometric singularities is amplified when they are associated with low solution regularities, e.g., tip-geometry effects in many fields. The present work introduces a matched interface and boundary (MIB) Galerkin method for solving two-dimensional (2D) elliptic PDEs with complex interfaces, geometric singularities and low solution regularities. The Cartesian grid based triangular elements are employed to avoid the time consuming mesh generation procedure. Consequently, the interface cuts through elements. To ensure the continuity of classic basis functions across the interface, two sets of overlapping elements, called MIB elements, are defined near the interface. As a result, differentiation can be computed near the interface as if there is no interface. Interpolation functions are constructed on MIB element spaces to smoothly extend function values across the interface. A set of lowest order interface jump conditions is enforced on the interface, which in turn, determines the interpolation functions. The performance of the proposed MIB Galerkin finite element method is validated by numerical experiments with a wide range of interface geometries, geometric singularities, low regularity solutions and grid resolutions. Extensive numerical studies confirm

  19. MIB Galerkin method for elliptic interface problems.

    PubMed

    Xia, Kelin; Zhan, Meng; Wei, Guo-Wei

    2014-12-15

    Material interfaces are omnipresent in the real-world structures and devices. Mathematical modeling of material interfaces often leads to elliptic partial differential equations (PDEs) with discontinuous coefficients and singular sources, which are commonly called elliptic interface problems. The development of high-order numerical schemes for elliptic interface problems has become a well defined field in applied and computational mathematics and attracted much attention in the past decades. Despite of significant advances, challenges remain in the construction of high-order schemes for nonsmooth interfaces, i.e., interfaces with geometric singularities, such as tips, cusps and sharp edges. The challenge of geometric singularities is amplified when they are associated with low solution regularities, e.g., tip-geometry effects in many fields. The present work introduces a matched interface and boundary (MIB) Galerkin method for solving two-dimensional (2D) elliptic PDEs with complex interfaces, geometric singularities and low solution regularities. The Cartesian grid based triangular elements are employed to avoid the time consuming mesh generation procedure. Consequently, the interface cuts through elements. To ensure the continuity of classic basis functions across the interface, two sets of overlapping elements, called MIB elements, are defined near the interface. As a result, differentiation can be computed near the interface as if there is no interface. Interpolation functions are constructed on MIB element spaces to smoothly extend function values across the interface. A set of lowest order interface jump conditions is enforced on the interface, which in turn, determines the interpolation functions. The performance of the proposed MIB Galerkin finite element method is validated by numerical experiments with a wide range of interface geometries, geometric singularities, low regularity solutions and grid resolutions. Extensive numerical studies confirm the

  20. Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems

    NASA Astrophysics Data System (ADS)

    Tryoen, J.; Le Maître, O.; Ndjinga, M.; Ern, A.

    2010-09-01

    This paper deals with stochastic spectral methods for uncertainty propagation and quantification in nonlinear hyperbolic systems of conservation laws. We consider problems with parametric uncertainty in initial conditions and model coefficients, whose solutions exhibit discontinuities in the spatial as well as in the stochastic variables. The stochastic spectral method relies on multi-resolution schemes where the stochastic domain is discretized using tensor-product stochastic elements supporting local polynomial bases. A Galerkin projection is used to derive a system of deterministic equations for the stochastic modes of the solution. Hyperbolicity of the resulting Galerkin system is analyzed. A finite volume scheme with a Roe-type solver is used for discretization of the spatial and time variables. An original technique is introduced for the fast evaluation of approximate upwind matrices, which is particularly well adapted to local polynomial bases. Efficiency and robustness of the overall method are assessed on the Burgers and Euler equations with shocks.

  1. Some recent developments of the discontinous Galerkin method for time-domain electromagnetics

    NASA Astrophysics Data System (ADS)

    Durochat, C.; Lanteri, S.; Moya, L.; Viquerat, J.; Descombes, S.; Scheid, C.

    2012-09-01

    We discuss about recent developments aiming at improving the accuracy, the flexibility and the computational performances of a high order discontinuous Galerkin time domain (DGTD) method for the simulation of time-domain electromagnetic wave propagation problems involving irregularly shaped objects and complex propagation media.

  2. An Adaptive Discontinuous Galerkin Method for Modeling Atmospheric Convection (Preprint)

    DTIC Science & Technology

    2011-04-13

    Giraldo and Volkmar Wirth 5 SENSITIVITY STUDIES One important question for each adaptive numerical model is: how accurate is the adaptive method? For...this criterion that is used later for some sensitivity studies . These studies include a comparison between a simulation on an adaptive mesh with a...simulation on a uniform mesh and a sensitivity study concerning the size of the refinement region. 5.1 Comparison Criterion For comparing different

  3. Visualization of Discontinuous Galerkin Based High-Order Methods

    DTIC Science & Technology

    2015-08-19

    Received Paper 2.00 3.00 4.00 Tiago Etiene, Daniel Jonsson, Timo Ropinski, Carlos Scheidegger, Joao L. D. Comba , Luis Gustavo Nonato, Robert M. Kirby...Visualization and Computer Graphics, Vol. 20, No. 1, pages 70-83, 2014. Tiago Etiene, Daniel Jonsson, Timo Ropinski, Carlos Scheidegger, Joao Comba , L...Vol. 20, No. 1, pages 70-83, 2014. Tiago Etiene, Daniel Jonsson, Timo Ropinski, Carlos Scheidegger, Joao Comba , L. Gustavo Nonato, Robert M. Kirby

  4. Adaptive enriched Galerkin methods for miscible displacement problems with entropy residual stabilization

    NASA Astrophysics Data System (ADS)

    Lee, Sanghyun; Wheeler, Mary F.

    2017-02-01

    We present a novel approach to the simulation of miscible displacement by employing adaptive enriched Galerkin finite element methods (EG) coupled with entropy residual stabilization for transport. In particular, numerical simulations of viscous fingering instabilities in heterogeneous porous media and Hele-Shaw cells are illustrated. EG is formulated by enriching the conforming continuous Galerkin finite element method (CG) with piecewise constant functions. The method provides locally and globally conservative fluxes, which are crucial for coupled flow and transport problems. Moreover, EG has fewer degrees of freedom in comparison with discontinuous Galerkin (DG) and an efficient flow solver has been derived which allows for higher order schemes. Dynamic adaptive mesh refinement is applied in order to reduce computational costs for large-scale three dimensional applications. In addition, entropy residual based stabilization for high order EG transport systems prevents spurious oscillations. Numerical tests are presented to show the capabilities of EG applied to flow and transport.

  5. Application of the Galerkin/least-squares formulation to the analysis of hypersonic flows. II - Flow past a double ellipse

    NASA Technical Reports Server (NTRS)

    Chalot, F.; Hughes, T. J. R.; Johan, Z.; Shakib, F.

    1991-01-01

    A finite element method for the compressible Navier-Stokes equations is introduced. The discretization is based on entropy variables. The methodology is developed within the framework of a Galerkin/least-squares formulation to which a discontinuity-capturing operator is added. Results for four test cases selected among those of the Workshop on Hypersonic Flows for Reentry Problems are presented.

  6. Discontinuous Spectral Difference Method for Conservation Laws on Unstructured Grids

    NASA Technical Reports Server (NTRS)

    Liu, Yen; Vinokur, Marcel; Wang, Z. J.

    2004-01-01

    A new, high-order, conservative, and efficient method for conservation laws on unstructured grids is developed. The concept of discontinuous and high-order local representations to achieve conservation and high accuracy is utilized in a manner similar to the Discontinuous Galerkin (DG) and the Spectral Volume (SV) methods, but while these methods are based on the integrated forms of the equations, the new method is based on the differential form to attain a simpler formulation and higher efficiency. A discussion on the Discontinuous Spectral Difference (SD) Method, locations of the unknowns and flux points and numerical results are also presented.

  7. Wavelet=Galerkin discretization of hyperbolic equations

    SciTech Connect

    Restrepo, J.M.; Leaf, G.K.

    1994-12-31

    The relative merits of the wavelet-Galerkin solution of hyperbolic partial differential equations, typical of geophysical problems, are quantitatively and qualitatively compared to traditional finite difference and Fourier-pseudo-spectral methods. The wavelet-Galerkin solution presented here is found to be a viable alternative to the two conventional techniques.

  8. Galerkin approximations for dissipative magnetohydrodynamics

    NASA Technical Reports Server (NTRS)

    Chen, Hudong; Shan, Xiaowen; Montgomery, David

    1990-01-01

    A Galerkin approximation scheme is proposed for voltage-driven, dissipative magnetohydrodynamics. The trial functions are exact eigenfunctions of the linearized continuum equations and represent helical deformations of the axisymmetric, zero-flow, driven steady state. The lowest nontrivial truncation is explored: one axisymmetric trial function and one helical trial function each for the magnetic and velocity fields. The system resembles the Lorenz approximation to Benard convection, but in the region of believed applicability, its dynamical behavior is rather different, including relaxation to a helically deformed state similar to those that have emerged in the much higher resolution computations of Dahlburg et al.

  9. Numerical and experimental validation of a particle Galerkin method for metal grinding simulation

    NASA Astrophysics Data System (ADS)

    Wu, C. T.; Bui, Tinh Quoc; Wu, Youcai; Luo, Tzui-Liang; Wang, Morris; Liao, Chien-Chih; Chen, Pei-Yin; Lai, Yu-Sheng

    2017-08-01

    In this paper, a numerical approach with an experimental validation is introduced for modelling high-speed metal grinding processes in 6061-T6 aluminum alloys. The derivation of the present numerical method starts with an establishment of a stabilized particle Galerkin approximation. A non-residual penalty term from strain smoothing is introduced as a means of stabilizing the particle Galerkin method. Additionally, second-order strain gradients are introduced to the penalized functional for the regularization of damage-induced strain localization problem. To handle the severe deformation in metal grinding simulation, an adaptive anisotropic Lagrangian kernel is employed. Finally, the formulation incorporates a bond-based failure criterion to bypass the prospective spurious damage growth issues in material failure and cutting debris simulation. A three-dimensional metal grinding problem is analyzed and compared with the experimental results to demonstrate the effectiveness and accuracy of the proposed numerical approach.

  10. Analysis of elastoplasticity problems using an improved complex variable element-free Galerkin method

    NASA Astrophysics Data System (ADS)

    Cheng, Yu-Min; Liu, Chao; Bai, Fu-Nong; Peng, Miao-Juan

    2015-10-01

    In this paper, based on the conjugate of the complex basis function, a new complex variable moving least-squares approximation is discussed. Then using the new approximation to obtain the shape function, an improved complex variable element-free Galerkin (ICVEFG) method is presented for two-dimensional (2D) elastoplasticity problems. Compared with the previous complex variable moving least-squares approximation, the new approximation has greater computational precision and efficiency. Using the penalty method to apply the essential boundary conditions, and using the constrained Galerkin weak form of 2D elastoplasticity to obtain the system equations, we obtain the corresponding formulae of the ICVEFG method for 2D elastoplasticity. Three selected numerical examples are presented using the ICVEFG method to show that the ICVEFG method has the advantages such as greater precision and computational efficiency over the conventional meshless methods. Project supported by the National Natural Science Foundation of China (Grant Nos. 11171208 and U1433104).

  11. Penalty-No Penalty Drop Policy.

    ERIC Educational Resources Information Center

    Muha, Joseph G.; Snyder, John H.

    This study sought to determine what effect the utilization of a penalty or no-penalty drop policy would have on retention of students at Pasadena City College. It was found that there was a significant difference both in grades received and the retention of students as a result of the adoption of a 16 week no-penalty drop policy at the college.…

  12. The Problem with Penalties

    ERIC Educational Resources Information Center

    Dueck, Myron

    2014-01-01

    Imposing a penalty for late or incomplete homework assignments, Dueck says, neither inspires learning nor provides accurate grades. Dueck lists four rules that a teacher must follow if penalties for inadequate homework are to be efficient in prodding students to do that work. The usual homework penalty structures violate each of these four rules.…

  13. The Problem with Penalties

    ERIC Educational Resources Information Center

    Dueck, Myron

    2014-01-01

    Imposing a penalty for late or incomplete homework assignments, Dueck says, neither inspires learning nor provides accurate grades. Dueck lists four rules that a teacher must follow if penalties for inadequate homework are to be efficient in prodding students to do that work. The usual homework penalty structures violate each of these four rules.…

  14. Meshless Galerkin least-squares method

    NASA Astrophysics Data System (ADS)

    Pan, X. F.; Zhang, X.; Lu, M. W.

    2005-02-01

    Collocation method and Galerkin method have been dominant in the existing meshless methods. Galerkin-based meshless methods are computational intensive, whereas collocation-based meshless methods suffer from instability. A new efficient meshless method, meshless Galerkin lest-squares method (MGLS), is proposed in this paper to combine the advantages of Galerkin method and collocation method. The problem domain is divided into two subdomains, the interior domain and boundary domain. Galerkin method is applied in the boundary domain, whereas the least-squares method is applied in the interior domain.The proposed scheme elliminates the posibilities of spurious solutions as that in the least-square method if an incorrect boundary conditions are used. To investigate the accuracy and efficiency of the proposed method, a cantilevered beam and an infinite plate with a central circular hole are analyzed in detail and numerical results are compared with those obtained by Galerkin-based meshless method (GBMM), collocation-based meshless method (CBMM) and meshless weighted least squares method (MWLS). Numerical studies show that the accuracy of the proposed MGLS is much higher than that of CBMM and is close to, even better than, that of GBMM, while the computational cost is much less than that of GBMM.

  15. A high-order element-based Galerkin Method for the global shallow water equations.

    SciTech Connect

    Nair, Ramachandran D.; Tufo, Henry M.; Levy, Michael Nathan

    2010-08-01

    The shallow water equations are used as a test for many atmospheric models because the solution mimics the horizontal aspects of atmospheric dynamics while the simplicity of the equations make them useful for numerical experiments. This study describes a high-order element-based Galerkin method for the global shallow water equations using absolute vorticity, divergence, and fluid depth (atmospheric thickness) as the prognostic variables, while the wind field is a diagnostic variable that can be calculated from the stream function and velocity potential (the Laplacians of which are the vorticity and divergence, respectively). The numerical method employed to solve the shallow water system is based on the discontinuous Galerkin and spectral element methods. The discontinuous Galerkin method, which is inherently conservative, is used to solve the equations governing two conservative variables - absolute vorticity and atmospheric thickness (mass). The spectral element method is used to solve the divergence equation and the Poisson equations for the velocity potential and the stream function. Time integration is done with an explicit strong stability-preserving second-order Runge-Kutta scheme and the wind field is updated directly from the vorticity and divergence at each stage, and the computational domain is the cubed sphere. A stable steady-state test is run and convergence results are provided, showing that the method is high-order accurate. Additionally, two tests without analytic solutions are run with comparable results to previous high-resolution runs found in the literature.

  16. A Galerkin-based formulation of the probability density evolution method for general stochastic finite element systems

    NASA Astrophysics Data System (ADS)

    Papadopoulos, Vissarion; Kalogeris, Ioannis

    2016-05-01

    The present paper proposes a Galerkin finite element projection scheme for the solution of the partial differential equations (pde's) involved in the probability density evolution method, for the linear and nonlinear static analysis of stochastic systems. According to the principle of preservation of probability, the probability density evolution of a stochastic system is expressed by its corresponding Fokker-Planck (FP) stochastic partial differential equation. Direct integration of the FP equation is feasible only for simple systems with a small number of degrees of freedom, due to analytical and/or numerical intractability. However, rewriting the FP equation conditioned to the random event description, a generalized density evolution equation (GDEE) can be obtained, which can be reduced to a one dimensional pde. Two Galerkin finite element method schemes are proposed for the numerical solution of the resulting pde's, namely a time-marching discontinuous Galerkin scheme and the StreamlineUpwind/Petrov Galerkin (SUPG) scheme. In addition, a reformulation of the classical GDEE is proposed, which implements the principle of probability preservation in space instead of time, making this approach suitable for the stochastic analysis of finite element systems. The advantages of the FE Galerkin methods and in particular the SUPG over finite difference schemes, like the modified Lax-Wendroff, which is the most frequently used method for the solution of the GDEE, are illustrated with numerical examples and explored further.

  17. DG-IMEX Stochastic Galerkin Schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings

    DOE PAGES

    Chen, Zheng; Liu, Liu; Mu, Lin

    2017-05-03

    In this paper, we consider the linear transport equation under diffusive scaling and with random inputs. The method is based on the generalized polynomial chaos approach in the stochastic Galerkin framework. Several theoretical aspects will be addressed. Additionally, a uniform numerical stability with respect to the Knudsen number ϵ, and a uniform in ϵ error estimate is given. For temporal and spatial discretizations, we apply the implicit–explicit scheme under the micro–macro decomposition framework and the discontinuous Galerkin method, as proposed in Jang et al. (SIAM J Numer Anal 52:2048–2072, 2014) for deterministic problem. Lastly, we provide a rigorous proof ofmore » the stochastic asymptotic-preserving (sAP) property. Extensive numerical experiments that validate the accuracy and sAP of the method are conducted.« less

  18. Application of the Galerkin/least-squares formulation to the analysis of hypersonic flows. I - Flow over a two-dimensional ramp

    NASA Technical Reports Server (NTRS)

    Chalot, F.; Hughes, T. J. R.; Johan, Z.; Shakib, F.

    1991-01-01

    An FEM for the compressible Navier-Stokes equations is introduced. The discretization is based on entropy variables. The methodology is developed within the framework of a Galerkin/least-squares formulation to which a discontinuity-capturing operator is added. Results for three test cases selected among those of the Workshop on Hypersonic Flows for Reentry Problems are presented.

  19. WEAK GALERKIN METHODS FOR SECOND ORDER ELLIPTIC INTERFACE PROBLEMS.

    PubMed

    Mu, Lin; Wang, Junping; Wei, Guowei; Ye, Xiu; Zhao, Shan

    2013-10-01

    Weak Galerkin methods refer to general finite element methods for partial differential equations (PDEs) in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic PDEs with discontinuous coefficients and interfaces. Theoretically, it is proved that high order numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element. Extensive numerical experiments have been carried to validate the WG-FEM for solving second order elliptic interface problems. High order of convergence is numerically confirmed in both L2 and L∞ norms for the piecewise linear WG-FEM. Special attention is paid to solve many interface problems, in which the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order [Formula: see text] to [Formula: see text] for the solution itself in L∞ norm. It is demonstrated that the WG-FEM of the lowest order, i.e., the piecewise constant WG-FEM, is capable of delivering numerical approximations that are of order [Formula: see text] to [Formula: see text] in the L∞ norm for C(1) or Lipschitz continuous interfaces associated with a C(1) or H(2) continuous solution.

  20. WEAK GALERKIN METHODS FOR SECOND ORDER ELLIPTIC INTERFACE PROBLEMS

    PubMed Central

    MU, LIN; WANG, JUNPING; WEI, GUOWEI; YE, XIU; ZHAO, SHAN

    2013-01-01

    Weak Galerkin methods refer to general finite element methods for partial differential equations (PDEs) in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic PDEs with discontinuous coefficients and interfaces. Theoretically, it is proved that high order numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element. Extensive numerical experiments have been carried to validate the WG-FEM for solving second order elliptic interface problems. High order of convergence is numerically confirmed in both L2 and L∞ norms for the piecewise linear WG-FEM. Special attention is paid to solve many interface problems, in which the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order O(h) to O(h1.5) for the solution itself in L∞ norm. It is demonstrated that the WG-FEM of the lowest order, i.e., the piecewise constant WG-FEM, is capable of delivering numerical approximations that are of order O(h1.75) to O(h2) in the L∞ norm for C1 or Lipschitz continuous interfaces associated with a C1 or H2 continuous solution. PMID:24072935

  1. A Review of Element-Based Galerkin Methods for Numerical Weather Prediction

    DTIC Science & Technology

    2015-04-01

    Discontinuous Galerkin Simone Marras1 · James F. Kelly2 · Margarida Moragues3 · Andreas Müller1 · Michal A. Kopera1 · Mariano Vázquez3,4 · Francis X...OMEGA model was presented in the work of Bacon et al. in 2000 [8]. Simulations of hurricane tracks by Gopalakrishnan et al. in 2002 [121] demonstrate...1981) 8. Bacon , D., Ahmad, N., Boybeyi, Z., Dunn, T., Hall, M., Lee, C., Sarma, R., Turner, M.: A dynamically adaptive weather and dispersion model

  2. A new family of stable elements for the Stokes problem based on a mixed Galerkin/least-squares finite element formulation

    NASA Technical Reports Server (NTRS)

    Franca, Leopoldo P.; Loula, Abimael F. D.; Hughes, Thomas J. R.; Miranda, Isidoro

    1989-01-01

    Adding to the classical Hellinger-Reissner formulation, a residual form of the equilibrium equation, a new Galerkin/least-squares finite element method is derived. It fits within the framework of a mixed finite element method and is stable for rather general combinations of stress and velocity interpolations, including equal-order discontinuous stress and continuous velocity interpolations which are unstable within the Galerkin approach. Error estimates are presented based on a generalization of the Babuska-Brezzi theory. Numerical results (not presented herein) have confirmed these estimates as well as the good accuracy and stability of the method.

  3. A new family of stable elements for the Stokes problem based on a mixed Galerkin/least-squares finite element formulation

    NASA Technical Reports Server (NTRS)

    Franca, Leopoldo P.; Loula, Abimael F. D.; Hughes, Thomas J. R.; Miranda, Isidoro

    1989-01-01

    Adding to the classical Hellinger-Reissner formulation, a residual form of the equilibrium equation, a new Galerkin/least-squares finite element method is derived. It fits within the framework of a mixed finite element method and is stable for rather general combinations of stress and velocity interpolations, including equal-order discontinuous stress and continuous velocity interpolations which are unstable within the Galerkin approach. Error estimates are presented based on a generalization of the Babuska-Brezzi theory. Numerical results (not presented herein) have confirmed these estimates as well as the good accuracy and stability of the method.

  4. Single-Ray Streaming Behavior for Discontinuous Finite Element Spatial Discretizations

    SciTech Connect

    Smedley-Stevenson, R.P

    2002-09-15

    This technical note compares the results for streaming along a single-ray direction from linear discontinuous finite element discretizations of the transport equation using both Galerkin and Petrov-Galerkin weight functions. The utility of a slope limiter to remove extrema from the transport solution is investigated as an alternative to mass lumping of the removal operator; the latter procedure introduces significant numerical diffusion and can destroy the fidelity of the solution. Results are presented for single-ray propagation in slab geometry and two-dimensional planar geometry.

  5. Penalty Inflation Adjustments for Civil Money Penalties. Interim Final Rule.

    PubMed

    2016-06-27

    In accordance with the Federal Civil Penalties Inflation Adjustment Act of 1990, as amended by the Debt Collection Improvement Act of 1996, and further amended by the Bipartisan Budget Act of 2015, section 701: Federal Civil Penalties Inflation Adjustment Act Improvements Act of 2015, this interim final rule incorporates the penalty inflation adjustments for the civil money penalties contained in the Social Security Act

  6. Nonnegative methods for bilinear discontinuous differencing of the SN equations on quadrilaterals

    SciTech Connect

    Maginot, Peter G.; Ragusa, Jean C.; Morel, Jim E.

    2016-12-22

    Historically, matrix lumping and ad hoc flux fixups have been the only methods used to eliminate or suppress negative angular flux solutions associated with the unlumped bilinear discontinuous (UBLD) finite element spatial discretization of the two-dimensional SN equations. Though matrix lumping inhibits negative angular flux solutions of the SN equations, it does not guarantee strictly positive solutions. In this paper, we develop and define a strictly nonnegative, nonlinear, Petrov-Galerkin finite element method that fully preserves the bilinear discontinuous spatial moments of the transport equation. Additionally, we define two ad hoc fixups that maintain particle balance and explicitly set negative nodes of the UBLD finite element solution to zero but use different auxiliary equations to fully define their respective solutions. We assess the ability to inhibit negative angular flux solutions and the accuracy of every spatial discretization that we consider using a glancing void test problem with a discontinuous solution known to stress numerical methods. Though significantly more computationally intense, the nonlinear Petrov-Galerkin scheme results in a strictly nonnegative solution and is a more accurate solution than all the other methods considered. One fixup, based on shape preserving, results in a strictly nonnegative final solution but has increased numerical diffusion relative to the Petrov-Galerkin scheme and is less accurate than the UBLD solution. The second fixup, which preserves as many spatial moments as possible while setting negative values of the unlumped solution to zero, is less accurate than the Petrov-Galerkin scheme but is more accurate than the other fixup. However, it fails to guarantee a strictly nonnegative final solution. As a result, the fully lumped bilinear discontinuous finite element solution is the least accurate method, with significantly more numerical diffusion than the Petrov-Galerkin scheme and both

  7. Nonnegative methods for bilinear discontinuous differencing of the SN equations on quadrilaterals

    DOE PAGES

    Maginot, Peter G.; Ragusa, Jean C.; Morel, Jim E.

    2016-12-22

    Historically, matrix lumping and ad hoc flux fixups have been the only methods used to eliminate or suppress negative angular flux solutions associated with the unlumped bilinear discontinuous (UBLD) finite element spatial discretization of the two-dimensional SN equations. Though matrix lumping inhibits negative angular flux solutions of the SN equations, it does not guarantee strictly positive solutions. In this paper, we develop and define a strictly nonnegative, nonlinear, Petrov-Galerkin finite element method that fully preserves the bilinear discontinuous spatial moments of the transport equation. Additionally, we define two ad hoc fixups that maintain particle balance and explicitly set negative nodesmore » of the UBLD finite element solution to zero but use different auxiliary equations to fully define their respective solutions. We assess the ability to inhibit negative angular flux solutions and the accuracy of every spatial discretization that we consider using a glancing void test problem with a discontinuous solution known to stress numerical methods. Though significantly more computationally intense, the nonlinear Petrov-Galerkin scheme results in a strictly nonnegative solution and is a more accurate solution than all the other methods considered. One fixup, based on shape preserving, results in a strictly nonnegative final solution but has increased numerical diffusion relative to the Petrov-Galerkin scheme and is less accurate than the UBLD solution. The second fixup, which preserves as many spatial moments as possible while setting negative values of the unlumped solution to zero, is less accurate than the Petrov-Galerkin scheme but is more accurate than the other fixup. However, it fails to guarantee a strictly nonnegative final solution. As a result, the fully lumped bilinear discontinuous finite element solution is the least accurate method, with significantly more numerical diffusion than the Petrov-Galerkin scheme and both fixups.« less

  8. A tessellated continuum approach to thermal analysis: discontinuity networks

    NASA Astrophysics Data System (ADS)

    Jiang, C.; Davey, K.; Prosser, R.

    2017-01-01

    Tessellated continuum mechanics is an approach for the representation of thermo-mechanical behaviour of porous media on tessellated continua. It involves the application of iteration function schemes using affine contraction and expansion maps, respectively, for the creation of porous fractal materials and associated tessellated continua. Highly complex geometries can be produced using a modest number of contraction mappings. The associated tessellations form the mesh in a numerical procedure. This paper tests the hypothesis that thermal analysis of porous structures can be achieved using a discontinuous Galerkin finite element method on a tessellation. Discontinuous behaviour is identified at a discontinuity network in a tessellation; its use is shown to provide a good representation of the physics relating to cellular heat exchanger designs. Results for different cellular designs (with corresponding tessellations) are contrasted against those obtained from direct analysis and very high accuracy is observed.

  9. 75 FR 5244 - Civil Penalties

    Federal Register 2010, 2011, 2012, 2013, 2014

    2010-02-02

    ... Monetary Penalty Inflation Adjustment Act of 1990, as amended by the Debt Collection Improvement Act of... penalties and to foster compliance with the law, the Federal Civil Monetary Penalty Inflation Adjustment Act... ``Act''), requires us and other Federal agencies to adjust civil penalties for inflation. Under...

  10. Teaching about the Death Penalty.

    ERIC Educational Resources Information Center

    Ryan, John Paul; Eden, John Michael

    1998-01-01

    Examines the reasons for the death penalty, the reasons why the death penalty attracts so much attention, whether the death penalty is applied consistently, and the evidence that the application of the death penalty may be racially biased. Provides an accompanying article on "Teaching Ideas" by Ronald A. Banaszak. (CMK)

  11. Teaching about the Death Penalty.

    ERIC Educational Resources Information Center

    Ryan, John Paul; Eden, John Michael

    1998-01-01

    Examines the reasons for the death penalty, the reasons why the death penalty attracts so much attention, whether the death penalty is applied consistently, and the evidence that the application of the death penalty may be racially biased. Provides an accompanying article on "Teaching Ideas" by Ronald A. Banaszak. (CMK)

  12. Discontinuous Finite Elements for a Hyperbolic Problem with Singular Coefficient: a Convergence Theory for 1-D Spherical Neutron Transport

    SciTech Connect

    Machorro, E. A.

    2010-09-07

    A theory of convergence is presented for the discontinuous Galerkin finite element method of solving the non-scattering spherically symmetric Boltzmann transport equation using piecewise constant test and trial functions. Results are then extended to higher order polynomial spaces. Comparisons of numerical properties were presented in earlier work.

  13. The Characteristic Galerkin Method for Hyperbolic Conservation Laws.

    NASA Astrophysics Data System (ADS)

    Childs, P. N.

    Available from UMI in association with The British Library. Requires signed TDF. The purpose of this thesis is to study Morton's characteristic Galerkin method for hyperbolic problems. The scheme arises through employing the method of characteristics within a finite element context. While initially based on a piecewise constant approximation space, an adaptive linear recovery process permits high resolution while maintaining stability; and furthermore, some degree of shock recovery is permitted. In the scalar case, a rigorous analysis is carried out for convergence of the scheme for an initial boundary value problem; and we give an assessment of various qualitative features of the method in the presence of discontinuities. The method is explicit, yet an important feature is the lack of a stability restriction on the timestep. We extend the method to one dimensional systems using various forms of flux splitting and investigate the questions of entropy satisfaction and optimal order accuracy. The finite element viewpoint allows the incorporation of various grid adaptation strategies. The results of a number of numerical experiments for compressible gas flow are presented through which to assess the method and enable a comparison with finite difference methods to be made. Finally, we consider the extension to multidimensional systems and to inhomogeneous equations.

  14. Eighth Amendment & Death Penalty.

    ERIC Educational Resources Information Center

    Shortall, Joseph M.; Merrill, Denise W.

    1987-01-01

    Presents a lesson on capital punishment for juveniles based on three hypothetical cases. The goal of the lesson is to have students understand the complexities of decisions regarding the death penalty for juveniles. (JDH)

  15. Eighth Amendment & Death Penalty.

    ERIC Educational Resources Information Center

    Shortall, Joseph M.; Merrill, Denise W.

    1987-01-01

    Presents a lesson on capital punishment for juveniles based on three hypothetical cases. The goal of the lesson is to have students understand the complexities of decisions regarding the death penalty for juveniles. (JDH)

  16. Maximum-entropy principle as Galerkin modelling paradigm

    NASA Astrophysics Data System (ADS)

    Noack, Bernd R.; Niven, Robert K.; Rowley, Clarence W.

    2012-11-01

    We show how the empirical Galerkin method, leading e.g. to POD models, can be derived from maximum-entropy principles building on Noack & Niven 2012 JFM. In particular, principles are proposed (1) for the Galerkin expansion, (2) for the Galerkin system identification, and (3) for the probability distribution of the attractor. Examples will illustrate the advantages of the entropic modelling paradigm. Partially supported by the ANR Chair of Excellence TUCOROM and an ADFA/UNSW Visiting Fellowship.

  17. Weak Galerkin finite element methods for Darcy flow: Anisotropy and heterogeneity

    SciTech Connect

    Lin, Guang; Liu, Jiangguo; Mu, Lin; Ye, Xiu

    2014-11-01

    This paper presents a family of weak Galerkin finite element methods (WGFEMs) for Darcy flow computation. The WGFEMs are new numerical methods that rely on the novel concept of discrete weak gradients. The WGFEMs solve for pressure unknowns both in element interiors and on the mesh skeleton. The numerical velocity is then obtained from the discrete weak gradient of the numerical pressure. The new methods are quite different than many existing numerical methods in that they are locally conservative by design, the resulting discrete linear systems are symmetric and positive-definite, and there is no need for tuning problem-dependent penalty factors. We test the WGFEMs on benchmark problems to demonstrate the strong potential of these new methods in handling strong anisotropy and heterogeneity in Darcy flow.

  18. A well-posed and stable stochastic Galerkin formulation of the incompressible Navier–Stokes equations with random data

    SciTech Connect

    Pettersson, Per; Nordström, Jan; Doostan, Alireza

    2016-02-01

    We present a well-posed stochastic Galerkin formulation of the incompressible Navier–Stokes equations with uncertainty in model parameters or the initial and boundary conditions. The stochastic Galerkin method involves representation of the solution through generalized polynomial chaos expansion and projection of the governing equations onto stochastic basis functions, resulting in an extended system of equations. A relatively low-order generalized polynomial chaos expansion is sufficient to capture the stochastic solution for the problem considered. We derive boundary conditions for the continuous form of the stochastic Galerkin formulation of the velocity and pressure equations. The resulting problem formulation leads to an energy estimate for the divergence. With suitable boundary data on the pressure and velocity, the energy estimate implies zero divergence of the velocity field. Based on the analysis of the continuous equations, we present a semi-discretized system where the spatial derivatives are approximated using finite difference operators with a summation-by-parts property. With a suitable choice of dissipative boundary conditions imposed weakly through penalty terms, the semi-discrete scheme is shown to be stable. Numerical experiments in the laminar flow regime corroborate the theoretical results and we obtain high-order accurate results for the solution variables and the velocity divergence converges to zero as the mesh is refined.

  19. Entropy Stable Spectral Collocation Schemes for the Navier-Stokes Equations: Discontinuous Interfaces

    NASA Technical Reports Server (NTRS)

    Carpenter, Mark H.; Fisher, Travis C.; Nielsen, Eric J.; Frankel, Steven H.

    2013-01-01

    Nonlinear entropy stability and a summation-by-parts framework are used to derive provably stable, polynomial-based spectral collocation methods of arbitrary order. The new methods are closely related to discontinuous Galerkin spectral collocation methods commonly known as DGFEM, but exhibit a more general entropy stability property. Although the new schemes are applicable to a broad class of linear and nonlinear conservation laws, emphasis herein is placed on the entropy stability of the compressible Navier-Stokes equations.

  20. 77 FR 70710 - Civil Penalties

    Federal Register 2010, 2011, 2012, 2013, 2014

    2012-11-27

    ... action is taken pursuant to the Federal Civil Monetary Penalty Inflation Adjustment Act of 1990, as..., adjust penalties based on inflation at least every four years. DATES: This rule is effective December 27... Penalty Inflation Adjustment Act of 1990 (28 U.S.C. 2461 Notes, Pub. L. 101- 410), as amended by the...

  1. 75 FR 49879 - Civil Penalties

    Federal Register 2010, 2011, 2012, 2013, 2014

    2010-08-16

    ... defraud. This action would be taken pursuant to the Federal Civil Monetary Penalty Inflation Adjustment... and, as warranted, adjust penalties based on inflation at least every four years. DATES: Comments on... Penalty Inflation Adjustment Act of 1990 (28 U.S.C. 2461, Notes, Pub. L. 101- 410), as amended by the...

  2. 75 FR 79978 - Civil Penalties

    Federal Register 2010, 2011, 2012, 2013, 2014

    2010-12-21

    ... the Federal Civil Monetary Penalty Inflation Adjustment Act of 1990, as amended by the Debt Collection Improvement Act of 1996, which requires NHTSA to review and, as warranted, adjust penalties based on inflation... Federal Civil Monetary Penalty Inflation Adjustment Act of 1990 (28 U.S.C. 2461, Notes, Pub. L. 101-...

  3. Numerically Tracking Contact Discontinuities with an Introduction for GPU Programming

    SciTech Connect

    Davis, Sean L

    2012-08-17

    We review some of the classic numerical techniques used to analyze contact discontinuities and compare their effectiveness. Several finite difference methods (the Lax-Wendroff method, a Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) method and a Monotone Upstream Scheme for Conservation Laws (MUSCL) scheme with an Artificial Compression Method (ACM)) as well as the finite element Streamlined Upwind Petrov-Galerkin (SUPG) method were considered. These methods were applied to solve the 2D advection equation. Based on our results we concluded that the MUSCL scheme produces the sharpest interfaces but can inappropriately steepen the solution. The SUPG method seems to represent a good balance between stability and interface sharpness without any inappropriate steepening. However, for solutions with discontinuities, the MUSCL scheme is superior. In addition, a preliminary implementation in a GPU program is discussed.

  4. Discontinuous dual-primal mixed finite elements for elliptic problems

    NASA Technical Reports Server (NTRS)

    Bottasso, Carlo L.; Micheletti, Stefano; Sacco, Riccardo

    2000-01-01

    We propose a novel discontinuous mixed finite element formulation for the solution of second-order elliptic problems. Fully discontinuous piecewise polynomial finite element spaces are used for the trial and test functions. The discontinuous nature of the test functions at the element interfaces allows to introduce new boundary unknowns that, on the one hand enforce the weak continuity of the trial functions, and on the other avoid the need to define a priori algorithmic fluxes as in standard discontinuous Galerkin methods. Static condensation is performed at the element level, leading to a solution procedure based on the sole interface unknowns. The resulting family of discontinuous dual-primal mixed finite element methods is presented in the one and two-dimensional cases. In the one-dimensional case, we show the equivalence of the method with implicit Runge-Kutta schemes of the collocation type exhibiting optimal behavior. Numerical experiments in one and two dimensions demonstrate the order accuracy of the new method, confirming the results of the analysis.

  5. The Death Penalty.

    ERIC Educational Resources Information Center

    Crockett, Mark

    1990-01-01

    Provides a lesson plan on the Eighth Amendment to the U.S. Constitution and the imposition of the death penalty. Focuses on the controversy concerning capital punishment and stimulates critical thinking in an analysis and discussion of eight hypothetical situations. Includes suggestions for readings, videotapes, and writing assignments. (NL)

  6. Death Penalty in America.

    ERIC Educational Resources Information Center

    Clifford, Amie L.

    1997-01-01

    Examines the legal and moral issues, controversies, and unique trial procedures involved with the death penalty. Discusses the 1972 landmark Supreme Court decision that resulted in many states abolishing this punishment, only to reintroduce it later with different provisions. Reviews the controversial case of Sam Sheppard. (MJP)

  7. Death Penalty in America.

    ERIC Educational Resources Information Center

    Clifford, Amie L.

    1997-01-01

    Examines the legal and moral issues, controversies, and unique trial procedures involved with the death penalty. Discusses the 1972 landmark Supreme Court decision that resulted in many states abolishing this punishment, only to reintroduce it later with different provisions. Reviews the controversial case of Sam Sheppard. (MJP)

  8. The Death Penalty.

    ERIC Educational Resources Information Center

    Crockett, Mark

    1990-01-01

    Provides a lesson plan on the Eighth Amendment to the U.S. Constitution and the imposition of the death penalty. Focuses on the controversy concerning capital punishment and stimulates critical thinking in an analysis and discussion of eight hypothetical situations. Includes suggestions for readings, videotapes, and writing assignments. (NL)

  9. A Galerkin formulation of the MIB method for three dimensional elliptic interface problems

    PubMed Central

    Xia, Kelin; Wei, Guo-Wei

    2014-01-01

    We develop a three dimensional (3D) Galerkin formulation of the matched interface and boundary (MIB) method for solving elliptic partial differential equations (PDEs) with discontinuous coefficients, i.e., the elliptic interface problem. The present approach builds up two sets of elements respectively on two extended subdomains which both include the interface. As a result, two sets of elements overlap each other near the interface. Fictitious solutions are defined on the overlapping part of the elements, so that the differentiation operations of the original PDEs can be discretized as if there was no interface. The extra coefficients of polynomial basis functions, which furnish the overlapping elements and solve the fictitious solutions, are determined by interface jump conditions. Consequently, the interface jump conditions are rigorously enforced on the interface. The present method utilizes Cartesian meshes to avoid the mesh generation in conventional finite element methods (FEMs). We implement the proposed MIB Galerkin method with three different elements, namely, rectangular prism element, five-tetrahedron element and six-tetrahedron element, which tile the Cartesian mesh without introducing any new node. The accuracy, stability and robustness of the proposed 3D MIB Galerkin are extensively validated over three types of elliptic interface problems. In the first type, interfaces are analytically defined by level set functions. These interfaces are relatively simple but admit geometric singularities. In the second type, interfaces are defined by protein surfaces, which are truly arbitrarily complex. The last type of interfaces originates from multiprotein complexes, such as molecular motors. Near second order accuracy has been confirmed for all of these problems. To our knowledge, it is the first time for an FEM to show a near second order convergence in solving the Poisson equation with realistic protein surfaces. Additionally, the present work offers the

  10. A Galerkin formulation of the MIB method for three dimensional elliptic interface problems.

    PubMed

    Xia, Kelin; Wei, Guo-Wei

    2014-10-01

    We develop a three dimensional (3D) Galerkin formulation of the matched interface and boundary (MIB) method for solving elliptic partial differential equations (PDEs) with discontinuous coefficients, i.e., the elliptic interface problem. The present approach builds up two sets of elements respectively on two extended subdomains which both include the interface. As a result, two sets of elements overlap each other near the interface. Fictitious solutions are defined on the overlapping part of the elements, so that the differentiation operations of the original PDEs can be discretized as if there was no interface. The extra coefficients of polynomial basis functions, which furnish the overlapping elements and solve the fictitious solutions, are determined by interface jump conditions. Consequently, the interface jump conditions are rigorously enforced on the interface. The present method utilizes Cartesian meshes to avoid the mesh generation in conventional finite element methods (FEMs). We implement the proposed MIB Galerkin method with three different elements, namely, rectangular prism element, five-tetrahedron element and six-tetrahedron element, which tile the Cartesian mesh without introducing any new node. The accuracy, stability and robustness of the proposed 3D MIB Galerkin are extensively validated over three types of elliptic interface problems. In the first type, interfaces are analytically defined by level set functions. These interfaces are relatively simple but admit geometric singularities. In the second type, interfaces are defined by protein surfaces, which are truly arbitrarily complex. The last type of interfaces originates from multiprotein complexes, such as molecular motors. Near second order accuracy has been confirmed for all of these problems. To our knowledge, it is the first time for an FEM to show a near second order convergence in solving the Poisson equation with realistic protein surfaces. Additionally, the present work offers the

  11. Comparison of two Galerkin quadrature methods

    DOE PAGES

    Morel, Jim E.; Warsa, James; Franke, Brian C.; ...

    2017-02-21

    Here, we compare two methods for generating Galerkin quadratures. In method 1, the standard SN method is used to generate the moment-to-discrete matrix and the discrete-to-moment matrix is generated by inverting the moment-to-discrete matrix. This is a particular form of the original Galerkin quadrature method. In method 2, which we introduce here, the standard SN method is used to generate the discrete-to-moment matrix and the moment-to-discrete matrix is generated by inverting the discrete-to-moment matrix. With an N-point quadrature, method 1 has the advantage that it preserves N eigenvalues and N eigenvectors of the scattering operator in a pointwise sense. Withmore » an N-point quadrature, method 2 has the advantage that it generates consistent angular moment equations from the corresponding SN equations while preserving N eigenvalues of the scattering operator. Our computational results indicate that these two methods are quite comparable for the test problem considered.« less

  12. Analysis of periodic 3D viscous flows using a quadratic discrete Galerkin boundary element method

    NASA Astrophysics Data System (ADS)

    Chan, Chiu Y.; Beris, Antony N.; Advani, Suresh G.

    1994-05-01

    A discrete Galerkin boundary element technique with a quadratic approximation of the variables was developed to simulate the three-dimensional (3D) viscous flow established in periodic assemblages of particles in suspensions and within a periodic porous medium. The Batchelor's unit-cell approach is used. The Galerkin formulation effectively handles the discontinuity in the traction arising in flow boundaries with edges or corners, such as the unit cell in this case. For an ellipsoidal dilute suspension over the range of aspect ratio studied (1 to 54), the numerical solutions of the rotational velocity of the particles and the viscosity correction were found to agree with the analytic values within 0.2% and 2% respectively, even with coarse meshes. In a suspension of cylindrical particles the calculated period of rotation agreed with the experimental data. However, Burgers' predictions for the correction to the suspension viscosity were found to be 30% too low and therefore the concept of the equivalent ellipsoidal ratio is judged to be inadequate. For pressure-driven flow through a fixed bed of fibers, the prediction on the permeability was shown to deviate by as much as 10% from the value calculated based on approximate permeability additivity rules using the corresponding values for planar flow past a periodic array of parallel cylinders. These applications show the versatility of the technique for studying viscous flows in complicated 3D geometries.

  13. 29 CFR 2570.86 - Reduction of penalty by other penalty assessments.

    Code of Federal Regulations, 2012 CFR

    2012-07-01

    ... Civil Penalties Under ERISA Section 502(l) § 2570.86 Reduction of penalty by other penalty assessments... ERISA section 502(i) and section 4975 of the Code. Prior to a reduction of penalty under this...

  14. 29 CFR 2570.86 - Reduction of penalty by other penalty assessments.

    Code of Federal Regulations, 2011 CFR

    2011-07-01

    ... Civil Penalties Under ERISA Section 502(l) § 2570.86 Reduction of penalty by other penalty assessments... ERISA section 502(i) and section 4975 of the Code. Prior to a reduction of penalty under this...

  15. 29 CFR 2570.86 - Reduction of penalty by other penalty assessments.

    Code of Federal Regulations, 2010 CFR

    2010-07-01

    ... Civil Penalties Under ERISA Section 502(l) § 2570.86 Reduction of penalty by other penalty assessments... ERISA section 502(i) and section 4975 of the Code. Prior to a reduction of penalty under this...

  16. 29 CFR 2570.86 - Reduction of penalty by other penalty assessments.

    Code of Federal Regulations, 2014 CFR

    2014-07-01

    ... Civil Penalties Under ERISA Section 502(l) § 2570.86 Reduction of penalty by other penalty assessments... ERISA section 502(i) and section 4975 of the Code. Prior to a reduction of penalty under this...

  17. 29 CFR 2570.86 - Reduction of penalty by other penalty assessments.

    Code of Federal Regulations, 2013 CFR

    2013-07-01

    ... Civil Penalties Under ERISA Section 502(l) § 2570.86 Reduction of penalty by other penalty assessments... ERISA section 502(i) and section 4975 of the Code. Prior to a reduction of penalty under this...

  18. Meshless Local Petrov-Galerkin Euler-Bernoulli Beam Problems: A Radial Basis Function Approach

    NASA Technical Reports Server (NTRS)

    Raju, I. S.; Phillips, D. R.; Krishnamurthy, T.

    2003-01-01

    A radial basis function implementation of the meshless local Petrov-Galerkin (MLPG) method is presented to study Euler-Bernoulli beam problems. Radial basis functions, rather than generalized moving least squares (GMLS) interpolations, are used to develop the trial functions. This choice yields a computationally simpler method as fewer matrix inversions and multiplications are required than when GMLS interpolations are used. Test functions are chosen as simple weight functions as in the conventional MLPG method. Compactly and noncompactly supported radial basis functions are considered. The non-compactly supported cubic radial basis function is found to perform very well. Results obtained from the radial basis MLPG method are comparable to those obtained using the conventional MLPG method for mixed boundary value problems and problems with discontinuous loading conditions.

  19. A hybridized formulation for the weak Galerkin mixed finite element method

    SciTech Connect

    Mu, Lin; Wang, Junping; Ye, Xiu

    2016-01-14

    This paper presents a hybridized formulation for the weak Galerkin mixed finite element method (WG-MFEM) which was introduced and analyzed in Wang and Ye (2014) for second order elliptic equations. The WG-MFEM method was designed by using discontinuous piecewise polynomials on finite element partitions consisting of polygonal or polyhedral elements of arbitrary shape. The key to WG-MFEM is the use of a discrete weak divergence operator which is defined and computed by solving inexpensive problems locally on each element. The hybridized formulation of this paper leads to a significantly reduced system of linear equations involving only the unknowns arising from the Lagrange multiplier in hybridization. Optimal-order error estimates are derived for the hybridized WG-MFEM approximations. In conclusion, some numerical results are reported to confirm the theory and a superconvergence for the Lagrange multiplier.

  20. A hybridized formulation for the weak Galerkin mixed finite element method

    DOE PAGES

    Mu, Lin; Wang, Junping; Ye, Xiu

    2016-01-14

    This paper presents a hybridized formulation for the weak Galerkin mixed finite element method (WG-MFEM) which was introduced and analyzed in Wang and Ye (2014) for second order elliptic equations. The WG-MFEM method was designed by using discontinuous piecewise polynomials on finite element partitions consisting of polygonal or polyhedral elements of arbitrary shape. The key to WG-MFEM is the use of a discrete weak divergence operator which is defined and computed by solving inexpensive problems locally on each element. The hybridized formulation of this paper leads to a significantly reduced system of linear equations involving only the unknowns arising frommore » the Lagrange multiplier in hybridization. Optimal-order error estimates are derived for the hybridized WG-MFEM approximations. In conclusion, some numerical results are reported to confirm the theory and a superconvergence for the Lagrange multiplier.« less