A fourth order accurate finite difference scheme for the computation of elastic waves
NASA Technical Reports Server (NTRS)
Bayliss, A.; Jordan, K. E.; Lemesurier, B. J.; Turkel, E.
1986-01-01
A finite difference for elastic waves is introduced. The model is based on the first order system of equations for the velocities and stresses. The differencing is fourth order accurate on the spatial derivatives and second order accurate in time. The model is tested on a series of examples including the Lamb problem, scattering from plane interf aces and scattering from a fluid-elastic interface. The scheme is shown to be effective for these problems. The accuracy and stability is insensitive to the Poisson ratio. For the class of problems considered here it is found that the fourth order scheme requires for two-thirds to one-half the resolution of a typical second order scheme to give comparable accuracy.
Accurate Finite Difference Algorithms
NASA Technical Reports Server (NTRS)
Goodrich, John W.
1996-01-01
Two families of finite difference algorithms for computational aeroacoustics are presented and compared. All of the algorithms are single step explicit methods, they have the same order of accuracy in both space and time, with examples up to eleventh order, and they have multidimensional extensions. One of the algorithm families has spectral like high resolution. Propagation with high order and high resolution algorithms can produce accurate results after O(10(exp 6)) periods of propagation with eight grid points per wavelength.
NASA Astrophysics Data System (ADS)
Wu, Kailiang; Tang, Huazhong
2015-10-01
The paper develops high-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamical (RHD) equations, built on the local Lax-Friedrichs splitting, the WENO reconstruction, the physical-constraints-preserving flux limiter, and the high-order strong stability preserving time discretization. They are extensions of the positivity-preserving finite difference WENO schemes for the non-relativistic Euler equations [20]. However, developing physical-constraints-preserving methods for the RHD system becomes much more difficult than the non-relativistic case because of the strongly coupling between the RHD equations, no explicit formulas of the primitive variables and the flux vectors with respect to the conservative vector, and one more physical constraint for the fluid velocity in addition to the positivity of the rest-mass density and the pressure. The key is to prove the convexity and other properties of the admissible state set and discover a concave function with respect to the conservative vector instead of the pressure which is an important ingredient to enforce the positivity-preserving property for the non-relativistic case. Several one- and two-dimensional numerical examples are used to demonstrate accuracy, robustness, and effectiveness of the proposed physical-constraints-preserving schemes in solving RHD problems with large Lorentz factor, or strong discontinuities, or low rest-mass density or pressure etc.
Upwind Compact Finite Difference Schemes
NASA Astrophysics Data System (ADS)
Christie, I.
1985-07-01
It was shown by Ciment, Leventhal, and Weinberg ( J. Comput. Phys.28 (1978), 135) that the standard compact finite difference scheme may break down in convection dominated problems. An upwinding of the method, which maintains the fourth order accuracy, is suggested and favorable numerical results are found for a number of test problems.
NASA Technical Reports Server (NTRS)
Yefet, Amir; Petropoulos, Peter G.
1999-01-01
We consider a divergence-free non-dissipative fourth-order explicit staggered finite difference scheme for the hyperbolic Maxwell's equations. Special one-sided difference operators are derived in order to implement the scheme near metal boundaries and dielectric interfaces. Numerical results show the scheme is long-time stable, and is fourth-order convergent over complex domains that include dielectric interfaces and perfectly conducting surfaces. We also examine the scheme's behavior near metal surfaces that are not aligned with the grid axes, and compare its accuracy to that obtained by the Yee scheme.
NASA Technical Reports Server (NTRS)
Harten, A.; Tal-Ezer, H.
1981-01-01
This paper presents a family of two-level five-point implicit schemes for the solution of one-dimensional systems of hyperbolic conservation laws, which generalized the Crank-Nicholson scheme to fourth order accuracy (4-4) in both time and space. These 4-4 schemes are nondissipative and unconditionally stable. Special attention is given to the system of linear equations associated with these 4-4 implicit schemes. The regularity of this system is analyzed and efficiency of solution-algorithms is examined. A two-datum representation of these 4-4 implicit schemes brings about a compactification of the stencil to three mesh points at each time-level. This compact two-datum representation is particularly useful in deriving boundary treatments. Numerical results are presented to illustrate some properties of the proposed scheme.
NASA Technical Reports Server (NTRS)
Harten, A.; Tal-Ezer, H.
1981-01-01
An implicit finite difference method of fourth order accuracy in space and time is introduced for the numerical solution of one-dimensional systems of hyperbolic conservation laws. The basic form of the method is a two-level scheme which is unconditionally stable and nondissipative. The scheme uses only three mesh points at level t and three mesh points at level t + delta t. The dissipative version of the basic method given is conditionally stable under the CFL (Courant-Friedrichs-Lewy) condition. This version is particularly useful for the numerical solution of problems with strong but nonstiff dynamic features, where the CFL restriction is reasonable on accuracy grounds. Numerical results are provided to illustrate properties of the proposed method.
Conservative properties of finite difference schemes for incompressible flow
NASA Technical Reports Server (NTRS)
Morinishi, Youhei
1995-01-01
The purpose of this research is to construct accurate finite difference schemes for incompressible unsteady flow simulations such as LES (large-eddy simulation) or DNS (direct numerical simulation). In this report, conservation properties of the continuity, momentum, and kinetic energy equations for incompressible flow are specified as analytical requirements for a proper set of discretized equations. Existing finite difference schemes in staggered grid systems are checked for satisfaction of the requirements. Proper higher order accurate finite difference schemes in a staggered grid system are then proposed. Plane channel flow is simulated using the proposed fourth order accurate finite difference scheme and the results compared with those of the second order accurate Harlow and Welch algorithm.
On nonstandard finite difference schemes in biosciences
NASA Astrophysics Data System (ADS)
Anguelov, R.; Dumont, Y.; Lubuma, J. M.-S.
2012-10-01
We design, analyze and implement nonstandard finite difference (NSFD) schemes for some differential models in biosciences. The NSFD schemes are reliable in three directions. They are topologically dynamically consistent for onedimensional models. They can replicate the global asymptotic stability of the disease-free equilibrium of the MSEIR model in epidemiology whenever the basic reproduction number is less than 1. They preserve the positivity and boundedness property of solutions of advection-reaction and reaction-diffusion equations.
Direct simulations of turbulent flow using finite-difference schemes
NASA Technical Reports Server (NTRS)
Rai, Man Mohan; Moin, Parviz
1989-01-01
A high-order accurate finite-difference approach is presented for calculating incompressible turbulent flow. The methods used include a kinetic energy conserving central difference scheme and an upwind difference scheme. The methods are evaluated in test cases for the evolution of small-amplitude disturbances and fully developed turbulent channel flow. It is suggested that the finite-difference approach can be applied to complex geometries more easilty than highly accurate spectral methods. It is concluded that the upwind scheme is a good candidate for direct simulations of turbulent flows over complex geometries.
TVD finite difference schemes and artificial viscosity
NASA Technical Reports Server (NTRS)
Davis, S. F.
1984-01-01
The total variation diminishing (TVD) finite difference scheme can be interpreted as a Lax-Wendroff scheme plus an upwind weighted artificial dissipation term. If a particular flux limiter is chosen and the requirement for upwind weighting is removed, an artificial dissipation term which is based on the theory of TVD schemes is obtained which does not contain any problem dependent parameters and which can be added to existing MacCormack method codes. Numerical experiments to examine the performance of this new method are discussed.
A positive finite-difference advection scheme
Hundsdorfer, W.; Koren, B.; Loon, M. van
1995-03-01
This paper examines a class of explicit finite-difference advection schemes derived along the method of lines. An important application field is large-scale atmospheric transport. The paper therefore focuses on the demand of positivity. For the spatial discretization, attention is confined to conservative schemes using five points per direction. The fourth-order central scheme and the family of {kappa}-schemes, comprising the second-order central, the second-order upwind, and the third-order upwind biased, are studied. Positivity is enforced through flux limiting. It is concluded that the limited third-order upwind discretization is the best candidate from the four examined. For the time integration attention is confined to a number of explicit Runge-Kutta methods of orders two to four. With regard to the demand of positivity, these integration methods turn out to behave almost equally and no best method could be identified. 16 refs., 4 figs., 4 tabs.
Compact finite difference schemes with spectral-like resolution
NASA Technical Reports Server (NTRS)
Lele, Sanjiva K.
1992-01-01
The present finite-difference schemes for the evaluation of first-order, second-order, and higher-order derivatives yield improved representation of a range of scales and may be used on nonuniform meshes. Various boundary conditions may be invoked, and both accurate interpolation and spectral-like filtering can be accomplished by means of schemes for derivatives at mid-cell locations. This family of schemes reduces to the Pade schemes when the maximal formal accuracy constraint is imposed with a specific computational stencil. Attention is given to illustrative applications of these schemes in fluid dynamics.
Accurate finite difference methods for time-harmonic wave propagation
NASA Technical Reports Server (NTRS)
Harari, Isaac; Turkel, Eli
1994-01-01
Finite difference methods for solving problems of time-harmonic acoustics are developed and analyzed. Multidimensional inhomogeneous problems with variable, possibly discontinuous, coefficients are considered, accounting for the effects of employing nonuniform grids. A weighted-average representation is less sensitive to transition in wave resolution (due to variable wave numbers or nonuniform grids) than the standard pointwise representation. Further enhancement in method performance is obtained by basing the stencils on generalizations of Pade approximation, or generalized definitions of the derivative, reducing spurious dispersion, anisotropy and reflection, and by improving the representation of source terms. The resulting schemes have fourth-order accurate local truncation error on uniform grids and third order in the nonuniform case. Guidelines for discretization pertaining to grid orientation and resolution are presented.
Finite-difference schemes for anisotropic diffusion
Es, Bram van; Koren, Barry; Blank, Hugo J. de
2014-09-01
In fusion plasmas diffusion tensors are extremely anisotropic due to the high temperature and large magnetic field strength. This causes diffusion, heat conduction, and viscous momentum loss, to effectively be aligned with the magnetic field lines. This alignment leads to different values for the respective diffusive coefficients in the magnetic field direction and in the perpendicular direction, to the extent that heat diffusion coefficients can be up to 10{sup 12} times larger in the parallel direction than in the perpendicular direction. This anisotropy puts stringent requirements on the numerical methods used to approximate the MHD-equations since any misalignment of the grid may cause the perpendicular diffusion to be polluted by the numerical error in approximating the parallel diffusion. Currently the common approach is to apply magnetic field-aligned coordinates, an approach that automatically takes care of the directionality of the diffusive coefficients. This approach runs into problems at x-points and at points where there is magnetic re-connection, since this causes local non-alignment. It is therefore useful to consider numerical schemes that are tolerant to the misalignment of the grid with the magnetic field lines, both to improve existing methods and to help open the possibility of applying regular non-aligned grids. To investigate this, in this paper several discretization schemes are developed and applied to the anisotropic heat diffusion equation on a non-aligned grid.
Asymptotically Correct Finite Difference Schemes for Highly Oscillatory ODEs
Arnold, Anton; Geier, Jens
2010-09-30
We are concerned with the numerical integration of ODE-initial value problems of the form {epsilon}{sup 2{phi}}{sub xx}+a(x){phi} = 0 with given a(x){>=}a{sub 0}>0 in the highly oscillatory regime 0<{epsilon}(appearing as a stationary Schroedinger equation, e.g.). In two steps we derive an accurate finite difference scheme that does not need to resolve each oscillation: With a WKB-ansatz the dominant oscillations are ''transformed out'', yielding a much smoother ODE. For the resulting oscillatory integrals we devise an asymptotic expansion both in {epsilon} and h. The resulting scheme typically has a step size restriction of h = o({radical}({epsilon})). If the phase of the WKB-transformation can be computed explicitly, then the scheme is asymptotically correct with an error bound of the order o({epsilon}{sup 3}h{sup 2}). As an application we present simulations of a 1D-model for ballistic quantum transport in a MOSFET (metal oxide semiconductor field-effect transistor).
Dispersion-relation-preserving finit difference schemes for computational acoustics
Tam, C.K.W.; Webb, J.C. )
1993-08-01
Acoustics problems are governed by the linearized Euler equations. According to wave propagation theory, the number of wave modes and their wave propagation characteristics are all encoded in the dispersion relation of the governing equations. Thus one is assured that the numerical solutions of high order finite difference scheme will have the same number of wave modes (namely, the acoustic, vorticity, and entropy waves), the same wave propagation characteristics (namely, nondispersive, nondissipative, and isotropic) and the same wave speeds as those of the solutions of the Euler equations if both systems of equations have the same dispersion relations. Finite difference schemes which have the same dispersion relations as the original partial differential equations are referred to as dispersion-relation-preserving (DRP) schemes. A way to construct time marching DRP schemes by optimizing the finite difference approximations of the space and time derivatives in the wave number and frequency space is proposed. The stability of these schemes is analyzed and a sufficient condition for numerical stability is established. A set of radiation and outflow boundary conditions compatible with the DRP schemes is constructed. These conditions are derived from the asymptotic solutions of the governing equations. The asymptotic solutions are found by the use of Fourier-Laplace transforms and the method of stationary phase. A sequence of numerical simulations has been carried out. These simulation are designed to test the effectiveness of the DRP schemes and the radiation and outflow boundary conditions. The computed solutions agree very favorably with the exact solutions. The radiation boundary conditions perform satisfactorily causing little acoustic reflections. The outflow boundary conditions are found to be quite transparent to outgoing disturbances even when the disturbances are made up of a combination of acoustic, vorticity, and entropy waves. 26 refs., 14 figs.
Versypt, Ashlee N. Ford; Braatz, Richard D.
2014-01-01
Two finite difference discretization schemes for approximating the spatial derivatives in the diffusion equation in spherical coordinates with variable diffusivity are presented and analyzed. The numerical solutions obtained by the discretization schemes are compared for five cases of the functional form for the variable diffusivity: (I) constant diffusivity, (II) temporally-dependent diffusivity, (III) spatially-dependent diffusivity, (IV) concentration-dependent diffusivity, and (V) implicitly-defined, temporally- and spatially-dependent diffusivity. Although the schemes have similar agreement to known analytical or semi-analytical solutions in the first four cases, in the fifth case for the variable diffusivity, one scheme produces a stable, physically reasonable solution, while the other diverges. We recommend the adoption of the more accurate and stable of these finite difference discretization schemes to numerically approximate the spatial derivatives of the diffusion equation in spherical coordinates for any functional form of variable diffusivity, especially cases where the diffusivity is a function of position. PMID:25642003
A semi-Lagrangian finite difference WENO scheme for scalar nonlinear conservation laws
NASA Astrophysics Data System (ADS)
Huang, Chieh-Sen; Arbogast, Todd; Hung, Chen-Hui
2016-10-01
For a nonlinear scalar conservation law in one-space dimension, we develop a locally conservative semi-Lagrangian finite difference scheme based on weighted essentially non-oscillatory reconstructions (SL-WENO). This scheme has the advantages of both WENO and semi-Lagrangian schemes. It is a locally mass conservative finite difference scheme, it is formally high-order accurate in space, it has small time truncation error, and it is essentially non-oscillatory. The scheme is nearly free of a CFL time step stability restriction for linear problems, and it has a relaxed CFL condition for nonlinear problems. The scheme can be considered as an extension of the SL-WENO scheme of Qiu and Shu (2011) [2] developed for linear problems. The new scheme is based on a standard sliding average formulation with the flux function defined using WENO reconstructions of (semi-Lagrangian) characteristic tracings of grid points. To handle nonlinear problems, we use an approximate, locally frozen trace velocity and a flux correction step. A special two-stage WENO reconstruction procedure is developed that is biased to the upstream direction. A Strang splitting algorithm is used for higher-dimensional problems. Numerical results are provided to illustrate the performance of the scheme and verify its formal accuracy. Included are applications to the Vlasov-Poisson and guiding-center models of plasma flow.
NASA Astrophysics Data System (ADS)
Kumar, Vivek; Raghurama Rao, S. V.
2008-04-01
Non-standard finite difference methods (NSFDM) introduced by Mickens [ Non-standard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994] are interesting alternatives to the traditional finite difference and finite volume methods. When applied to linear hyperbolic conservation laws, these methods reproduce exact solutions. In this paper, the NSFDM is first extended to hyperbolic systems of conservation laws, by a novel utilization of the decoupled equations using characteristic variables. In the second part of this paper, the NSFDM is studied for its efficacy in application to nonlinear scalar hyperbolic conservation laws. The original NSFDMs introduced by Mickens (1994) were not in conservation form, which is an important feature in capturing discontinuities at the right locations. Mickens [Construction and analysis of a non-standard finite difference scheme for the Burgers-Fisher equations, Journal of Sound and Vibration 257 (4) (2002) 791-797] recently introduced a NSFDM in conservative form. This method captures the shock waves exactly, without any numerical dissipation. In this paper, this algorithm is tested for the case of expansion waves with sonic points and is found to generate unphysical expansion shocks. As a remedy to this defect, we use the strategy of composite schemes [R. Liska, B. Wendroff, Composite schemes for conservation laws, SIAM Journal of Numerical Analysis 35 (6) (1998) 2250-2271] in which the accurate NSFDM is used as the basic scheme and localized relaxation NSFDM is used as the supporting scheme which acts like a filter. Relaxation schemes introduced by Jin and Xin [The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Communications in Pure and Applied Mathematics 48 (1995) 235-276] are based on relaxation systems which replace the nonlinear hyperbolic conservation laws by a semi-linear system with a stiff relaxation term. The relaxation parameter ( λ) is chosen locally
A fifth-order finite difference scheme for hyperbolic equations on block-adaptive curvilinear grids
NASA Astrophysics Data System (ADS)
Chen, Yuxi; Tóth, Gábor; Gombosi, Tamas I.
2016-01-01
We present a new fifth-order accurate finite difference method for hyperbolic equations on block-adaptive curvilinear grids. The scheme employs the 5th order accurate monotonicity preserving limiter MP5 to construct high order accurate face fluxes. The fifth-order accuracy of the spatial derivatives is ensured by a flux correction step. The method is generalized to curvilinear grids with a free-stream preserving discretization. It is also extended to block-adaptive grids using carefully designed ghost cell interpolation algorithms. Only three layers of ghost cells are required, and the grid blocks can be as small as 6 × 6 × 6 cells. Dynamic grid refinement and coarsening are also fifth-order accurate. All interpolation algorithms employ a general limiter based on the principles of the MP5 limiter. The finite difference scheme is fully conservative on static uniform grids. Conservation is only maintained at the truncation error level at grid resolution changes and during grid adaptation, but our numerical tests indicate that the results are still very accurate. We demonstrate the capabilities of the new method on a number of numerical tests, including smooth but non-linear problems as well as simulations involving discontinuities.
On the monotonicity of multidimensional finite difference schemes
NASA Astrophysics Data System (ADS)
Kovyrkina, O.; Ostapenko, V.
2016-10-01
The classical concept of monotonicity, introduced by Godunov for linear one-dimensional difference schemes, is extended to multidimensional case. Necessary and sufficient conditions of monotonicity are obtained for linear multidimensional difference schemes of first order. The constraints on the numerical viscosity are given that ensure the monotonicity of a difference scheme in the multidimensional case. It is proposed a modification of the second order multidimensional CABARET scheme that preserves the monotonicity of one-dimensional discrete solutions and, as a result, ensures higher smoothness in the computation of multidimensional discontinuous solutions. The results of two-dimensional test computations illustrating the advantages of the modified CABARET scheme are presented.
NASA Technical Reports Server (NTRS)
Abramopoulos, Frank
1988-01-01
The conditions under which finite difference schemes for the shallow water equations can conserve both total energy and potential enstrophy are considered. A method of deriving such schemes using operator formalism is developed. Several such schemes are derived for the A-, B- and C-grids. The derived schemes include second-order schemes and pseudo-fourth-order schemes. The simplest B-grid pseudo-fourth-order schemes are presented.
High-Order Entropy Stable Finite Difference Schemes for Nonlinear Conservation Laws: Finite Domains
NASA Technical Reports Server (NTRS)
Fisher, Travis C.; Carpenter, Mark H.
2013-01-01
Developing stable and robust high-order finite difference schemes requires mathematical formalism and appropriate methods of analysis. In this work, nonlinear entropy stability is used to derive provably stable high-order finite difference methods with formal boundary closures for conservation laws. Particular emphasis is placed on the entropy stability of the compressible Navier-Stokes equations. A newly derived entropy stable weighted essentially non-oscillatory finite difference method is used to simulate problems with shocks and a conservative, entropy stable, narrow-stencil finite difference approach is used to approximate viscous terms.
Finite-difference scheme for the numerical solution of the Schroedinger equation
NASA Technical Reports Server (NTRS)
Mickens, Ronald E.; Ramadhani, Issa
1992-01-01
A finite-difference scheme for numerical integration of the Schroedinger equation is constructed. Asymptotically (r goes to infinity), the method gives the exact solution correct to terms of order r exp -2.
Converged accelerated finite difference scheme for the multigroup neutron diffusion equation
Terranova, N.; Mostacci, D.; Ganapol, B. D.
2013-07-01
Computer codes involving neutron transport theory for nuclear engineering applications always require verification to assess improvement. Generally, analytical and semi-analytical benchmarks are desirable, since they are capable of high precision solutions to provide accurate standards of comparison. However, these benchmarks often involve relatively simple problems, usually assuming a certain degree of abstract modeling. In the present work, we show how semi-analytical equivalent benchmarks can be numerically generated using convergence acceleration. Specifically, we investigate the error behavior of a 1D spatial finite difference scheme for the multigroup (MG) steady-state neutron diffusion equation in plane geometry. Since solutions depending on subsequent discretization can be envisioned as terms of an infinite sequence converging to the true solution, extrapolation methods can accelerate an iterative process to obtain the limit before numerical instability sets in. The obtained results have been compared to the analytical solution to the 1D multigroup diffusion equation when available, using FORTRAN as the computational language. Finally, a slowing down problem has been solved using a cascading source update, showing how a finite difference scheme performs for ultra-fine groups (104 groups) in a reasonable computational time using convergence acceleration. (authors)
High-order conservative finite difference GLM-MHD schemes for cell-centered MHD
NASA Astrophysics Data System (ADS)
Mignone, Andrea; Tzeferacos, Petros; Bodo, Gianluigi
2010-08-01
We present and compare third- as well as fifth-order accurate finite difference schemes for the numerical solution of the compressible ideal MHD equations in multiple spatial dimensions. The selected methods lean on four different reconstruction techniques based on recently improved versions of the weighted essentially non-oscillatory (WENO) schemes, monotonicity preserving (MP) schemes as well as slope-limited polynomial reconstruction. The proposed numerical methods are highly accurate in smooth regions of the flow, avoid loss of accuracy in proximity of smooth extrema and provide sharp non-oscillatory transitions at discontinuities. We suggest a numerical formulation based on a cell-centered approach where all of the primary flow variables are discretized at the zone center. The divergence-free condition is enforced by augmenting the MHD equations with a generalized Lagrange multiplier yielding a mixed hyperbolic/parabolic correction, as in Dedner et al. [J. Comput. Phys. 175 (2002) 645-673]. The resulting family of schemes is robust, cost-effective and straightforward to implement. Compared to previous existing approaches, it completely avoids the CPU intensive workload associated with an elliptic divergence cleaning step and the additional complexities required by staggered mesh algorithms. Extensive numerical testing demonstrate the robustness and reliability of the proposed framework for computations involving both smooth and discontinuous features.
Nonlinear wave propagation using three different finite difference schemes (category 2 application)
NASA Technical Reports Server (NTRS)
Pope, D. Stuart; Hardin, J. C.
1995-01-01
Three common finite difference schemes are used to examine the computation of one-dimensional nonlinear wave propagation. The schemes are studied for their responses to numerical parameters such as time step selection, boundary condition implementation, and discretization of governing equations. The performance of the schemes is compared and various numerical phenomena peculiar to each is discussed.
A free surface capturing discretization for the staggered grid finite difference scheme
NASA Astrophysics Data System (ADS)
Duretz, T.; May, D. A.; Yamato, P.
2016-03-01
The coupling that exists between surface processes and deformation within both the shallow crust and the deeper mantle-lithosphere has stimulated the development of computational geodynamic models that incorporate a free surface boundary condition. We introduce a treatment of this boundary condition that is suitable for staggered grid, finite difference schemes employing a structured Eulerian mesh. Our interface capturing treatment discretizes the free surface boundary condition via an interface that conforms with the edges of control volumes (e.g. a `staircase' representation) and requires only local stencil modifications to be performed. Comparisons with analytic solutions verify that the method is first-order accurate. Additional intermodel comparisons are performed between known reference models to further validate our free surface approximation. Lastly, we demonstrate the applicability of a multigrid solver to our free surface methodology and demonstrate that the local stencil modifications do not strongly influence the convergence of the iterative solver.
NASA Astrophysics Data System (ADS)
Bhattacharya, Amitabh
2013-11-01
An efficient algorithm for simulating Stokes flow around particles is presented here, in which a second order Finite Difference method (FDM) is coupled to a Boundary Integral method (BIM). This method utilizes the strong points of FDM (i.e. localized stencil) and BIM (i.e. accurate representation of particle surface). Specifically, in each iteration, the flow field away from the particles is solved on a Cartesian FDM grid, while the traction on the particle surface (given the the velocity of the particle) is solved using BIM. The two schemes are coupled by matching the solution in an intermediate region between the particle and surrounding fluid. We validate this method by solving for flow around an array of cylinders, and find good agreement with Hasimoto's (J. Fluid Mech. 1959) analytical results.
Projection methods for incompressible flow problems with WENO finite difference schemes
NASA Astrophysics Data System (ADS)
de Frutos, Javier; John, Volker; Novo, Julia
2016-03-01
Weighted essentially non-oscillatory (WENO) finite difference schemes have been recommended in a competitive study of discretizations for scalar evolutionary convection-diffusion equations [20]. This paper explores the applicability of these schemes for the simulation of incompressible flows. To this end, WENO schemes are used in several non-incremental and incremental projection methods for the incompressible Navier-Stokes equations. Velocity and pressure are discretized on the same grid. A pressure stabilization Petrov-Galerkin (PSPG) type of stabilization is introduced in the incremental schemes to account for the violation of the discrete inf-sup condition. Algorithmic aspects of the proposed schemes are discussed. The schemes are studied on several examples with different features. It is shown that the WENO finite difference idea can be transferred to the simulation of incompressible flows. Some shortcomings of the methods, which are due to the splitting in projection schemes, become also obvious.
Li, Y.
1997-05-15
This paper proposes some new wavenumber-extended high-order upwind-biased schemes. The dispersion and dissipation errors of upwind-biased finite-difference schemes are assessed and compared by means of a Fourier analysis of the difference schemes. Up to 11th-order upwind-biased schemes are analyzed. It is shown that both the upwind-biased scheme of order 2N - 1 and the corresponding centered differencing scheme of order 2N have the same dispersion characteristics; thus the former can be considered to be the latter plus a correction that reduces the numerical dissipation. The new second-order wavenumber-extended scheme is tested and compared with some well-known schemes. The range of wavenumbers that are accurately treated by the upwind-biased schemes is improved by using additional constraints from the Fourier analysis to construct the new schemes. The anisotropic behavior of the dispersion and dissipation errors is also analyzed for both the conventional and the new wavenumber-extended upwind-biased finite-difference schemes.
NASA Astrophysics Data System (ADS)
Cunha, G.; Redonnet, S.
2014-04-01
The present article aims at highlighting the strengths and weaknesses of the so-called spectral-like optimized (explicit central) finite-difference schemes, when the latter are used for numerically approximating spatial derivatives in aeroacoustics evolution problems. With that view, we first remind how differential operators can be approximated using explicit central finite-difference schemes. The possible spectral-like optimization of the latter is then discussed, the advantages and drawbacks of such an optimization being theoretically studied, before they are numerically quantified. For doing so, two popular spectral-like optimized schemes are assessed via a direct comparison against their standard counterparts, such a comparative exercise being conducted for several academic test cases. At the end, general conclusions are drawn, which allows us discussing the way spectral-like optimized schemes shall be preferred (or not) to standard ones, when it comes to simulate real-life aeroacoustics problems.
A perturbational h[sup 4] exponential finite difference scheme for the convective diffusion equation
Chen, G.Q.; Gao, Z. ); Yang, Z.F. )
1993-01-01
A perturbational h[sup 4] compact exponential finite difference scheme with diagonally dominant coefficient matrix and upwind effect is developed for the convective diffusion equation. Perturbations of second order are exerted on the convective coefficients and source term of an h[sup 2] exponential finite difference scheme proposed in this paper based on a transformation to eliminate the upwind effect of the convective diffusion equation. Four numerical examples including one- to three-dimensional model equations of fluid flow and a problem of natural convective heat transfer are given to illustrate the excellent behavior of the present exponential schemes. Besides, the h[sup 4] accuracy of the perturbational scheme is verified using double precision arithmetic.
A multigrid algorithm for the cell-centered finite difference scheme
NASA Technical Reports Server (NTRS)
Ewing, Richard E.; Shen, Jian
1993-01-01
In this article, we discuss a non-variational V-cycle multigrid algorithm based on the cell-centered finite difference scheme for solving a second-order elliptic problem with discontinuous coefficients. Due to the poor approximation property of piecewise constant spaces and the non-variational nature of our scheme, one step of symmetric linear smoothing in our V-cycle multigrid scheme may fail to be a contraction. Again, because of the simple structure of the piecewise constant spaces, prolongation and restriction are trivial; we save significant computation time with very promising computational results.
Lie group invariant finite difference schemes for the neutron diffusion equation
Jaegers, P.J.
1994-06-01
Finite difference techniques are used to solve a variety of differential equations. For the neutron diffusion equation, the typical local truncation error for standard finite difference approximation is on the order of the mesh spacing squared. To improve the accuracy of the finite difference approximation of the diffusion equation, the invariance properties of the original differential equation have been incorporated into the finite difference equations. Using the concept of an invariant difference operator, the invariant difference approximations of the multi-group neutron diffusion equation were determined in one-dimensional slab and two-dimensional Cartesian coordinates, for multiple region problems. These invariant difference equations were defined to lie upon a cell edged mesh as opposed to the standard difference equations, which lie upon a cell centered mesh. Results for a variety of source approximations showed that the invariant difference equations were able to determine the eigenvalue with greater accuracy, for a given mesh spacing, than the standard difference approximation. The local truncation errors for these invariant difference schemes were found to be highly dependent upon the source approximation used, and the type of source distribution played a greater role in determining the accuracy of the invariant difference scheme than the local truncation error.
Application of the symplectic finite-difference time-domain scheme to electromagnetic simulation
Sha, Wei . E-mail: ws108@ahu.edu.cn; Huang, Zhixiang; Wu, Xianliang; Chen, Mingsheng
2007-07-01
An explicit fourth-order finite-difference time-domain (FDTD) scheme using the symplectic integrator is applied to electromagnetic simulation. A feasible numerical implementation of the symplectic FDTD (SFDTD) scheme is specified. In particular, new strategies for the air-dielectric interface treatment and the near-to-far-field (NFF) transformation are presented. By using the SFDTD scheme, both the radiation and the scattering of three-dimensional objects are computed. Furthermore, the energy-conserving characteristic hold for the SFDTD scheme is verified under long-term simulation. Numerical results suggest that the SFDTD scheme is more efficient than the traditional FDTD method and other high-order methods, and can save computational resources.
2D numerical simulation of the MEP energy-transport model with a finite difference scheme
Romano, V. . E-mail: romano@dmi.unict.it
2007-02-10
A finite difference scheme of Scharfetter-Gummel type is used to simulate a consistent energy-transport model for electron transport in semiconductors devices, free of any fitting parameters, formulated on the basis of the maximum entropy principle. Simulations of silicon n{sup +}-n-n{sup +} diodes, 2D-MESFET and 2D-MOSFET and comparisons with the results obtained by a direct simulation of the Boltzmann transport equation and with other energy-transport models, known in the literature, show the validity of the model and the robustness of the numerical scheme.
Ceotto, Michele; Zhuang, Yu; Hase, William L
2013-02-01
This paper shows how a compact finite difference Hessian approximation scheme can be proficiently implemented into semiclassical initial value representation molecular dynamics. Effects of the approximation on the monodromy matrix calculation are tested by propagating initial sampling distributions to determine power spectra for analytic potential energy surfaces and for "on the fly" carbon dioxide direct dynamics. With the approximation scheme the computational cost is significantly reduced, making ab initio direct semiclassical dynamics computationally more feasible and, at the same time, properly reproducing important quantum effects inherent in the monodromy matrix and the pre-exponential factor of the semiclassical propagator. PMID:23406107
Accelerated direct semiclassical molecular dynamics using a compact finite difference Hessian scheme
NASA Astrophysics Data System (ADS)
Ceotto, Michele; Zhuang, Yu; Hase, William L.
2013-02-01
This paper shows how a compact finite difference Hessian approximation scheme can be proficiently implemented into semiclassical initial value representation molecular dynamics. Effects of the approximation on the monodromy matrix calculation are tested by propagating initial sampling distributions to determine power spectra for analytic potential energy surfaces and for "on the fly" carbon dioxide direct dynamics. With the approximation scheme the computational cost is significantly reduced, making ab initio direct semiclassical dynamics computationally more feasible and, at the same time, properly reproducing important quantum effects inherent in the monodromy matrix and the pre-exponential factor of the semiclassical propagator.
Real-space finite difference scheme for the von Neumann equation with the Dirac Hamiltonian
NASA Astrophysics Data System (ADS)
Schreilechner, Magdalena; Pötz, Walter
2016-07-01
A finite difference scheme for the numerical treatment of the von Neumann equation for the (2+1)D Dirac Hamiltonian is presented. It is based on a sequential left-right (ket-bra) application of a staggered space-time scheme for the pure-state Dirac equation and offers a numerical treatment of the general mixed-state dynamics of an isolated quantum system within the von Neumann equation. Thereby this direct scheme inherits all the favorable features of the finite-difference scheme for the pure-state Dirac equation, such as the single-cone energy-momentum dispersion, convergence conditions, and scaling behavior. A conserved functional is identified. Moreover this scheme is shown to conserve both Hermiticity and positivity. Numerical tests comprise a numerical analysis of stability, as well as the simulation of a mixed-state time-evolution of Gaussian wave functions, illustrating Zitterbewegung and transverse current oscillations. Imaginary-potential absorbing boundary conditions and parameters which pertain to topological insulator surface states were used in the numerical simulations.
An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation
NASA Astrophysics Data System (ADS)
Li, Xiao; Qiao, ZhongHua; Zhang, Hui
2016-09-01
In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation is solved numerically by using the finite difference method in combination with a convex splitting technique of the energy functional. For the non-stochastic case, we develop an unconditionally energy stable difference scheme which is proved to be uniquely solvable. For the stochastic case, by adopting the same splitting of the energy functional, we construct a similar and uniquely solvable difference scheme with the discretized stochastic term. The resulted schemes are nonlinear and solved by Newton iteration. For the long time simulation, an adaptive time stepping strategy is developed based on both first- and second-order derivatives of the energy. Numerical experiments are carried out to verify the energy stability, the efficiency of the adaptive time stepping and the effect of the stochastic term.
NASA Technical Reports Server (NTRS)
Fisher, Travis C.; Carpenter, Mark H.; Yamaleev, Nail K.; Frankel, Steven H.
2009-01-01
A general strategy exists for constructing Energy Stable Weighted Essentially Non Oscillatory (ESWENO) finite difference schemes up to eighth-order on periodic domains. These ESWENO schemes satisfy an energy norm stability proof for both continuous and discontinuous solutions of systems of linear hyperbolic equations. Herein, boundary closures are developed for the fourth-order ESWENO scheme that maintain wherever possible the WENO stencil biasing properties, while satisfying the summation-by-parts (SBP) operator convention, thereby ensuring stability in an L2 norm. Second-order, and third-order boundary closures are developed that achieve stability in diagonal and block norms, respectively. The global accuracy for the second-order closures is three, and for the third-order closures is four. A novel set of non-uniform flux interpolation points is necessary near the boundaries to simultaneously achieve 1) accuracy, 2) the SBP convention, and 3) WENO stencil biasing mechanics.
A truncated implicit high-order finite-difference scheme combined with boundary conditions
NASA Astrophysics Data System (ADS)
Chang, Suo-Liang; Liu, Yang
2013-03-01
In this paper, first we calculate finite-difference coefficients of implicit finitedifference methods (IFDM) for the first- and second-order derivatives on normal grids and firstorder derivatives on staggered grids and find that small coefficients of high-order IFDMs exist. Dispersion analysis demonstrates that omitting these small coefficients can retain approximately the same order accuracy but greatly reduce computational costs. Then, we introduce a mirrorimage symmetric boundary condition to improve IFDMs accuracy and stability and adopt the hybrid absorbing boundary condition (ABC) to reduce unwanted reflections from the model boundary. Last, we give elastic wave modeling examples for homogeneous and heterogeneous models to demonstrate the advantages of the proposed scheme.
NASA Astrophysics Data System (ADS)
Guan, Zhen; Heinonen, Vili; Lowengrub, John; Wang, Cheng; Wise, Steven M.
2016-09-01
In this paper we construct an energy stable finite difference scheme for the amplitude expansion equations for the two-dimensional phase field crystal (PFC) model. The equations are formulated in a periodic hexagonal domain with respect to the reciprocal lattice vectors to achieve a provably unconditionally energy stable and solvable scheme. To our knowledge, this is the first such energy stable scheme for the PFC amplitude equations. The convexity of each part in the amplitude equations is analyzed, in both the semi-discrete and fully-discrete cases. Energy stability is based on a careful convexity analysis for the energy (in both the spatially continuous and discrete cases). As a result, unique solvability and unconditional energy stability are available for the resulting scheme. Moreover, we show that the scheme is point-wise stable for any time and space step sizes. An efficient multigrid solver is devised to solve the scheme, and a few numerical experiments are presented, including grain rotation and shrinkage and grain growth studies, as examples of the strength and robustness of the proposed scheme and solver.
A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws
NASA Astrophysics Data System (ADS)
Zhu, Jun; Qiu, Jianxian
2016-08-01
In this paper a new simple fifth order weighted essentially non-oscillatory (WENO) scheme is presented in the finite difference framework for solving the hyperbolic conservation laws. The new WENO scheme is a convex combination of a fourth degree polynomial with two linear polynomials in a traditional WENO fashion. This new fifth order WENO scheme uses the same five-point information as the classical fifth order WENO scheme [14,20], could get less absolute truncation errors in L1 and L∞ norms, and obtain the same accuracy order in smooth region containing complicated numerical solution structures simultaneously escaping nonphysical oscillations adjacent strong shocks or contact discontinuities. The associated linear weights are artificially set to be any random positive numbers with the only requirement that their sum equals one. New nonlinear weights are proposed for the purpose of sustaining the optimal fifth order accuracy. The new WENO scheme has advantages over the classical WENO scheme [14,20] in its simplicity and easy extension to higher dimensions. Some benchmark numerical tests are performed to illustrate the capability of this new fifth order WENO scheme.
NASA Technical Reports Server (NTRS)
Noor, A. K.; Stephens, W. B.
1973-01-01
Several finite difference schemes are applied to the stress and free vibration analysis of homogeneous isotropic and layered orthotropic shells of revolution. The study is based on a form of the Sanders-Budiansky first-approximation linear shell theory modified such that the effects of shear deformation and rotary inertia are included. A Fourier approach is used in which all the shell stress resultants and displacements are expanded in a Fourier series in the circumferential direction, and the governing equations reduce to ordinary differential equations in the meridional direction. While primary attention is given to finite difference schemes used in conjunction with first order differential equation formulation, comparison is made with finite difference schemes used with other formulations. These finite difference discretization models are compared with respect to simplicity of application, convergence characteristics, and computational efficiency. Numerical studies are presented for the effects of variations in shell geometry and lamination parameters on the accuracy and convergence of the solutions obtained by the different finite difference schemes. On the basis of the present study it is shown that the mixed finite difference scheme based on the first order differential equation formulation and two interlacing grids for the different fundamental unknowns combines a number of advantages over other finite difference schemes previously reported in the literature.
NASA Astrophysics Data System (ADS)
Tsai, T. C.; Yu, H.-S.; Hsieh, M.-S.; Lai, S. H.; Yang, Y.-H.
2015-11-01
Nowadays most of supercomputers are based on the frame of PC cluster; therefore, the efficiency of parallel computing is of importance especially with the increasing computing scale. This paper proposes a high-order implicit predictor-corrector central finite difference (iPCCFD) scheme and demonstrates its high efficiency in parallel computing. Of special interests are the large scale numerical studies such as the magnetohydrodynamic (MHD) simulations in the planetary magnetosphere. An iPCCFD scheme is developed based on fifth-order central finite difference method and fourth-order implicit predictor-corrector method in combination with elimination-of-the-round-off-errors (ERE) technique. We examine several numerical studies such as one-dimensional Brio-Wu shock tube problem, two-dimensional Orszag-Tang vortex system, vortex type K-H instability, kink type K-H instability, field loop advection, and blast wave. All the simulation results are consistent with many literatures. iPCCFD can minimize the numerical instabilities and noises along with the additional diffusion terms. All of our studies present relatively small numerical errors without employing any divergence-free reconstruction. In particular, we obtain fairly stable results in the two-dimensional Brio-Wu shock tube problem which well conserves ∇ ṡ B = 0 throughout the simulation. The ERE technique removes the accumulation of roundoff errors in the uniform or non-disturbed system. We have also shown that iPCCFD is characterized by the high order of accuracy and the low numerical dissipation in the circularly polarized Alfvén wave tests. The proposed iPCCFD scheme is a parallel-efficient and high precision numerical scheme for solving the MHD equations in hyperbolic conservation systems.
Second-order accurate nonoscillatory schemes for scalar conservation laws
NASA Technical Reports Server (NTRS)
Huynh, Hung T.
1989-01-01
Explicit finite difference schemes for the computation of weak solutions of nonlinear scalar conservation laws is presented and analyzed. These schemes are uniformly second-order accurate and nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time.
Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes
NASA Astrophysics Data System (ADS)
Don, Wai-Sun; Borges, Rafael
2013-10-01
In the reconstruction step of (2r-1) order weighted essentially non-oscillatory conservative finite difference schemes (WENO) for solving hyperbolic conservation laws, nonlinear weights αk and ωk, such as the WENO-JS weights by Jiang et al. and the WENO-Z weights by Borges et al., are designed to recover the formal (2r-1) order (optimal order) of the upwinded central finite difference scheme when the solution is sufficiently smooth. The smoothness of the solution is determined by the lower order local smoothness indicators βk in each substencil. These nonlinear weight formulations share two important free parameters in common: the power p, which controls the amount of numerical dissipation, and the sensitivity ε, which is added to βk to avoid a division by zero in the denominator of αk. However, ε also plays a role affecting the order of accuracy of WENO schemes, especially in the presence of critical points. It was recently shown that, for any design order (2r-1), ε should be of Ω(Δx2) (Ω(Δxm) means that ε⩾CΔxm for some C independent of Δx, as Δx→0) for the WENO-JS scheme to achieve the optimal order, regardless of critical points. In this paper, we derive an alternative proof of the sufficient condition using special properties of βk. Moreover, it is unknown if the WENO-Z scheme should obey the same condition on ε. Here, using same special properties of βk, we prove that in fact the optimal order of the WENO-Z scheme can be guaranteed with a much weaker condition ε=Ω(Δxm), where m(r,p)⩾2 is the optimal sensitivity order, regardless of critical points. Both theoretical results are confirmed numerically on smooth functions with arbitrary order of critical points. This is a highly desirable feature, as illustrated with the Lax problem and the Mach 3 shock-density wave interaction of one dimensional Euler equations, for a smaller ε allows a better essentially non-oscillatory shock capturing as it does not over-dominate over the size of
Optimized finite-difference (DRP) schemes perform poorly for decaying or growing oscillations
NASA Astrophysics Data System (ADS)
Brambley, E. J.
2016-11-01
Computational aeroacoustics often use finite difference schemes optimized to require relatively few points per wavelength; such optimized schemes are often called Dispersion Relation Preserving (DRP). Similar techniques are also used outside aeroacoustics. Here the question is posed: what is the equivalent of points per wavelength for growing or decaying waves, and how well are such waves resolved numerically? Such non-constant-amplitude waves are common in aeroacoustics, such as the exponential decay caused by acoustic linings, the O (1 / r) decay of an expanding spherical wave, and the decay of high-azimuthal-order modes in the radial direction towards the centre of a cylindrical duct. It is shown that optimized spatial derivatives perform poorly for waves that are not of constant amplitude, under performing maximal-order schemes. An equivalent criterion to points per wavelength is proposed for non-constant-amplitude oscillations, reducing to the standard definition for constant-amplitude oscillations and valid even for pure growth or decay with no oscillation. Using this definition, coherent statements about points per wavelength necessary for a given accuracy can be made for maximal-order schemes applied to non-constant-amplitude oscillations. These features are illustrated through a numerical example of a one-dimensional wave propagating through a damping region.
Bensa, Julien; Bilbao, Stefan; Kronland-Martinet, Richard; Smith, Julius O
2003-08-01
A model of transverse piano string vibration, second order in time, which models frequency-dependent loss and dispersion effects is presented here. This model has many desirable properties, in particular that it can be written as a well-posed initial-boundary value problem (permitting stable finite difference schemes) and that it may be directly related to a digital waveguide model, a digital filter-based algorithm which can be used for musical sound synthesis. Techniques for the extraction of model parameters from experimental data over the full range of the grand piano are discussed, as is the link between the model parameters and the filter responses in a digital waveguide. Simulations are performed. Finally, the waveguide model is extended to the case of several coupled strings. PMID:12942987
Nonstandard finite difference scheme for SIRS epidemic model with disease-related death
NASA Astrophysics Data System (ADS)
Fitriah, Z.; Suryanto, A.
2016-04-01
It is well known that SIRS epidemic with disease-related death can be described by a system of nonlinear ordinary differential equations (NL ODEs). This model has two equilibrium points where their existence and stability properties are determined by the basic reproduction number [1]. Besides the qualitative properties, it is also often needed to solve the system of NL ODEs. Euler method and 4th order Runge-Kutta (RK4) method are often used to solve the system of NL ODEs. However, both methods may produce inconsistent qualitative properties of the NL ODEs such as converging to wrong equilibrium point, etc. In this paper we apply non-standard finite difference (NSFD) scheme (see [2,3]) to approximate the solution of SIRS epidemic model with disease-related death. It is shown that the discrete system obtained by NSFD scheme is dynamically consistent with the continuous model. By our numerical simulations, we find that the solutions of NSFD scheme are always positive, bounded and convergent to the correct equilibrium point for any step size of integration (h), while those of Euler or RK4 method have the same properties only for relatively small h.
NASA Astrophysics Data System (ADS)
Christlieb, Andrew J.; Rossmanith, James A.; Tang, Qi
2014-07-01
In this work we develop a class of high-order finite difference weighted essentially non-oscillatory (FD-WENO) schemes for solving the ideal magnetohydrodynamic (MHD) equations in 2D and 3D. The philosophy of this work is to use efficient high-order WENO spatial discretizations with high-order strong stability-preserving Runge-Kutta (SSP-RK) time-stepping schemes. Numerical results have shown that with such methods we are able to resolve solution structures that are only visible at much higher grid resolutions with lower-order schemes. The key challenge in applying such methods to ideal MHD is to control divergence errors in the magnetic field. We achieve this by augmenting the base scheme with a novel high-order constrained transport approach that updates the magnetic vector potential. The predicted magnetic field from the base scheme is replaced by a divergence-free magnetic field that is obtained from the curl of this magnetic potential. The non-conservative weakly hyperbolic system that the magnetic vector potential satisfies is solved using a version of FD-WENO developed for Hamilton-Jacobi equations. The resulting numerical method is endowed with several important properties: (1) all quantities, including all components of the magnetic field and magnetic potential, are treated as point values on the same mesh (i.e., there is no mesh staggering); (2) both the spatial and temporal orders of accuracy are fourth-order; (3) no spatial integration or multidimensional reconstructions are needed in any step; and (4) special limiters in the magnetic vector potential update are used to control unphysical oscillations in the magnetic field. Several 2D and 3D numerical examples are presented to verify the order of accuracy on smooth test problems and to show high-resolution on test problems that involve shocks.
High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains
NASA Astrophysics Data System (ADS)
Fisher, Travis C.; Carpenter, Mark H.
2013-11-01
Nonlinear entropy stability is used to derive provably stable high-order finite difference operators including boundary closure stencils, for the compressible Navier-Stokes equations. A comparison technique is used to derive a new Entropy Stable Weighted Essentially Non-Oscillatory (SSWENO) finite difference method, appropriate for simulations of problems with shocks. Viscous terms are approximated using conservative, entropy stable, narrow-stencil finite difference operators. The efficacy of the new discrete operators is demonstrated using both smooth and discontinuous test cases.
A simple parallel prefix algorithm for compact finite-difference schemes
NASA Technical Reports Server (NTRS)
Sun, Xian-He; Joslin, Ronald D.
1993-01-01
A compact scheme is a discretization scheme that is advantageous in obtaining highly accurate solutions. However, the resulting systems from compact schemes are tridiagonal systems that are difficult to solve efficiently on parallel computers. Considering the almost symmetric Toeplitz structure, a parallel algorithm, simple parallel prefix (SPP), is proposed. The SPP algorithm requires less memory than the conventional LU decomposition and is highly efficient on parallel machines. It consists of a prefix communication pattern and AXPY operations. Both the computation and the communication can be truncated without degrading the accuracy when the system is diagonally dominant. A formal accuracy study was conducted to provide a simple truncation formula. Experimental results were measured on a MasPar MP-1 SIMD machine and on a Cray 2 vector machine. Experimental results show that the simple parallel prefix algorithm is a good algorithm for the compact scheme on high-performance computers.
Landing-gear noise prediction using high-order finite difference schemes
NASA Astrophysics Data System (ADS)
Liu, Wen; Wook Kim, Jae; Zhang, Xin; Angland, David; Caruelle, Bastien
2013-07-01
Aerodynamic noise from a generic two-wheel landing-gear model is predicted by a CFD/FW-H hybrid approach. The unsteady flow-field is computed using a compressible Navier-Stokes solver based on high-order finite difference schemes and a fully structured grid. The calculated time history of the surface pressure data is used in an FW-H solver to predict the far-field noise levels. Both aerodynamic and aeroacoustic results are compared to wind tunnel measurements and are found to be in good agreement. The far-field noise was found to vary with the 6th power of the free-stream velocity. Individual contributions from three components, i.e. wheels, axle and strut of the landing-gear model are also investigated to identify the relative contribution to the total noise by each component. It is found that the wheels are the dominant noise source in general. Strong vortex shedding from the axle is the second major contributor to landing-gear noise. This work is part of Airbus LAnding Gear nOise database for CAA validatiON (LAGOON) program with the general purpose of evaluating current CFD/CAA and experimental techniques for airframe noise prediction.
Fuzzy logic to improve efficiency of finite element and finite difference schemes
Garcia, M.D.; Heger, A.S.
1994-05-01
This paper explores possible applications of logic in the areas of finite element and finite difference methods applied to engineering design problems. The application of fuzzy logic to both front-end selection of computational options and within the numerical computation itself are proposed. Further, possible methods of overcoming these limitations through the application of methods are explored. Decision strategy is a fundamental limitation in performing finite element calculations, such as selecting the optimum coarseness of the grid, numerical integration algorithm, element type, implicit versus explicit schemes, and the like. This is particularly true of novice analysts who are confronted with a myriad of choices in performing a calculation. The advantage of having the myriad of options available to the analyst is, however, that it improves and optimizes the design process if the appropriate ones are selected. Unfortunately, the optimum choices are not always apparent and only through the process of elimination or prior extensive experience can the optimum choices or combination of choices be selected. The knowledge of expert analysts could be integrated into a fuzzy ``front-end`` rule-based package to optimize the design process. The use of logic to capture the heuristic and human knowledge for selecting optimum solution strategies sets the framework for these proposed strategies.
A High Order Finite Difference Scheme with Sharp Shock Resolution for the Euler Equations
NASA Technical Reports Server (NTRS)
Gerritsen, Margot; Olsson, Pelle
1996-01-01
We derive a high-order finite difference scheme for the Euler equations that satisfies a semi-discrete energy estimate, and present an efficient strategy for the treatment of discontinuities that leads to sharp shock resolution. The formulation of the semi-discrete energy estimate is based on a symmetrization of the Euler equations that preserves the homogeneity of the flux vector, a canonical splitting of the flux derivative vector, and the use of difference operators that satisfy a discrete analogue to the integration by parts procedure used in the continuous energy estimate. Around discontinuities or sharp gradients, refined grids are created on which the discrete equations are solved after adding a newly constructed artificial viscosity. The positioning of the sub-grids and computation of the viscosity are aided by a detection algorithm which is based on a multi-scale wavelet analysis of the pressure grid function. The wavelet theory provides easy to implement mathematical criteria to detect discontinuities, sharp gradients and spurious oscillations quickly and efficiently.
NASA Astrophysics Data System (ADS)
Liu, Zhe; Lin, Lei; Xie, Lian; Gao, Huiwang
2016-10-01
To improve the efficiency of the terrain-following σ-coordinate non-hydrostatic ocean model, a partially implicit finite difference (PIFD) scheme is proposed. By using explicit terms instead of implicit terms to discretize the parts of the vertical dynamic pressure gradient derived from the σ-coordinate transformation, the coefficient matrix of the discrete Poisson equation that the dynamic pressure satisfies can be simplified from 15 diagonals to 7 diagonals. The PIFD scheme is shown to run stably when it is applied to simulate five benchmark cases, namely, a standing wave in a basin, a surface solitary wave, a lock-exchange problem, a periodic wave over a bar and a tidally induced internal wave. Compared with the conventional fully implicit finite difference (FIFD) scheme, the PIFD scheme produces simulation results of equivalent accuracy at only 40-60% of the computational cost. The PIFD scheme demonstrates strong applicability and can be easily implemented in σ-coordinate ocean models.
NASA Astrophysics Data System (ADS)
Wang, Cheng; Dong, XinZhuang; Shu, Chi-Wang
2015-10-01
For numerical simulation of detonation, computational cost using uniform meshes is large due to the vast separation in both time and space scales. Adaptive mesh refinement (AMR) is advantageous for problems with vastly different scales. This paper aims to propose an AMR method with high order accuracy for numerical investigation of multi-dimensional detonation. A well-designed AMR method based on finite difference weighted essentially non-oscillatory (WENO) scheme, named as AMR&WENO is proposed. A new cell-based data structure is used to organize the adaptive meshes. The new data structure makes it possible for cells to communicate with each other quickly and easily. In order to develop an AMR method with high order accuracy, high order prolongations in both space and time are utilized in the data prolongation procedure. Based on the message passing interface (MPI) platform, we have developed a workload balancing parallel AMR&WENO code using the Hilbert space-filling curve algorithm. Our numerical experiments with detonation simulations indicate that the AMR&WENO is accurate and has a high resolution. Moreover, we evaluate and compare the performance of the uniform mesh WENO scheme and the parallel AMR&WENO method. The comparison results provide us further insight into the high performance of the parallel AMR&WENO method.
NASA Technical Reports Server (NTRS)
Mickens, Ronald E.
1989-01-01
A family of conditionally stable, forward Euler finite difference equations can be constructed for the simplest equation of Schroedinger type, namely u sub t - iu sub xx. Generalization of this result to physically realistic Schroedinger type equations is presented.
NASA Astrophysics Data System (ADS)
Dalguer, L. A.; Day, S. M.
2006-12-01
Accuracy in finite difference (FD) solutions to spontaneous rupture problems is controlled principally by the scheme used to represent the fault discontinuity, and not by the grid geometry used to represent the continuum. We have numerically tested three fault representation methods, the Thick Fault (TF) proposed by Madariaga et al (1998), the Stress Glut (SG) described by Andrews (1999), and the Staggered-Grid Split-Node (SGSN) methods proposed by Dalguer and Day (2006), each implemented in a the fourth-order velocity-stress staggered-grid (VSSG) FD scheme. The TF and the SG methods approximate the discontinuity through inelastic increments to stress components ("inelastic-zone" schemes) at a set of stress grid points taken to lie on the fault plane. With this type of scheme, the fault surface is indistinguishable from an inelastic zone with a thickness given by a spatial step dx for the SG, and 2dx for the TF model. The SGSN method uses the traction-at-split-node (TSN) approach adapted to the VSSG FD. This method represents the fault discontinuity by explicitly incorporating discontinuity terms at velocity nodes in the grid, with interactions between the "split nodes" occurring exclusively through the tractions (frictional resistance) acting between them. These tractions in turn are controlled by the jump conditions and a friction law. Our 3D tests problem solutions show that the inelastic-zone TF and SG methods show much poorer performance than does the SGSN formulation. The SG inelastic-zone method achieved solutions that are qualitatively meaningful and quantitatively reliable to within a few percent. The TF inelastic-zone method did not achieve qualitatively agreement with the reference solutions to the 3D test problem, and proved to be sufficiently computationally inefficient that it was not feasible to explore convergence quantitatively. The SGSN method gives very accurate solutions, and is also very efficient. Reliable solution of the rupture time is
A staggered mesh finite difference scheme for the computation of hypersonic Euler flows
NASA Technical Reports Server (NTRS)
Sanders, Richard
1991-01-01
A shock capturing finite difference method for systems of hyperbolic conservation laws is presented which avoids the need to solve Riemann problems while being competitive in performance with other current methods. A staggered spatial mesh is employed, so that complicated nonlinear waves generated at cell interfaces are averaged over cell interiors at the next time level. The full method combines to form a conservative version of the modified method of characteristics. The advantages of the method are discussed, and numerical results are presented for the two-dimensional double ellipse problem.
NASA Astrophysics Data System (ADS)
Li, Y.; Han, B.; Métivier, L.; Brossier, R.
2016-09-01
We investigate an optimal fourth-order staggered-grid finite-difference scheme for 3D frequency-domain viscoelastic wave modeling. An anti-lumped mass strategy is incorporated to minimize the numerical dispersion. The optimal finite-difference coefficients and the mass weighting coefficients are obtained by minimizing the misfit between the normalized phase velocities and the unity. An iterative damped least-squares method, the Levenberg-Marquardt algorithm, is utilized for the optimization. Dispersion analysis shows that the optimal fourth-order scheme presents less grid dispersion and anisotropy than the conventional fourth-order scheme with respect to different Poisson's ratios. Moreover, only 3.7 grid-points per minimum shear wavelength are required to keep the error of the group velocities below 1%. The memory cost is then greatly reduced due to a coarser sampling. A parallel iterative method named CARP-CG is used to solve the large ill-conditioned linear system for the frequency-domain modeling. Validations are conducted with respect to both the analytic viscoacoustic and viscoelastic solutions. Compared with the conventional fourth-order scheme, the optimal scheme generates wavefields having smaller error under the same discretization setups. Profiles of the wavefields are presented to confirm better agreement between the optimal results and the analytic solutions.
NASA Astrophysics Data System (ADS)
Tan, Sirui; Huang, Lianjie
2014-11-01
For modeling scalar-wave propagation in geophysical problems using finite-difference schemes, optimizing the coefficients of the finite-difference operators can reduce numerical dispersion. Most optimized finite-difference schemes for modeling seismic-wave propagation suppress only spatial but not temporal dispersion errors. We develop a novel optimized finite-difference scheme for numerical scalar-wave modeling to control dispersion errors not only in space but also in time. Our optimized scheme is based on a new stencil that contains a few more grid points than the standard stencil. We design an objective function for minimizing relative errors of phase velocities of waves propagating in all directions within a given range of wavenumbers. Dispersion analysis and numerical examples demonstrate that our optimized finite-difference scheme is computationally up to 2.5 times faster than the optimized schemes using the standard stencil to achieve the similar modeling accuracy for a given 2D or 3D problem. Compared with the high-order finite-difference scheme using the same new stencil, our optimized scheme reduces 50 percent of the computational cost to achieve the similar modeling accuracy. This new optimized finite-difference scheme is particularly useful for large-scale 3D scalar-wave modeling and inversion.
Tan, Sirui; Huang, Lianjie
2014-11-01
For modeling scalar-wave propagation in geophysical problems using finite-difference schemes, optimizing the coefficients of the finite-difference operators can reduce numerical dispersion. Most optimized finite-difference schemes for modeling seismic-wave propagation suppress only spatial but not temporal dispersion errors. We develop a novel optimized finite-difference scheme for numerical scalar-wave modeling to control dispersion errors not only in space but also in time. Our optimized scheme is based on a new stencil that contains a few more grid points than the standard stencil. We design an objective function for minimizing relative errors of phase velocities of waves propagating in all directions within a given range of wavenumbers. Dispersion analysis and numerical examples demonstrate that our optimized finite-difference scheme is computationally up to 2.5 times faster than the optimized schemes using the standard stencil to achieve the similar modeling accuracy for a given 2D or 3D problem. Compared with the high-order finite-difference scheme using the same new stencil, our optimized scheme reduces 50 percent of the computational cost to achieve the similar modeling accuracy. This new optimized finite-difference scheme is particularly useful for large-scale 3D scalar-wave modeling and inversion.
NASA Astrophysics Data System (ADS)
Oberman, Adam M.; Salvador, Tiago
2015-03-01
We build a simple and general class of finite difference schemes for first order Hamilton-Jacobi (HJ) Partial Differential Equations. These filtered schemes are convergent to the unique viscosity solution of the equation. The schemes are accurate: we implement second, third and fourth order accurate schemes in one dimension and second order accurate schemes in two dimensions, indicating how to build higher order ones. They are also explicit, which means they can be solved using the fast sweeping method. The accuracy of the method is validated with computational results for the eikonal equation and other HJ equations in one and two dimensions, using filtered schemes made from standard centered differences, higher order upwinding and ENO interpolation.
Son, Sang-Kil
2011-03-01
We introduce a new numerical grid-based method on unstructured grids in the three-dimensional real-space to investigate the electronic structure of polyatomic molecules. The Voronoi-cell finite difference (VFD) method realizes a discrete Laplacian operator based on Voronoi cells and their natural neighbors, featuring high adaptivity and simplicity. To resolve multicenter Coulomb singularity in all-electron calculations of polyatomic molecules, this method utilizes highly adaptive molecular grids which consist of spherical atomic grids. It provides accurate and efficient solutions for the Schroedinger equation and the Poisson equation with the all-electron Coulomb potentials regardless of the coordinate system and the molecular symmetry. For numerical examples, we assess accuracy of the VFD method for electronic structures of one-electron polyatomic systems, and apply the method to the density-functional theory for many-electron polyatomic molecules.
NASA Astrophysics Data System (ADS)
Etemadsaeed, Leila; Moczo, Peter; Kristek, Jozef; Ansari, Anooshiravan; Kristekova, Miriam
2016-10-01
We investigate the problem of finite-difference approximations of the velocity-stress formulation of the equation of motion and constitutive law on the staggered grid (SG) and collocated grid (CG). For approximating the first spatial and temporal derivatives, we use three approaches: Taylor expansion (TE), dispersion-relation preserving (DRP), and combined TE-DRP. The TE and DRP approaches represent two fundamental extremes. We derive useful formulae for DRP and TE-DRP approximations. We compare accuracy of the numerical wavenumbers and numerical frequencies of the basic TE, DRP and TE-DRP approximations. Based on the developed approximations, we construct and numerically investigate 14 basic TE, DRP and TE-DRP finite-difference schemes on SG and CG. We find that (1) the TE second-order in time, TE fourth-order in space, 2-point in time, 4-point in space SG scheme (that is the standard (2,4) VS SG scheme, say TE-2-4-2-4-SG) is the best scheme (of the 14 investigated) for large fractions of the maximum possible time step, or, in other words, in a homogeneous medium; (2) the TE second-order in time, combined TE-DRP second-order in space, 2-point in time, 4-point in space SG scheme (say TE-DRP-2-2-2-4-SG) is the best scheme for small fractions of the maximum possible time step, or, in other words, in models with large velocity contrasts if uniform spatial grid spacing and time step are used. The practical conclusion is that in computer codes based on standard TE-2-4-2-4-SG, it is enough to redefine the values of the approximation coefficients by those of TE-DRP-2-2-2-4-SG for increasing accuracy of modelling in models with large velocity contrast between rock and sediments.
Temporal and spatial inconsistencies of time-split finite-difference schemes
NASA Technical Reports Server (NTRS)
Dwoyer, D. L.; Thames, F. C.
1981-01-01
The properties of an implicit time-split algorithm, which utilizes locally one dimensional spatial steps, are examined using the two-dimensional heat conduction equation as the test problem. Both temporal and spatial inconsistencies inherent in the scheme are identified. A consistent, implicit splitting approach is developed. The relationship between this method and other time-split implicit schemes is explained, and stability problems encountered with the method in three dimensions are discussed.
The stability of numerical boundary treatments for compact high-order finite-difference schemes
NASA Technical Reports Server (NTRS)
Carpenter, Mark H.; Gottlieb, David; Abarbanel, Saul
1991-01-01
The stability characteristics of various compact fourth and sixth order spatial operators are assessed using the theory of Gustafsson, Kreiss and Sundstrom (G-K-S) for the semi-discrete Initial Boundary Value Problem (IBVP). These results are then generalized to the fully discrete case using a recently developed theory of Kreiss. In all cases, favorable comparisons are obtained between the G-K-S theory, eigenvalue determination, and numerical simulation. The conventional definition of stability is then sharpened to include only those spatial discretizations that are asymptotically stable. It is shown that many of the higher order schemes which are G-K-S stable are not asymptotically stable. A series of compact fourth and sixth order schemes, which are both asymptotically and G-K-S stable for the scalar case, are then developed.
NASA Astrophysics Data System (ADS)
Kim, Jae Wook
2013-05-01
This paper proposes a novel systematic approach for the parallelization of pentadiagonal compact finite-difference schemes and filters based on domain decomposition. The proposed approach allows a pentadiagonal banded matrix system to be split into quasi-disjoint subsystems by using a linear-algebraic transformation technique. As a result the inversion of pentadiagonal matrices can be implemented within each subdomain in an independent manner subject to a conventional halo-exchange process. The proposed matrix transformation leads to new subdomain boundary (SB) compact schemes and filters that require three halo terms to exchange with neighboring subdomains. The internode communication overhead in the present approach is equivalent to that of standard explicit schemes and filters based on seven-point discretization stencils. The new SB compact schemes and filters demand additional arithmetic operations compared to the original serial ones. However, it is shown that the additional cost becomes sufficiently low by choosing optimal sizes of their discretization stencils. Compared to earlier published results, the proposed SB compact schemes and filters successfully reduce parallelization artifacts arising from subdomain boundaries to a level sufficiently negligible for sophisticated aeroacoustic simulations without degrading parallel efficiency. The overall performance and parallel efficiency of the proposed approach are demonstrated by stringent benchmark tests.
Uniformly high order accurate essentially non-oscillatory schemes 3
NASA Technical Reports Server (NTRS)
Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S. R.
1986-01-01
In this paper (a third in a series) the construction and the analysis of essentially non-oscillatory shock capturing methods for the approximation of hyperbolic conservation laws are presented. Also presented is a hierarchy of high order accurate schemes which generalizes Godunov's scheme and its second order accurate MUSCL extension to arbitrary order of accuracy. The design involves an essentially non-oscillatory piecewise polynomial reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem, and averaging of this approximate solution over each cell. The reconstruction algorithm is derived from a new interpolation technique that when applied to piecewise smooth data gives high-order accuracy whenever the function is smooth but avoids a Gibbs phenomenon at discontinuities. Unlike standard finite difference methods this procedure uses an adaptive stencil of grid points and consequently the resulting schemes are highly nonlinear.
NASA Astrophysics Data System (ADS)
Arsoski, V. V.; Čukarić, N. A.; Tadić, M. Ž.; Peeters, F. M.
2015-12-01
The electron states in axially symmetric quantum wires are computed by means of the effective-mass Schrödinger equation, which is written in cylindrical coordinates φ, ρ, and z. We show that a direct discretization of the Schrödinger equation by central finite differences leads to a non-symmetric Hamiltonian matrix. Because diagonalization of such matrices is more complex it is advantageous to transform it in a symmetric form. This can be done by the Liouville-like transformation proposed by Rizea et al. (2008), which replaces the wave function ψ(ρ) with the function F(ρ) = ψ(ρ) √{ ρ } and transforms the Hamiltonian accordingly. Even though a symmetric Hamiltonian matrix is produced by this procedure, the computed wave functions are found to be inaccurate near the origin, and the accuracy of the energy levels is not very high. In order to improve on this, we devised a finite-difference scheme which discretizes the Schrödinger equation in the first step, and then applies the Liouville-like transformation to the difference equation. Such a procedure gives a symmetric Hamiltonian matrix, resulting in an accuracy comparable to the one obtained with the finite element method. The superior efficiency of the new finite-difference scheme (FDM) is demonstrated for a few ρ-dependent one-dimensional potentials which are usually employed to model the electron states in free-standing and core-shell quantum wires. The new scheme is compared with the other FDM schemes for solving the effective-mass Schrödinger equation, and is found to deliver energy levels with much smaller numerical error for all the analyzed potentials. It also gives more accurate results than the scheme of Rizea et al., except for the ground state of an infinite rectangular potential in freestanding quantum wires. Moreover, the PT symmetry is invoked to explain similarities and differences between the considered FDM schemes.
NASA Technical Reports Server (NTRS)
Carpenter, Mark H.; Gottlieb, David; Abarbanel, Saul
1993-01-01
We present a systematic method for constructing boundary conditions (numerical and physical) of the required accuracy, for compact (Pade-like) high-order finite-difference schemes for hyperbolic systems. First, a roper summation-by-parts formula is found for the approximate derivative. A 'simultaneous approximation term' (SAT) is then introduced to treat the boundary conditions. This procedure leads to time-stable schemes even in the system case. An explicit construction of the fourth-order compact case is given. Numerical studies are presented to verify the efficacy of the approach.
NASA Technical Reports Server (NTRS)
Abarbanel, S.; Gottlieb, D.
1976-01-01
The paper considers the leap-frog finite-difference method (Kreiss and Oliger, 1973) for systems of partial differential equations of the form du/dt = dF/dx + dG/dy + dH/dz, where d denotes partial derivative, u is a q-component vector and a function of x, y, z, and t, and the vectors F, G, and H are functions of u only. The original leap-frog algorithm is shown to admit a modification that improves on the stability conditions for two and three dimensions by factors of 2 and 2.8, respectively, thereby permitting larger time steps. The scheme for three dimensions is considered optimal in the sense that it combines simple averaging and large time steps.
NASA Astrophysics Data System (ADS)
Byun, Jaeseung; Bodony, Daniel; Pantano, Carlos
2014-11-01
Improved order-of-accuracy discretizations often require careful consideration of their numerical stability. We report on new high-order finite difference schemes using Summation-By-Parts (SBP) operators along with the Simultaneous-Approximation-Terms (SAT) boundary condition treatment for first and second-order spatial derivatives with variable coefficients. In particular, we present a highly accurate operator for SBP-SAT-based approximations of second-order derivatives with variable coefficients for Dirichlet and Neumann boundary conditions. These terms are responsible for approximating the physical dissipation of kinetic and thermal energy in a simulation, and contain grid metrics when the grid is curvilinear. Analysis using the Laplace transform method shows that strong stability is ensured with Dirichlet boundary conditions while weaker stability is obtained for Neumann boundary conditions. Furthermore, the benefits of the scheme is shown in the direct numerical simulation (DNS) of a Mach 1.5 compressible turbulent supersonic jet using curvilinear grids and skew-symmetric discretization. Particularly, we show that the improved methods allow minimization of the numerical filter often employed in these simulations and we discuss the qualities of the simulation.
NASA Technical Reports Server (NTRS)
Madavan, Nateri K.
1995-01-01
The work in this report was conducted at NASA Ames Research Center during the period from August 1993 to January 1995 deals with the direct numerical simulation of transitional and turbulent flow at low Mach numbers using high-order-accurate finite-difference techniques. A computation of transition to turbulence of the spatially-evolving boundary layer on a heated flat plate in the presence of relatively high freestream turbulence was performed. The geometry and flow conditions were chosen to match earlier experiments. The development of the momentum and thermal boundary layers was documented. Velocity and temperature profiles, as well as distributions of skin friction, surface heat transfer rate, Reynolds shear stress, and turbulent heat flux were shown to compare well with experiment. The numerical method used here can be applied to complex geometries in a straightforward manner.
NASA Technical Reports Server (NTRS)
Madavan, Nateri K.
1995-01-01
This report deals with the direct numerical simulation of transitional and turbulent flow at low Mach numbers using high-order-accurate finite-difference techniques. A computation of transition to turbulence of the spatially-evolving boundary layer on a heated flat plate in the presence of relatively high freestream turbulence was performed. The geometry and flow conditions were chosen to match earlier experiments. The development of the momentum and thermal boundary layers was documented. Velocity and temperature profiles, as well as distributions of skin friction, surface heat transfer rate, Reynolds shear stress, and turbulent heat flux, were shown to compare well with experiment. The results indicate that the essential features of the transition process have been captured. The numerical method used here can be applied to complex geometries in a straightforward manner.
A high order accurate difference scheme for complex flow fields
Dexun Fu; Yanwen Ma
1997-06-01
A high order accurate finite difference method for direct numerical simulation of coherent structure in the mixing layers is presented. The reason for oscillation production in numerical solutions is analyzed. It is caused by a nonuniform group velocity of wavepackets. A method of group velocity control for the improvement of the shock resolution is presented. In numerical simulation the fifth-order accurate upwind compact difference relation is used to approximate the derivatives in the convection terms of the compressible N-S equations, a sixth-order accurate symmetric compact difference relation is used to approximate the viscous terms, and a three-stage R-K method is used to advance in time. In order to improve the shock resolution the scheme is reconstructed with the method of diffusion analogy which is used to control the group velocity of wavepackets. 18 refs., 12 figs., 1 tab.
NASA Technical Reports Server (NTRS)
Steger, J. L.; Dougherty, F. C.; Benek, J. A.
1983-01-01
A mesh system composed of multiple overset body-conforming grids is described for adapting finite-difference procedures to complex aircraft configurations. In this so-called 'chimera mesh,' a major grid is generated about a main component of the configuration and overset minor grids are used to resolve all other features. Methods for connecting overset multiple grids and modifications of flow-simulation algorithms are discussed. Computational tests in two dimensions indicate that the use of multiple overset grids can simplify the task of grid generation without an adverse effect on flow-field algorithms and computer code complexity.
A compact fourth-order finite difference scheme for the three-dimensional Cahn-Hilliard equation
NASA Astrophysics Data System (ADS)
Li, Yibao; Lee, Hyun Geun; Xia, Binhu; Kim, Junseok
2016-03-01
This work extends the previous two-dimensional compact scheme for the Cahn-Hilliard equation (Lee et al., 2014) to three-dimensional space. The proposed scheme, derived by combining a compact formula and a linearly stabilized splitting scheme, has second-order accuracy in time and fourth-order accuracy in space. The discrete system is conservative and practically stable. We also implement the compact scheme in a three-dimensional adaptive mesh refinement framework. The resulting system of discrete equations is solved by using a multigrid. We demonstrate the performance of our proposed algorithm by several numerical experiments.
Navier-Stokes simulations of blade-vortex interaction using high-order accurate upwind schemes
NASA Technical Reports Server (NTRS)
Rai, Man Mohan
1987-01-01
Conventional, spatially second-order-accurate, finite-difference schemes are much too dissipative for calculations involving vortices that travel large distances (relative to some measure of the size of the vortex). This study presents a fifth-order-accurate upwind-biased scheme that preserves vortex structure for much longer times than existing second-order-accurate central and upwind difference schemes. Vortex calculations demonstrating this aspect of the fifth-order scheme are also presented. The method is then applied to the blade-vortex interaction problem. Results for strong interactions wherein the vortex impinges directly on the airfoil or a shock associated with the airfoil are presented. None of these calculations required any modeling of the shape, size, and trajectory of the interacting vortex.
NASA Astrophysics Data System (ADS)
Kumar, A.; Rudy, D. H.; Drummond, J. P.; Harris, J. E.
1982-08-01
Several two- and three-dimensional external and internal flow problems solved on the STAR-100 and CYBER-203 vector processing computers are described. The flow field was described by the full Navier-Stokes equations which were then solved by explicit finite-difference algorithms. Problem results and computer system requirements are presented. Program organization and data base structure for three-dimensional computer codes which will eliminate or improve on page faulting, are discussed. Storage requirements for three-dimensional codes are reduced by calculating transformation metric data in each step. As a result, in-core grid points were increased in number by 50% to 150,000, with a 10% execution time increase. An assessment of current and future machine requirements shows that even on the CYBER-205 computer only a few problems can be solved realistically. Estimates reveal that the present situation is more storage limited than compute rate limited, but advancements in both storage and speed are essential to realistically calculate three-dimensional flow.
NASA Technical Reports Server (NTRS)
Kumar, A.; Rudy, D. H.; Drummond, J. P.; Harris, J. E.
1982-01-01
Several two- and three-dimensional external and internal flow problems solved on the STAR-100 and CYBER-203 vector processing computers are described. The flow field was described by the full Navier-Stokes equations which were then solved by explicit finite-difference algorithms. Problem results and computer system requirements are presented. Program organization and data base structure for three-dimensional computer codes which will eliminate or improve on page faulting, are discussed. Storage requirements for three-dimensional codes are reduced by calculating transformation metric data in each step. As a result, in-core grid points were increased in number by 50% to 150,000, with a 10% execution time increase. An assessment of current and future machine requirements shows that even on the CYBER-205 computer only a few problems can be solved realistically. Estimates reveal that the present situation is more storage limited than compute rate limited, but advancements in both storage and speed are essential to realistically calculate three-dimensional flow.
NASA Astrophysics Data System (ADS)
Ono, Tomoya; Egami, Yoshiyuki; Hirose, Kikuji
2012-11-01
We demonstrate an efficient nonequilibrium Green's function transport calculation procedure based on the real-space finite-difference method. The direct inversion of matrices for obtaining the self-energy terms of electrodes is computationally demanding in the real-space method because the matrix dimension corresponds to the number of grid points in the unit cell of electrodes, which is much larger than that of sites in the tight-binding approach. The procedure using the ratio matrices of the overbridging boundary-matching technique [Y. Fujimoto and K. Hirose, Phys. Rev. BPRBMDO1098-012110.1103/PhysRevB.67.195315 67, 195315 (2003)], which is related to the wave functions of a couple of grid planes in the matching regions, greatly reduces the computational effort to calculate self-energy terms without losing mathematical strictness. In addition, the present procedure saves computational time to obtain the Green's function of the semi-infinite system required in the Landauer-Büttiker formula. Moreover, the compact expression to relate Green's functions and scattering wave functions, which provide a real-space picture of the scattering process, is introduced. An example of the calculated results is given for the transport property of the BN ring connected to (9,0) carbon nanotubes. The wave-function matching at the interface reveals that the rotational symmetry of wave functions with respect to the tube axis plays an important role in electron transport. Since the states coming from and going to electrodes show threefold rotational symmetry, the states in the vicinity of the Fermi level, the wave function of which exhibits fivefold symmetry, do not contribute to the electron transport through the BN ring.
Uniformly second-order-accurate essentially nonoscillatory schemes for the Euler equations
NASA Astrophysics Data System (ADS)
Yang, J. Y.
1990-12-01
Two time-level explicit and implicit finite-difference shock-capturing schemes based on the characteristic flux difference splitting method and the modified flux approach with the essentially nonoscillatory (ENO) property of Harten and Osher have been developed for the two-dimensional Euler equations. The methods are conservative, uniformly second-order accurate in time and space, even at local extrema. General coordinate systems are used to treat complex geometries. Standard alternating direction implicit approximate factorization is used for constructing implicit schemes. Numerical results have been obtained for unsteady shock wave reflection around general two-dimensional blunt bodies and for steady transonic flows over a circular arc bump in a channel. Properties of ENO schemes as applied to two-dimensional flows with multiple embedded discontinuities are discussed. Comparisons of the performance between the present ENO schemes and the previous total variation diminishing schemes is also included.
Second-order accurate difference schemes on highly irregular meshes
Manteuffel, T.A.; White, A.B. Jr.
1988-01-01
In this paper compact-as-possible second-order accurate difference schemes will be constructed for boundary-value problems of arbitrary order on highly irregular meshes. It will be shown that for equations of order (K) these schemes will have truncation error of order (3/endash/K). This phenomena is known as supraconvergence. 7 refs.
NASA Astrophysics Data System (ADS)
Song, Xiaolei; Fomel, Sergey; Ying, Lexing
2013-05-01
We introduce a novel finite-difference (FD) approach for seismic wave extrapolation in time. We derive the coefficients of the finite-difference operator from a lowrank approximation of the space-wavenumber, wave-propagator matrix. Applying the technique of lowrank finite-differences, we also improve the finite difference scheme of the two-way Fourier finite differences (FFD). We call the new operator lowrank Fourier finite differences (LFFD). Both the lowrank FD and lowrank FFD methods can be applied to enhance accuracy in seismic imaging by reverse-time migration. Numerical examples confirm the validity of the proposed technique.
Adaptive finite difference for seismic wavefield modelling in acoustic media.
Yao, Gang; Wu, Di; Debens, Henry Alexander
2016-01-01
Efficient numerical seismic wavefield modelling is a key component of modern seismic imaging techniques, such as reverse-time migration and full-waveform inversion. Finite difference methods are perhaps the most widely used numerical approach for forward modelling, and here we introduce a novel scheme for implementing finite difference by introducing a time-to-space wavelet mapping. Finite difference coefficients are then computed by minimising the difference between the spatial derivatives of the mapped wavelet and the finite difference operator over all propagation angles. Since the coefficients vary adaptively with different velocities and source wavelet bandwidths, the method is capable to maximise the accuracy of the finite difference operator. Numerical examples demonstrate that this method is superior to standard finite difference methods, while comparable to Zhang's optimised finite difference scheme. PMID:27491333
Adaptive finite difference for seismic wavefield modelling in acoustic media
NASA Astrophysics Data System (ADS)
Yao, Gang; Wu, Di; Debens, Henry Alexander
2016-08-01
Efficient numerical seismic wavefield modelling is a key component of modern seismic imaging techniques, such as reverse-time migration and full-waveform inversion. Finite difference methods are perhaps the most widely used numerical approach for forward modelling, and here we introduce a novel scheme for implementing finite difference by introducing a time-to-space wavelet mapping. Finite difference coefficients are then computed by minimising the difference between the spatial derivatives of the mapped wavelet and the finite difference operator over all propagation angles. Since the coefficients vary adaptively with different velocities and source wavelet bandwidths, the method is capable to maximise the accuracy of the finite difference operator. Numerical examples demonstrate that this method is superior to standard finite difference methods, while comparable to Zhang’s optimised finite difference scheme.
Adaptive finite difference for seismic wavefield modelling in acoustic media
Yao, Gang; Wu, Di; Debens, Henry Alexander
2016-01-01
Efficient numerical seismic wavefield modelling is a key component of modern seismic imaging techniques, such as reverse-time migration and full-waveform inversion. Finite difference methods are perhaps the most widely used numerical approach for forward modelling, and here we introduce a novel scheme for implementing finite difference by introducing a time-to-space wavelet mapping. Finite difference coefficients are then computed by minimising the difference between the spatial derivatives of the mapped wavelet and the finite difference operator over all propagation angles. Since the coefficients vary adaptively with different velocities and source wavelet bandwidths, the method is capable to maximise the accuracy of the finite difference operator. Numerical examples demonstrate that this method is superior to standard finite difference methods, while comparable to Zhang’s optimised finite difference scheme. PMID:27491333
Adaptive finite difference for seismic wavefield modelling in acoustic media.
Yao, Gang; Wu, Di; Debens, Henry Alexander
2016-08-05
Efficient numerical seismic wavefield modelling is a key component of modern seismic imaging techniques, such as reverse-time migration and full-waveform inversion. Finite difference methods are perhaps the most widely used numerical approach for forward modelling, and here we introduce a novel scheme for implementing finite difference by introducing a time-to-space wavelet mapping. Finite difference coefficients are then computed by minimising the difference between the spatial derivatives of the mapped wavelet and the finite difference operator over all propagation angles. Since the coefficients vary adaptively with different velocities and source wavelet bandwidths, the method is capable to maximise the accuracy of the finite difference operator. Numerical examples demonstrate that this method is superior to standard finite difference methods, while comparable to Zhang's optimised finite difference scheme.
Applications of an exponential finite difference technique
NASA Technical Reports Server (NTRS)
Handschuh, Robert F.; Keith, Theo G., Jr.
1988-01-01
An exponential finite difference scheme first presented by Bhattacharya for one dimensional unsteady heat conduction problems in Cartesian coordinates was extended. The finite difference algorithm developed was used to solve the unsteady diffusion equation in one dimensional cylindrical coordinates and was applied to two and three dimensional conduction problems in Cartesian coordinates. Heat conduction involving variable thermal conductivity was also investigated. The method was used to solve nonlinear partial differential equations in one and two dimensional Cartesian coordinates. Predicted results are compared to exact solutions where available or to results obtained by other numerical methods.
Practical Schemes for Accurate Forces in Quantum Monte Carlo.
Moroni, S; Saccani, S; Filippi, C
2014-11-11
While the computation of interatomic forces has become a well-established practice within variational Monte Carlo (VMC), the use of the more accurate Fixed-Node Diffusion Monte Carlo (DMC) method is still largely limited to the computation of total energies on structures obtained at a lower level of theory. Algorithms to compute exact DMC forces have been proposed in the past, and one such scheme is also put forward in this work, but remain rather impractical due to their high computational cost. As a practical route to DMC forces, we therefore revisit here an approximate method, originally developed in the context of correlated sampling and named here the Variational Drift-Diffusion (VD) approach. We thoroughly investigate its accuracy by checking the consistency between the approximate VD force and the derivative of the DMC potential energy surface for the SiH and C2 molecules and employ a wide range of wave functions optimized in VMC to assess its robustness against the choice of trial function. We find that, for all but the poorest wave function, the discrepancy between force and energy is very small over all interatomic distances, affecting the equilibrium bond length obtained with the VD forces by less than 0.004 au. Furthermore, when the VMC forces are approximate due to the use of a partially optimized wave function, the DMC forces have smaller errors and always lead to an equilibrium distance in better agreement with the experimental value. We also show that the cost of computing the VD forces is only slightly larger than the cost of calculating the DMC energy. Therefore, the VD approximation represents a robust and efficient approach to compute accurate DMC forces, superior to the VMC counterparts.
New finite difference formulas for numerical differentiation
NASA Astrophysics Data System (ADS)
Khan, Ishtiaq Rasool; Ohba, Ryoji
2000-12-01
Conventional numerical differentiation formulas based on interpolating polynomials, operators and lozenge diagrams can be simplified to one of the finite difference approximations based on Taylor series, and closed-form expressions of these finite difference formulas have already been presented. In this paper, we present new finite difference formulas, which are more accurate than the available ones, especially for the oscillating functions having frequency components near the Nyquist frequency. Closed-form expressions of the new formulas are given for arbitrary order. A comparison of the previously available three types of approximations is given with the presented formulas. A computer program written in MATHEMATICA, based on new formulas is given in the appendix for numerical differentiation of a function at a specified mesh point.
NASA Technical Reports Server (NTRS)
Venkatachari, Balaji Shankar; Streett, Craig L.; Chang, Chau-Lyan; Friedlander, David J.; Wang, Xiao-Yen; Chang, Sin-Chung
2016-01-01
Despite decades of development of unstructured mesh methods, high-fidelity time-accurate simulations are still predominantly carried out on structured, or unstructured hexahedral meshes by using high-order finite-difference, weighted essentially non-oscillatory (WENO), or hybrid schemes formed by their combinations. In this work, the space-time conservation element solution element (CESE) method is used to simulate several flow problems including supersonic jet/shock interaction and its impact on launch vehicle acoustics, and direct numerical simulations of turbulent flows using tetrahedral meshes. This paper provides a status report for the continuing development of the space-time conservation element solution element (CESE) numerical and software framework under the Revolutionary Computational Aerosciences (RCA) project. Solution accuracy and large-scale parallel performance of the numerical framework is assessed with the goal of providing a viable paradigm for future high-fidelity flow physics simulations.
High resolution schemes for hyperbolic conservation laws
NASA Technical Reports Server (NTRS)
Harten, A.
1983-01-01
A class of new explicit second order accurate finite difference schemes for the computation of weak solutions of hyperbolic conservation laws is presented. These highly nonlinear schemes are obtained by applying a nonoscillatory first order accurate scheme to an appropriately modified flux function. The so-derived second order accurate schemes achieve high resolution while preserving the robustness of the original nonoscillatory first order accurate scheme. Numerical experiments are presented to demonstrate the performance of these new schemes.
Accurate Monotonicity - Preserving Schemes With Runge-Kutta Time Stepping
NASA Technical Reports Server (NTRS)
Suresh, A.; Huynh, H. T.
1997-01-01
A new class of high-order monotonicity-preserving schemes for the numerical solution of conservation laws is presented. The interface value in these schemes is obtained by limiting a higher-order polynominal reconstruction. The limiting is designed to preserve accuracy near extrema and to work well with Runge-Kutta time stepping. Computational efficiency is enhanced by a simple test that determines whether the limiting procedure is needed. For linear advection in one dimension, these schemes are shown as well as the Euler equations also confirm their high accuracy, good shock resolution, and computational efficiency.
Uniformly high-order accurate non-oscillatory schemes, 1
NASA Technical Reports Server (NTRS)
Harten, A.; Osher, S.
1985-01-01
The construction and the analysis of nonoscillatory shock capturing methods for the approximation of hyperbolic conservation laws was begun. These schemes share many desirable properties with total variation diminishing schemes (TVD), but TVD schemes have at most first order accuracy, in the sense of truncation error, at extreme of the solution. A uniformly second order approximation was constucted, which is nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time. This is achieved via a nonoscillatory piecewise linear reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem, and averaging of this approximate solution over each cell.
Cotta, R.M.; Gerk, J.E.V. . Dept. de Engenharia Mecanica)
1994-06-01
The integral transform method is employed in conjunction with second-order-accurate explicit finite-differences schemes, to handle accurately a class of parabolic-hyperbolic problems that appear in connection with transient forced convection inside ducts. The integral transformation process eliminates the independent variables in which the diffusion phenomena predominate. A system of coupled hyperbolic equations then results, involving time and the space coordinates in which convection is dominant, which is solved numerically through a modified upwind second-order finite-difference scheme. Stability and convergence characteristics of the proposed mixed approach are also examined. Typical applications in two- and three-dimensional geometries are considered, for both slug and laminar flow situations.
Finite-difference model for 3-D flow in bays and estuaries
Smith, Peter E.; Larock, Bruce E.; ,
1993-01-01
This paper describes a semi-implicit finite-difference model for the numerical solution of three-dimensional flow in bays and estuaries. The model treats the gravity wave and vertical diffusion terms in the governing equations implicitly, and other terms explicitly. The model achieves essentially second-order accurate and stable solutions in strongly nonlinear problems by using a three-time-level leapfrog-trapezoidal scheme for the time integration.
Highly accurate adaptive finite element schemes for nonlinear hyperbolic problems
NASA Astrophysics Data System (ADS)
Oden, J. T.
1992-08-01
This document is a final report of research activities supported under General Contract DAAL03-89-K-0120 between the Army Research Office and the University of Texas at Austin from July 1, 1989 through June 30, 1992. The project supported several Ph.D. students over the contract period, two of which are scheduled to complete dissertations during the 1992-93 academic year. Research results produced during the course of this effort led to 6 journal articles, 5 research reports, 4 conference papers and presentations, 1 book chapter, and two dissertations (nearing completion). It is felt that several significant advances were made during the course of this project that should have an impact on the field of numerical analysis of wave phenomena. These include the development of high-order, adaptive, hp-finite element methods for elastodynamic calculations and high-order schemes for linear and nonlinear hyperbolic systems. Also, a theory of multi-stage Taylor-Galerkin schemes was developed and implemented in the analysis of several wave propagation problems, and was configured within a general hp-adaptive strategy for these types of problems. Further details on research results and on areas requiring additional study are given in the Appendix.
Weighted average finite difference methods for fractional diffusion equations
NASA Astrophysics Data System (ADS)
Yuste, S. B.
2006-07-01
A class of finite difference methods for solving fractional diffusion equations is considered. These methods are an extension of the weighted average methods for ordinary (non-fractional) diffusion equations. Their accuracy is of order (Δ x) 2 and Δ t, except for the fractional version of the Crank-Nicholson method, where the accuracy with respect to the timestep is of order (Δ t) 2 if a second-order approximation to the fractional time-derivative is used. Their stability is analyzed by means of a recently proposed procedure akin to the standard von Neumann stability analysis. A simple and accurate stability criterion valid for different discretization schemes of the fractional derivative, arbitrary weight factor, and arbitrary order of the fractional derivative, is found and checked numerically. Some examples are provided in which the new methods' numerical solutions are obtained and compared against exact solutions.
Tam, C.K.W.; Webb, J.C. )
1994-07-01
In this paper finite-difference solutions of the Helmholtz equation in an open domain are considered. By using a second-order central difference scheme and the Bayliss-Turkel radiation boundary condition, reasonably accurate solutions can be obtained when the number of grid points per acoustic wavelength used is large. However, when a smaller number of grid points per wavelength is used excessive reflections occur which tend to overwhelm the computed solutions. Excessive reflections are due to the incompatibility between the governing finite difference equation and the Bayliss-Turkel radiation boundary condition. The Bayliss-Turkel radiation boundary condition was developed from the asymptotic solution of the partial differential equation. To obtain compatibility, the radiation boundary condition should be constructed from the asymptotic solution of the finite difference equation instead. Examples are provided using the improved radiation boundary condition based on the asymptotic solution of the governing finite difference equation. The computed results are free of reflections even when only five grid points per wavelength are used. The improved radiation boundary condition has also been tested for problems with complex acoustic sources and sources embedded in a uniform mean flow. The present method of developing a radiation boundary condition is also applicable to higher order finite difference schemes. In all these cases no reflected waves could be detected. The use of finite difference approximation inevitably introduces anisotropy into the governing field equation. The effect of anisotropy is to distort the directional distribution of the amplitude and phase of the computed solution. It can be quite large when the number of grid points per wavelength used in the computation is small. A way to correct this effect is proposed. 15 refs., 15 figs.
The Benard problem: A comparison of finite difference and spectral collocation eigen value solutions
NASA Technical Reports Server (NTRS)
Skarda, J. Raymond Lee; Mccaughan, Frances E.; Fitzmaurice, Nessan
1995-01-01
The application of spectral methods, using a Chebyshev collocation scheme, to solve hydrodynamic stability problems is demonstrated on the Benard problem. Implementation of the Chebyshev collocation formulation is described. The performance of the spectral scheme is compared with that of a 2nd order finite difference scheme. An exact solution to the Marangoni-Benard problem is used to evaluate the performance of both schemes. The error of the spectral scheme is at least seven orders of magnitude smaller than finite difference error for a grid resolution of N = 15 (number of points used). The performance of the spectral formulation far exceeded the performance of the finite difference formulation for this problem. The spectral scheme required only slightly more effort to set up than the 2nd order finite difference scheme. This suggests that the spectral scheme may actually be faster to implement than higher order finite difference schemes.
NASA Astrophysics Data System (ADS)
Aristova, E. N.; Rogov, B. V.; Chikitkin, A. V.
2016-06-01
A hybrid scheme is proposed for solving the nonstationary inhomogeneous transport equation. The hybridization procedure is based on two baseline schemes: (1) a bicompact one that is fourth-order accurate in all space variables and third-order accurate in time and (2) a monotone first-order accurate scheme from the family of short characteristic methods with interpolation over illuminated faces. It is shown that the first-order accurate scheme has minimal dissipation, so it is called optimal. The solution of the hybrid scheme depends locally on the solutions of the baseline schemes at each node of the space-time grid. A monotonization procedure is constructed continuously and uniformly in all mesh cells so as to keep fourth-order accuracy in space and third-order accuracy in time in domains where the solution is smooth, while maintaining a high level of accuracy in domains of discontinuous solution. Due to its logical simplicity and uniformity, the algorithm is well suited for supercomputer simulation.
Fast solvers for finite difference approximations for the Stokes and Navier-Stokes equations
Shin, D.
1992-01-01
The authors consider several methods for solving the linear equations arising from finite difference discretizations of the Stokes equations. The pressure equation method presented here for the first time, apparently, and the method, presented by Bramble and Pasciak, are shown to have computational effort that grows slowly with the number of grid points. The methods work with second-order accurate discretizations. Computational results are shown for both the Stokes and incompressible Navier-Stokes at low Reynolds number. The inf-sup conditions resulting from three finite difference approximations of the Stokes equations are proven. These conditions are used to prove that the Schur complement Q[sub h] of the linear system generated by each of these approximations is bounded uniformly away from zero. For the pressure equation method, this guarantees that the conjugate gradient method applied to Q[sub h] converges in a finite number of iterations which is independent of mesh size. The fact that Q[sub h] is bounded below is used to prove convergence estimates for the solutions generated by these finite difference approximations. One of the estimates is for a staggered grid and the estimate of the scheme shows that both the pressure and the velocity parts of the solution are second-order accurate. Iterative methods are compared by the use of the regularized central differencing introduced by Strikwerda. Several finite difference approximations of the Stokes equations by the SOR method are compared and the excellence of the approximations by the regularized central differencing over the other finite difference approximation is mentioned. This difference gives rise to a linear equation with a matrix which is slightly non-symmetric. The convergence of the typical steepest descent method and conjugate gradient method, which is almost as same as the typical conjugate gradient method, applied to slightly non-symmetric positive definite matrices are proven.
A semi-implicit finite difference model for three-dimensional tidal circulation,
Casulli, V.; Cheng, R.T.
1992-01-01
A semi-implicit finite difference formulation for the numerical solution of three-dimensional tidal circulation is presented. The governing equations are the three-dimensional Reynolds equations in which the pressure is assumed to be hydrostatic. A minimal degree of implicitness has been introduced in the finite difference formula so that in the absence of horizontal viscosity the resulting algorithm is unconditionally stable at a minimal computational cost. When only one vertical layer is specified this method reduces, as a particular case, to a semi-implicit scheme for the solutions of the corresponding two-dimensional shallow water equations. The resulting two- and three-dimensional algorithm is fast, accurate and mass conservative. This formulation includes the simulation of flooding and drying of tidal flats, and is fully vectorizable for an efficient implementation on modern vector computers.
Fourth-Order Accurate IDO Scheme Using Gradient-Staggered Interpolation
NASA Astrophysics Data System (ADS)
Imai, Yohsuke; Aoki, Takayuki
An Interpolated Differential Operator (IDO) scheme using a new interpolation function is proposed. The gradient of the dependent variable is calculated at the position shifted by a half grid size from that of the physical value. A fourth-order Hermite-interpolation function is constructed locally using both the value and the gradient defined at staggered positions. The numerical solutions for the Poisson, diffusion, advection and wave equations have fourth-order accuracy in space. In particular, for the Poisson and diffusion equations, the Gradient-Staggered (G-S) IDO scheme shows better accuracy than the original IDO scheme. As a practical application, the Direct Numerical Simulation (DNS) for two-dimensional isotropic homogeneous turbulence is examined and a comparable result with that of the original IDO scheme is obtained. The G-S IDO scheme clearly contributes to high-accurate computations for solving partial differential equations in computational mechanics.
ACCURATE ORBITAL INTEGRATION OF THE GENERAL THREE-BODY PROBLEM BASED ON THE D'ALEMBERT-TYPE SCHEME
Minesaki, Yukitaka
2013-03-15
We propose an accurate orbital integration scheme for the general three-body problem that retains all conserved quantities except angular momentum. The scheme is provided by an extension of the d'Alembert-type scheme for constrained autonomous Hamiltonian systems. Although the proposed scheme is merely second-order accurate, it can precisely reproduce some periodic, quasiperiodic, and escape orbits. The Levi-Civita transformation plays a role in designing the scheme.
Accurate Orbital Integration of the General Three-body Problem Based on the d'Alembert-type Scheme
NASA Astrophysics Data System (ADS)
Minesaki, Yukitaka
2013-03-01
We propose an accurate orbital integration scheme for the general three-body problem that retains all conserved quantities except angular momentum. The scheme is provided by an extension of the d'Alembert-type scheme for constrained autonomous Hamiltonian systems. Although the proposed scheme is merely second-order accurate, it can precisely reproduce some periodic, quasiperiodic, and escape orbits. The Levi-Civita transformation plays a role in designing the scheme.
Finite-Difference Algorithms For Computing Sound Waves
NASA Technical Reports Server (NTRS)
Davis, Sanford
1993-01-01
Governing equations considered as matrix system. Method variant of method described in "Scheme for Finite-Difference Computations of Waves" (ARC-12970). Present method begins with matrix-vector formulation of fundamental equations, involving first-order partial derivatives of primitive variables with respect to space and time. Particular matrix formulation places time and spatial coordinates on equal footing, so governing equations considered as matrix system and treated as unit. Spatial and temporal discretizations not treated separately as in other finite-difference methods, instead treated together by linking spatial-grid interval and time step via common scale factor related to speed of sound.
High-resolution schemes for hyperbolic conservation laws
NASA Technical Reports Server (NTRS)
Harten, A.
1982-01-01
A class of new explicit second order accurate finite difference schemes for the computation of weak solutions of hyperbolic conservation laws is presented. These highly nonlinear schemes are obtained by applying a nonoscillatory first order accurae scheme to an appropriately modified flux function. The so derived second order accurate schemes achieve high resolution while preserving the robustness of the original nonoscillatory first order accurate scheme.
Extending geometric conservation law to cell-centered finite difference methods on stationary grids
NASA Astrophysics Data System (ADS)
Liao, Fei; Ye, Zhengyin; Zhang, Lingxia
2015-03-01
In a wide range of high-order high-resolution schemes, the finite difference method (FDM) is a suitable selection for accurate numerical calculations because it efficiently reduces dispersion and dissipation errors. FDM is easier to perform to obtain high-order capabilities than the finite volume method (FVM). Most FDMs are node-centered; such techniques include weighted essentially non-oscillatory schemes (WENO) [1], weighted compact nonlinear schemes (WCNS) [2,3], dissipative compact schemes (DCS) [4], and compact central schemes [5,6]. WENO represents a class of nonlinear high-order high-resolution shock-capture schemes derived by Shu [1]; this technique can be successfully used in multiscale flow simulation problems. WCNS is another nonlinear high-order shock-capture scheme derived by Deng and Zhang. WCNS uses interpolation and not reconstruction to obtain half-node values and features a better spectral resolution than WENO. Deng et al. [4] further developed linear DCS with a free parameter to control upwind tendency and thus decrease the dissipation of upwind schemes. Furthermore, compact central scheme proposed by Lele [5] and developed by Visbal and Gaitonde [6] plays a dominant role for research on large eddy simulation and direct numerical simulation because of its ultra-high-order and spectral-like resolution.
NASA Astrophysics Data System (ADS)
Ohkawauchi, K.; Takenaka, H.
2006-12-01
We propose a new method for the calculation of seismic wave propagation using the interpolated differential operator (IDO, Aoki,1997) which is a numerical method for solving the partial differential equations and is based on a high accurate interpolation of the profile for the independent variables over a local area. It improves the accuracy of wave computation with high accuracy because the local interpolation can represent high order behavior of wave field between grid points. In addition, locality of this approach makes possible treatment of boundary conditions exactly. In this study, we address computation of plane-wave responses of vertically heterogeneous structure models. We then solve the elastodynamic equation for plane wave derived by Tanaka and Takenaka (2005). The equations to be solved in our method are not only velocity-stress equations but also the corresponding ones integrated over each cell between adjacent grid points. We use two staggered-grid systems which can be non-uniform, and then discretize the governing equations using a finite-difference scheme of second-order accurate in time, and the second-order Hermite interpolation in space. In this method, the second-order Hermite interpolation of particle velocity or stress is obtained from the values at the adjacent two grid points and the integration value at the cell between the grid points. The time marching of the original and integrated quantities are proceeded, and in the following time step the quantities are computed on the alternative grid system to that used in the current time step. In implementation of a free-surface boundary condition, all field quantities locate just on the free surface. Their computational accuracy is the same order as those in the other spatial domain. We also implement the interface condition in a similarly way to the free surface condition. We used some simple models to test the scheme. The results showed that the waveforms calculated by our method fit the
NASA Astrophysics Data System (ADS)
Lee, Dongwook
2013-06-01
In this paper, we extend the unsplit staggered mesh scheme (USM) for 2D magnetohydrodynamics (MHD) [D. Lee, A.E. Deane, An unsplit staggered mesh scheme for multidimensional magnetohydrodynamics, J. Comput. Phys. 228 (2009) 952-975] to a full 3D MHD scheme. The scheme is a finite-volume Godunov method consisting of a constrained transport (CT) method and an efficient and accurate single-step, directionally unsplit multidimensional data reconstruction-evolution algorithm, which extends Colella's original 2D corner transport upwind (CTU) method [P. Colella, Multidimensional upwind methods for hyperbolic conservation laws, J. Comput. Phys. 87 (1990) 446-466]. We present two types of data reconstruction-evolution algorithms for 3D: (1) a reduced CTU scheme and (2) a full CTU scheme. The reduced 3D CTU scheme is a variant of a simple 3D extension of Collela's 2D CTU method and is considered as a direct extension from the 2D USM scheme. The full 3D CTU scheme is our primary 3D solver which includes all multidimensional cross-derivative terms for stability. The latter method is logically analogous to the 3D unsplit CTU method by Saltzman [J. Saltzman, An unsplit 3D upwind method for hyperbolic conservation laws, J. Comput. Phys. 115 (1994) 153-168]. The major novelties in our algorithms are twofold. First, we extend the reduced CTU scheme to the full CTU scheme which is able to run with CFL numbers close to unity. Both methods utilize the transverse update technique developed in the 2D USM algorithm to account for transverse fluxes without solving intermediate Riemann problems, which in turn gives cost-effective 3D methods by reducing the total number of Riemann solves. The proposed algorithms are simple and efficient especially when including multidimensional MHD terms that maintain in-plane magnetic field dynamics. Second, we introduce a new CT scheme that makes use of proper upwind information in taking averages of electric fields. Our 3D USM schemes can be easily
NASA Astrophysics Data System (ADS)
Choi, S.-J.; Giraldo, F. X.; Kim, J.; Shin, S.
2014-06-01
The non-hydrostatic (NH) compressible Euler equations of dry atmosphere are solved in a simplified two dimensional (2-D) slice framework employing a spectral element method (SEM) for the horizontal discretization and a finite difference method (FDM) for the vertical discretization. The SEM uses high-order nodal basis functions associated with Lagrange polynomials based on Gauss-Lobatto-Legendre (GLL) quadrature points. The FDM employs a third-order upwind biased scheme for the vertical flux terms and a centered finite difference scheme for the vertical derivative terms and quadrature. The Euler equations used here are in a flux form based on the hydrostatic pressure vertical coordinate, which are the same as those used in the Weather Research and Forecasting (WRF) model, but a hybrid sigma-pressure vertical coordinate is implemented in this model. We verified the model by conducting widely used standard benchmark tests: the inertia-gravity wave, rising thermal bubble, density current wave, and linear hydrostatic mountain wave. The results from those tests demonstrate that the horizontally spectral element vertically finite difference model is accurate and robust. By using the 2-D slice model, we effectively show that the combined spatial discretization method of the spectral element and finite difference method in the horizontal and vertical directions, respectively, offers a viable method for the development of a NH dynamical core.
A time-accurate high-resolution TVD scheme for solving the Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Kim, Hyun Dae; Liu, Nan-Suey
1992-01-01
A total variation diminishing (TVD) scheme has been developed and incorporated into an existing time-accurate high-resolution Navier-Stokes code. The accuracy and the robustness of the resulting solution procedure have been assessed by performing many calculations in four different areas: shock tube flows, regular shock reflection, supersonic boundary layer, and shock boundary layer interactions. These numerical results compare well with corresponding exact solutions or experimental data.
A time-accurate high-resolution TVD scheme for solving the Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Kim, Hyun D.; Liu, Nan-Suey
1993-01-01
A total variation diminishing (TVD) scheme has been developed and incorporated into an existing time-accurate high-resolution Navier-Stokes code. The accuracy and the robustness of the resulting solution procedure have been assessed by performing many calculations in four different areas: shock tube flows, regular shock reflection, supersonic boundary layer, and shock boundary layer interactions. These numerical results compare well with corresponding exact solutions or experimental data.
Accurate scoring of non-uniform sampling schemes for quantitative NMR
Aoto, Phillip C.; Fenwick, R. Bryn; Kroon, Gerard J. A.; Wright, Peter E.
2014-01-01
Non-uniform sampling (NUS) in NMR spectroscopy is a recognized and powerful tool to minimize acquisition time. Recent advances in reconstruction methodologies are paving the way for the use of NUS in quantitative applications, where accurate measurement of peak intensities is crucial. The presence or absence of NUS artifacts in reconstructed spectra ultimately determines the success of NUS in quantitative NMR. The quality of reconstructed spectra from NUS acquired data is dependent upon the quality of the sampling scheme. Here we demonstrate that the best performing sampling schemes make up a very small percentage of the total randomly generated schemes. A scoring method is found to accurately predict the quantitative similarity between reconstructed NUS spectra and those of fully sampled spectra. We present an easy-to-use protocol to batch generate and rank optimal Poisson-gap NUS schedules for use with 2D NMR with minimized noise and accurate signal reproduction, without the need for the creation of synthetic spectra. PMID:25063954
A time accurate finite volume high resolution scheme for three dimensional Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Liou, Meng-Sing; Hsu, Andrew T.
1989-01-01
A time accurate, three-dimensional, finite volume, high resolution scheme for solving the compressible full Navier-Stokes equations is presented. The present derivation is based on the upwind split formulas, specifically with the application of Roe's (1981) flux difference splitting. A high-order accurate (up to the third order) upwind interpolation formula for the inviscid terms is derived to account for nonuniform meshes. For the viscous terms, discretizations consistent with the finite volume concept are described. A variant of second-order time accurate method is proposed that utilizes identical procedures in both the predictor and corrector steps. Avoiding the definition of midpoint gives a consistent and easy procedure, in the framework of finite volume discretization, for treating viscous transport terms in the curvilinear coordinates. For the boundary cells, a new treatment is introduced that not only avoids the use of 'ghost cells' and the associated problems, but also satisfies the tangency conditions exactly and allows easy definition of viscous transport terms at the first interface next to the boundary cells. Numerical tests of steady and unsteady high speed flows show that the present scheme gives accurate solutions.
A Review of High-Order and Optimized Finite-Difference Methods for Simulating Linear Wave Phenomena
NASA Technical Reports Server (NTRS)
Zingg, David W.
1996-01-01
This paper presents a review of high-order and optimized finite-difference methods for numerically simulating the propagation and scattering of linear waves, such as electromagnetic, acoustic, or elastic waves. The spatial operators reviewed include compact schemes, non-compact schemes, schemes on staggered grids, and schemes which are optimized to produce specific characteristics. The time-marching methods discussed include Runge-Kutta methods, Adams-Bashforth methods, and the leapfrog method. In addition, the following fourth-order fully-discrete finite-difference methods are considered: a one-step implicit scheme with a three-point spatial stencil, a one-step explicit scheme with a five-point spatial stencil, and a two-step explicit scheme with a five-point spatial stencil. For each method studied, the number of grid points per wavelength required for accurate simulation of wave propagation over large distances is presented. Recommendations are made with respect to the suitability of the methods for specific problems and practical aspects of their use, such as appropriate Courant numbers and grid densities. Avenues for future research are suggested.
A comparative investigation on recurrence formulae in finite difference methods
NASA Astrophysics Data System (ADS)
Haberland, Christoph; Lahrmann, Andreas
1988-06-01
To solve the transient heat conduction equation, the Pade approximation is introduced into the finite-difference method. But if the time step is chosen too large relative to the element size, the Euler method and the Crank-Nicolson solution lead to significant oscillations. In contrast, the full implicit scheme does not show this oscillatory behavior, but is more inaccurate. Compared to these time-stepping algorithms the weighted-time-step method presented here is seen to offer definite advantages.
Time accurate application of the MacCormack 2-4 scheme on massively parallel computers
NASA Technical Reports Server (NTRS)
Hudson, Dale A.; Long, Lyle N.
1995-01-01
Many recent computational efforts in turbulence and acoustics research have used higher order numerical algorithms. One popular method has been the explicit MacCormack 2-4 scheme. The MacCormack 2-4 scheme is second order accurate in time and fourth order accurate in space, and is stable for CFL's below 2/3. Current research has shown that the method can give accurate results but does exhibit significant Gibbs phenomena at sharp discontinuities. The impact of adding Jameson type second, third, and fourth order artificial viscosity was examined here. Category 2 problems, the nonlinear traveling wave and the Riemann problem, were computed using a CFL number of 0.25. This research has found that dispersion errors can be significantly reduced or nearly eliminated by using a combination of second and third order terms in the damping. Use of second and fourth order terms reduced the magnitude of dispersion errors but not as effectively as the second and third order combination. The program was coded using Thinking Machine's CM Fortran, a variant of Fortran 90/High Performance Fortran, and was executed on a 2K CM-200. Simple extrapolation boundary conditions were used for both problems.
Finite-difference computations of rotor loads
NASA Technical Reports Server (NTRS)
Caradonna, F. X.; Tung, C.
1985-01-01
This paper demonstrates the current and future potential of finite-difference methods for solving real rotor problems which now rely largely on empiricism. The demonstration consists of a simple means of combining existing finite-difference, integral, and comprehensive loads codes to predict real transonic rotor flows. These computations are performed for hover and high-advance-ratio flight. Comparisons are made with experimental pressure data.
Finite-difference computations of rotor loads
NASA Technical Reports Server (NTRS)
Caradonna, F. X.; Tung, C.
1985-01-01
The current and future potential of finite difference methods for solving real rotor problems which now rely largely on empiricism are demonstrated. The demonstration consists of a simple means of combining existing finite-difference, integral, and comprehensive loads codes to predict real transonic rotor flows. These computations are performed for hover and high-advanced-ratio flight. Comparisons are made with experimental pressure data.
Geometrically invariant and high capacity image watermarking scheme using accurate radial transform
NASA Astrophysics Data System (ADS)
Singh, Chandan; Ranade, Sukhjeet K.
2013-12-01
Angular radial transform (ART) is a region based descriptor and possesses many attractive features such as rotation invariance, low computational complexity and resilience to noise which make them more suitable for invariant image watermarking than that of many transform domain based image watermarking techniques. In this paper, we introduce ART for fast and geometrically invariant image watermarking scheme with high embedding capacity. We also develop an accurate and fast framework for the computation of ART coefficients based on Gaussian quadrature numerical integration, 8-way symmetry/anti-symmetry properties and recursive relations for the calculation of sinusoidal kernel functions. ART coefficients so computed are then used for embedding the binary watermark using dither modulation. Experimental studies reveal that the proposed watermarking scheme not only provides better robustness against geometric transformations and other signal processing distortions, but also has superior advantages over the existing ones in terms of embedding capacity, speed and visual imperceptibility.
An accurate scheme to solve cluster dynamics equations using a Fokker-Planck approach
NASA Astrophysics Data System (ADS)
Jourdan, T.; Stoltz, G.; Legoll, F.; Monasse, L.
2016-10-01
We present a numerical method to accurately simulate particle size distributions within the formalism of rate equation cluster dynamics. This method is based on a discretization of the associated Fokker-Planck equation. We show that particular care has to be taken to discretize the advection part of the Fokker-Planck equation, in order to avoid distortions of the distribution due to numerical diffusion. For this purpose we use the Kurganov-Noelle-Petrova scheme coupled with the monotonicity-preserving reconstruction MP5, which leads to very accurate results. The interest of the method is highlighted in the case of loop coarsening in aluminum. We show that the choice of the models to describe the energetics of loops does not significantly change the normalized loop distribution, while the choice of the models for the absorption coefficients seems to have a significant impact on it.
A non-linear constrained optimization technique for the mimetic finite difference method
Manzini, Gianmarco; Svyatskiy, Daniil; Bertolazzi, Enrico; Frego, Marco
2014-09-30
This is a strategy for the construction of monotone schemes in the framework of the mimetic finite difference method for the approximation of diffusion problems on unstructured polygonal and polyhedral meshes.
Development of highly accurate approximate scheme for computing the charge transfer integral.
Pershin, Anton; Szalay, Péter G
2015-08-21
The charge transfer integral is a key parameter required by various theoretical models to describe charge transport properties, e.g., in organic semiconductors. The accuracy of this important property depends on several factors, which include the level of electronic structure theory and internal simplifications of the applied formalism. The goal of this paper is to identify the performance of various approximate approaches of the latter category, while using the high level equation-of-motion coupled cluster theory for the electronic structure. The calculations have been performed on the ethylene dimer as one of the simplest model systems. By studying different spatial perturbations, it was shown that while both energy split in dimer and fragment charge difference methods are equivalent with the exact formulation for symmetrical displacements, they are less efficient when describing transfer integral along the asymmetric alteration coordinate. Since the "exact" scheme was found computationally expensive, we examine the possibility to obtain the asymmetric fluctuation of the transfer integral by a Taylor expansion along the coordinate space. By exploring the efficiency of this novel approach, we show that the Taylor expansion scheme represents an attractive alternative to the "exact" calculations due to a substantial reduction of computational costs, when a considerably large region of the potential energy surface is of interest. Moreover, we show that the Taylor expansion scheme, irrespective of the dimer symmetry, is very accurate for the entire range of geometry fluctuations that cover the space the molecule accesses at room temperature. PMID:26298117
Development of highly accurate approximate scheme for computing the charge transfer integral
Pershin, Anton; Szalay, Péter G.
2015-08-21
The charge transfer integral is a key parameter required by various theoretical models to describe charge transport properties, e.g., in organic semiconductors. The accuracy of this important property depends on several factors, which include the level of electronic structure theory and internal simplifications of the applied formalism. The goal of this paper is to identify the performance of various approximate approaches of the latter category, while using the high level equation-of-motion coupled cluster theory for the electronic structure. The calculations have been performed on the ethylene dimer as one of the simplest model systems. By studying different spatial perturbations, it was shown that while both energy split in dimer and fragment charge difference methods are equivalent with the exact formulation for symmetrical displacements, they are less efficient when describing transfer integral along the asymmetric alteration coordinate. Since the “exact” scheme was found computationally expensive, we examine the possibility to obtain the asymmetric fluctuation of the transfer integral by a Taylor expansion along the coordinate space. By exploring the efficiency of this novel approach, we show that the Taylor expansion scheme represents an attractive alternative to the “exact” calculations due to a substantial reduction of computational costs, when a considerably large region of the potential energy surface is of interest. Moreover, we show that the Taylor expansion scheme, irrespective of the dimer symmetry, is very accurate for the entire range of geometry fluctuations that cover the space the molecule accesses at room temperature.
Orbital Advection by Interpolation: A Fast and Accurate Numerical Scheme for Super-Fast MHD Flows
Johnson, B M; Guan, X; Gammie, F
2008-04-11
In numerical models of thin astrophysical disks that use an Eulerian scheme, gas orbits supersonically through a fixed grid. As a result the timestep is sharply limited by the Courant condition. Also, because the mean flow speed with respect to the grid varies with position, the truncation error varies systematically with position. For hydrodynamic (unmagnetized) disks an algorithm called FARGO has been developed that advects the gas along its mean orbit using a separate interpolation substep. This relaxes the constraint imposed by the Courant condition, which now depends only on the peculiar velocity of the gas, and results in a truncation error that is more nearly independent of position. This paper describes a FARGO-like algorithm suitable for evolving magnetized disks. Our method is second order accurate on a smooth flow and preserves {del} {center_dot} B = 0 to machine precision. The main restriction is that B must be discretized on a staggered mesh. We give a detailed description of an implementation of the code and demonstrate that it produces the expected results on linear and nonlinear problems. We also point out how the scheme might be generalized to make the integration of other supersonic/super-fast flows more efficient. Although our scheme reduces the variation of truncation error with position, it does not eliminate it. We show that the residual position dependence leads to characteristic radial variations in the density over long integrations.
Finite-difference lattice-Boltzmann methods for binary fluids.
Xu, Aiguo
2005-06-01
We investigate two-fluid Bhatnagar-Gross-Krook (BGK) kinetic methods for binary fluids. The developed theory works for asymmetric as well as symmetric systems. For symmetric systems it recovers Sirovich's theory and is summarized in models A and B. For asymmetric systems it contributes models C, D, and E which are especially useful when the total masses and/or local temperatures of the two components are greatly different. The kinetic models are discretized based on an octagonal discrete velocity model. The discrete-velocity kinetic models and the continuous ones are required to describe the same hydrodynamic equations. The combination of a discrete-velocity kinetic model and an appropriate finite-difference scheme composes a finite-difference lattice Boltzmann method. The validity of the formulated methods is verified by investigating (i) uniform relaxation processes, (ii) isothermal Couette flow, and (iii) diffusion behavior. PMID:16089910
Finite elements and finite differences for transonic flow calculations
NASA Technical Reports Server (NTRS)
Hafez, M. M.; Murman, E. M.; Wellford, L. C.
1978-01-01
The paper reviews the chief finite difference and finite element techniques used for numerical solution of nonlinear mixed elliptic-hyperbolic equations governing transonic flow. The forms of the governing equations for unsteady two-dimensional transonic flow considered are the Euler equation, the full potential equation in both conservative and nonconservative form, the transonic small-disturbance equation in both conservative and nonconservative form, and the hodograph equations for the small-disturbance case and the full-potential case. Finite difference methods considered include time-dependent methods, relaxation methods, semidirect methods, and hybrid methods. Finite element methods include finite element Lax-Wendroff schemes, implicit Galerkin method, mixed variational principles, dual iterative procedures, optimal control methods and least squares.
NASA Astrophysics Data System (ADS)
Lan, Haiqiang; Zhang, Zhongjie
2011-06-01
The finite-difference (FD) method is a powerful tool in seismic wave field modelling for understanding seismic wave propagation in the Earth's interior and interpreting the real seismic data. The accuracy of FD modelling partly depends on the implementation of the free-surface (i.e. traction-free) condition. In the past 40 years, at least six kinds of free-surface boundary condition approximate schemes (such as one-sided, centred finite-difference, composed, new composed, implicit and boundary-modified approximations) have been developed in FD second-order elastodynamic simulation. Herein we simulate seismic wave fields in homogeneous and lateral heterogeneous models using these free-surface boundary condition approximate schemes and evaluate their stability and applicability by comparing with corresponding analytical solutions, and then quantitatively evaluate the accuracies of different approximate schemes from the misfit of the amplitude and phase between the numerical and analytical results. Our results confirm that the composed scheme becomes unstable for the Vs/Vp ratio less than 0.57, and suggest that (1) the one-sided scheme is only accurate to first order and therefore introduces serious errors for the shorter wavelengths, other schemes are all of second-order precision; (2) the new composed, implicit and boundary-modified schemes are stable even when the Vs/Vp ratio is less than 0.2; (3) the implicit and boundary-modified schemes are able to deal with laterally varying (heterogeneous) free surface; (4) in the corresponding stability range, the one-sided scheme shows remarkable errors in both phase and amplitude compared to analytical solution (which means larger errors in travel-time and reflection strength), the other five approximate schemes show better performance in travel-time (phase) than strength (amplitude).
Ewing, R.E.; Saevareid, O.; Shen, J.
1994-12-31
A multigrid algorithm for the cell-centered finite difference on equilateral triangular grids for solving second-order elliptic problems is proposed. This finite difference is a four-point star stencil in a two-dimensional domain and a five-point star stencil in a three dimensional domain. According to the authors analysis, the advantages of this finite difference are that it is an O(h{sup 2})-order accurate numerical scheme for both the solution and derivatives on equilateral triangular grids, the structure of the scheme is perhaps the simplest, and its corresponding multigrid algorithm is easily constructed with an optimal convergence rate. They are interested in relaxation of the equilateral triangular grid condition to certain general triangular grids and the application of this multigrid algorithm as a numerically reasonable preconditioner for the lowest-order Raviart-Thomas mixed triangular finite element method. Numerical test results are presented to demonstrate their analytical results and to investigate the applications of this multigrid algorithm on general triangular grids.
NASA Technical Reports Server (NTRS)
Strong, Stuart L.; Meade, Andrew J., Jr.
1992-01-01
Preliminary results are presented of a finite element/finite difference method (semidiscrete Galerkin method) used to calculate compressible boundary layer flow about airfoils, in which the group finite element scheme is applied to the Dorodnitsyn formulation of the boundary layer equations. The semidiscrete Galerkin (SDG) method promises to be fast, accurate and computationally efficient. The SDG method can also be applied to any smoothly connected airfoil shape without modification and possesses the potential capability of calculating boundary layer solutions beyond flow separation. Results are presented for low speed laminar flow past a circular cylinder and past a NACA 0012 airfoil at zero angle of attack at a Mach number of 0.5. Also shown are results for compressible flow past a flat plate for a Mach number range of 0 to 10 and results for incompressible turbulent flow past a flat plate. All numerical solutions assume an attached boundary layer.
Finite difference schemes for second order systems describing black holes
Motamed, Mohammad; Kreiss, H-O.; Babiuc, M.; Winicour, J.; Szilagyi, B.
2006-06-15
In the harmonic description of general relativity, the principal part of Einstein's equations reduces to 10 curved space wave equations for the components of the space-time metric. We present theorems regarding the stability of several evolution-boundary algorithms for such equations when treated in second order differential form. The theorems apply to a model black hole space-time consisting of a spacelike inner boundary excising the singularity, a timelike outer boundary and a horizon in between. These algorithms are implemented as stable, convergent numerical codes and their performance is compared in a 2-dimensional excision problem.
Algorithmic vs. finite difference Jacobians for infrared atmospheric radiative transfer
NASA Astrophysics Data System (ADS)
Schreier, Franz; Gimeno García, Sebastián; Vasquez, Mayte; Xu, Jian
2015-10-01
Jacobians, i.e. partial derivatives of the radiance and transmission spectrum with respect to the atmospheric state parameters to be retrieved from remote sensing observations, are important for the iterative solution of the nonlinear inverse problem. Finite difference Jacobians are easy to implement, but computationally expensive and possibly of dubious quality; on the other hand, analytical Jacobians are accurate and efficient, but the implementation can be quite demanding. GARLIC, our "Generic Atmospheric Radiation Line-by-line Infrared Code", utilizes algorithmic differentiation (AD) techniques to implement derivatives w.r.t. atmospheric temperature and molecular concentrations. In this paper, we describe our approach for differentiation of the high resolution infrared and microwave spectra and provide an in-depth assessment of finite difference approximations using "exact" AD Jacobians as a reference. The results indicate that the "standard" two-point finite differences with 1 K and 1% perturbation for temperature and volume mixing ratio, respectively, can exhibit substantial errors, and central differences are significantly better. However, these deviations do not transfer into the truncated singular value decomposition solution of a least squares problem. Nevertheless, AD Jacobians are clearly recommended because of the superior speed and accuracy.
2013-01-01
In the paper, we consider the problem of pricing options in wide classes of Lévy processes. We propose a general approach to the numerical methods based on a finite difference approximation for the generalized Black-Scholes equation. The goal of the paper is to incorporate the Wiener-Hopf factorization into finite difference methods for pricing options in Lévy models with jumps. The method is applicable for pricing barrier and American options. The pricing problem is reduced to the sequence of linear algebraic systems with a dense Toeplitz matrix; then the Wiener-Hopf factorization method is applied. We give an important probabilistic interpretation based on the infinitely divisible distributions theory to the Laurent operators in the correspondent factorization identity. Notice that our algorithm has the same complexity as the ones which use the explicit-implicit scheme, with a tridiagonal matrix. However, our method is more accurate. We support the advantage of the new method in terms of accuracy and convergence by using numerical experiments. PMID:24489518
A coarse-mesh nodal method-diffusive-mesh finite difference method
Joo, H.; Nichols, W.R.
1994-05-01
Modern nodal methods have been successfully used for conventional light water reactor core analyses where the homogenized, node average cross sections (XSs) and the flux discontinuity factors (DFs) based on equivalence theory can reliably predict core behavior. For other types of cores and other geometries characterized by tightly-coupled, heterogeneous core configurations, the intranodal flux shapes obtained from a homogenized nodal problem may not accurately portray steep flux gradients near fuel assembly interfaces or various reactivity control elements. This may require extreme values of DFs (either very large, very small, or even negative) to achieve a desired solution accuracy. Extreme values of DFs, however, can disrupt the convergence of the iterative methods used to solve for the node average fluxes, and can lead to a difficulty in interpolating adjacent DF values. Several attempts to remedy the problem have been made, but nothing has been satisfactory. A new coarse-mesh nodal scheme called the Diffusive-Mesh Finite Difference (DMFD) technique, as contrasted with the coarse-mesh finite difference (CMFD) technique, has been developed to resolve this problem. This new technique and the development of a few-group, multidimensional kinetics computer program are described in this paper.
A coarse-mesh nodal method, the diffusive-mesh finite difference method
Joo, H.; Nichols, W.R.
1994-12-31
Modern nodal methods have been successfully used for conventional light water reactor core analyses where the homogenized, node average cross section (XSs) and the flux discontinuity factors (DFs) based on equivalence theory can reliably predict core behavior. For other types of cores and other geometries characterized by tightly coupled, heterogeneous core configurations, the intranodal flux shapes obtained from a homogenized nodal problem may not accurately portray steep flux gradients near fuel assembly interfaces or various reactivity control elements. This may require extreme values of DFs (either very large, very small, or even negative) to achieve a desired solution accuracy. Extreme values of DFs, however, can disrupt the convergence of the iterative methods used to solve for the node average fluxes and can lead to difficulty in interpolating adjacent DF values. Several attempts to remedy the problem have been made, but nothing has been satisfactory. A new coarse-mesh nodal scheme called the diffusive-mesh finite difference (DMFD) technique, as contrasted with the coarse-mesh finite difference (CMFD) technique, has been developed to resolve this problem. This new technique and the development of a few-group, multidimensional kinetics computer program are described in this paper.
NASA Astrophysics Data System (ADS)
Beilina, Larisa
2016-08-01
We present domain decomposition finite element/finite difference method for the solution of hyperbolic equation. The domain decomposition is performed such that finite elements and finite differences are used in different subdomains of the computational domain: finite difference method is used on the structured part of the computational domain and finite elements on the unstructured part of the domain. Explicit discretizations for both methods are constructed such that the finite element and the finite difference schemes coincide on the common structured overlapping layer between computational subdomains. Then the resulting approach can be considered as a pure finite element scheme which avoids instabilities at the interfaces. We derive an energy estimate for the underlying hyperbolic equation with absorbing boundary conditions and illustrate efficiency of the domain decomposition method on the reconstruction of the conductivity function in three dimensions.
Optimized Finite-Difference Coefficients for Hydroacoustic Modeling
NASA Astrophysics Data System (ADS)
Preston, L. A.
2014-12-01
Responsible utilization of marine renewable energy sources through the use of current energy converter (CEC) and wave energy converter (WEC) devices requires an understanding of the noise generation and propagation from these systems in the marine environment. Acoustic noise produced by rotating turbines, for example, could adversely affect marine animals and human-related marine activities if not properly understood and mitigated. We are utilizing a 3-D finite-difference acoustic simulation code developed at Sandia that can accurately propagate noise in the complex bathymetry in the near-shore to open ocean environment. As part of our efforts to improve computation efficiency in the large, high-resolution domains required in this project, we investigate the effects of using optimized finite-difference coefficients on the accuracy of the simulations. We compare accuracy and runtime of various finite-difference coefficients optimized via criteria such as maximum numerical phase speed error, maximum numerical group speed error, and L-1 and L-2 norms of weighted numerical group and phase speed errors over a given spectral bandwidth. We find that those coefficients optimized for L-1 and L-2 norms are superior in accuracy to those based on maximal error and can produce runtimes of 10% of the baseline case, which uses Taylor Series finite-difference coefficients at the Courant time step limit. We will present comparisons of the results for the various cases evaluated as well as recommendations for utilization of the cases studied. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.
A Second-Order Accurate, Component-Wise TVD Scheme for Nonlinear, Hyperbolic Conservation Laws
NASA Astrophysics Data System (ADS)
Yu, Heng; Liu, Yu-Ping
2001-10-01
In this paper, we present a two-step, component-wise TVD scheme for nonlinear, hyperbolic conservation laws, which is obtained by combining the schemes of Mac Cormack and Warming-Beam. The scheme does not necessitate the characteristic decompositions of the usual TVD schemes. It employs component-wise limiting; hence the programming is much simpler, especially for complicated coupled systems. For Euler systems of conservation laws, we found the scheme is two times faster in computation than the usual TVD schemes based on field-by-field decomposition limiting. A lot of numerical results show primarily the value of the new method.
Kim, S.
1994-12-31
Parallel iterative procedures based on domain decomposition techniques are defined and analyzed for the numerical solution of wave propagation by finite element and finite difference methods. For finite element methods, in a Lagrangian framework, an efficient way for choosing the algorithm parameter as well as the algorithm convergence are indicated. Some heuristic arguments for finding the algorithm parameter for finite difference schemes are addressed. Numerical results are presented to indicate the effectiveness of the methods.
A parallel finite-difference method for computational aerodynamics
NASA Technical Reports Server (NTRS)
Swisshelm, Julie M.
1989-01-01
A finite-difference scheme for solving complex three-dimensional aerodynamic flow on parallel-processing supercomputers is presented. The method consists of a basic flow solver with multigrid convergence acceleration, embedded grid refinements, and a zonal equation scheme. Multitasking and vectorization have been incorporated into the algorithm. Results obtained include multiprocessed flow simulations from the Cray X-MP and Cray-2. Speedups as high as 3.3 for the two-dimensional case and 3.5 for segments of the three-dimensional case have been achieved on the Cray-2. The entire solver attained a factor of 2.7 improvement over its unitasked version on the Cray-2. The performance of the parallel algorithm on each machine is analyzed.
NASA Technical Reports Server (NTRS)
Yee, H. C.; Kutler, P.
1983-01-01
A one-parameter family of explicit and implicit second-order-accurate, entropy satisfying, total variation diminishing (TVD) schemes was developed by Harten. These TVD schemes were the property of not generating spurious oscillations for one-dimensional nonlinear scalar hyperbolic conservation laws and constant coefficient hyperbolic systems. Application of these methods to one- and two-dimensional fluid flows containing shocks (in Cartesian coordinates) yields highly accurate nonoscillatory numerical solutions. The goal of this work is to expand these methods to the multidimensional Euler equations in generalized coordinate systems. Some numerical results of shock waves impinging on cylindrical bodies are compared with MacCormack's method.
Finite difference computation of blast diffraction
NASA Astrophysics Data System (ADS)
Hillier, R.; Graham, J. M. R.
1985-07-01
This paper discusses the use of numerical finite difference methods for predicting flow fields in which a shock or blast wave is diffracted at a sharp edge. Three different types of method are studied: Donor Cell differencing with and without Flux Corrected Transport, a Finite Volume method with an explicit artificial viscosity and Runge-Kutta time stepping, and a second order upwind method based on the solution of a Riemann wave problem at cell interfaces. In the case of weak shock waves a comparison is made with the flow field predicted by acoustic theory including flow separation. Results for stronger shocks are also presented.
Software suite for finite difference method models.
Arola, T; Hannula, M; Narra, N; Malmivuo, J; Hyttinen, J
2006-01-01
We have developed a software suite for finite difference method (FDM) model construction, visualization and quasi-static simulation to be used in bioelectric field modeling. The aim of the software is to provide a full path from medical image data to simulation of bioelectric phenomena and results visualization. It is written in Java and can be run on various platforms while still supporting all features included. The software can be distributed across a network utilizing dedicated servers for calculation intensive tasks. Supported visualization modes are both two- and three-dimensional modes. PMID:17946057
Seismic imaging using finite-differences and parallel computers
Ober, C.C.
1997-12-31
A key to reducing the risks and costs of associated with oil and gas exploration is the fast, accurate imaging of complex geologies, such as salt domes in the Gulf of Mexico and overthrust regions in US onshore regions. Prestack depth migration generally yields the most accurate images, and one approach to this is to solve the scalar wave equation using finite differences. As part of an ongoing ACTI project funded by the US Department of Energy, a finite difference, 3-D prestack, depth migration code has been developed. The goal of this work is to demonstrate that massively parallel computers can be used efficiently for seismic imaging, and that sufficient computing power exists (or soon will exist) to make finite difference, prestack, depth migration practical for oil and gas exploration. Several problems had to be addressed to get an efficient code for the Intel Paragon. These include efficient I/O, efficient parallel tridiagonal solves, and high single-node performance. Furthermore, to provide portable code the author has been restricted to the use of high-level programming languages (C and Fortran) and interprocessor communications using MPI. He has been using the SUNMOS operating system, which has affected many of his programming decisions. He will present images created from two verification datasets (the Marmousi Model and the SEG/EAEG 3D Salt Model). Also, he will show recent images from real datasets, and point out locations of improved imaging. Finally, he will discuss areas of current research which will hopefully improve the image quality and reduce computational costs.
Wodrich, Matthew D; Corminboeuf, Clémence; Wheeler, Steven E
2012-04-01
Detailed knowledge of hydrocarbon radical thermochemistry is critical for understanding diverse chemical phenomena, ranging from combustion processes to organic reaction mechanisms. Unfortunately, experimental thermochemical data for many radical species tend to have large errors or are lacking entirely. Here we develop procedures for deriving high-quality thermochemical data for hydrocarbon radicals by extending Wheeler et al.'s "generalized bond separation reaction" (GBSR) scheme (J. Am. Chem. Soc., 2009, 131, 2547). Moreover, we show that the existing definition of hyperhomodesmotic reactions is flawed. This is because transformation reactions, in which one molecule each from the predefined sets of products and reactants can be converted to a different product and reactant molecule, are currently allowed. This problem is corrected via a refined definition of hyperhomodesmotic reactions in which there are equal numbers of carbon-carbon bond types inclusive of carbon hybridization and number of hydrogens attached. Ab initio and density functional theory (DFT) computations using the expanded GBSRs are applied to a newly derived test set of 27 hydrocarbon radicals (HCR27). Greatly reduced errors in computed reaction enthalpies are seen for hyperhomodesmotic and other highly balanced reactions classes, which benefit from increased matching of hybridization and bonding requirements. The best performing DFT methods for hyperhomodesmotic reactions, M06-2X and B97-dDsC, give average deviations from benchmark computations of only 0.31 and 0.44 (±0.90 and ±1.56 at the 95% confidence level) kcal/mol, respectively, over the test set. By exploiting the high degree of error cancellation provided by hyperhomodesmotic reactions, accurate thermochemical data for hydrocarbon radicals (e.g., enthalpies of formation) can be computed using relatively inexpensive computational methods.
Finite-difference modeling of Biot's poroelastic equations across all frequencies
Masson, Y.J.; Pride, S.R.
2009-10-22
An explicit time-stepping finite-difference scheme is presented for solving Biot's equations of poroelasticity across the entire band of frequencies. In the general case for which viscous boundary layers in the pores must be accounted for, the time-domain version of Darcy's law contains a convolution integral. It is shown how to efficiently and directly perform the convolution so that the Darcy velocity can be properly updated at each time step. At frequencies that are low enough compared to the onset of viscous boundary layers, no memory terms are required. At higher frequencies, the number of memory terms required is the same as the number of time points it takes to sample accurately the wavelet being used. In practice, we never use more than 20 memory terms and often considerably fewer. Allowing for the convolution makes the scheme even more stable (even larger time steps might be used) than it is when the convolution is entirely neglected. The accuracy of the scheme is confirmed by comparing numerical examples to exact analytic results.
TUNED FINITE-DIFFERENCE DIFFUSION OPERATORS
Maron, Jason; Low, Mordecai-Mark Mac E-mail: mordecai@amnh.org
2009-05-15
Finite-difference simulations of fluid dynamics and magnetohydrodynamics generally require an explicit diffusion operator, either to maintain stability by attenuating grid-scale structure, or to implement physical diffusivities such as viscosity or resistivity. If the goal is stability only, the diffusion must act at the grid scale, but should affect structure at larger scales as little as possible. For physical diffusivities the diffusion scale depends on the problem, and diffusion may act at larger scales as well. Diffusivity can undesirably limit the computational time step in both cases. We construct tuned finite-difference diffusion operators that minimally limit the time step while acting as desired near the diffusion scale. Such operators reach peak values at the diffusion scale rather than at the grid scale, but behave as standard operators at larger scales. These operators will be useful for simulations with high magnetic diffusivity or kinematic viscosity such as in the simulation of astrophysical dynamos with magnetic Prandtl number far from unity, or for numerical stabilization using hyperdiffusivity.
Application of a new finite difference algorithm for computational aeroacoustics
NASA Technical Reports Server (NTRS)
Goodrich, John W.
1995-01-01
Acoustic problems have become extremely important in recent years because of research efforts such as the High Speed Civil Transport program. Computational aeroacoustics (CAA) requires a faithful representation of wave propagation over long distances, and needs algorithms that are accurate and boundary conditions that are unobtrusive. This paper applies a new finite difference method and boundary algorithm to the Linearized Euler Equations (LEE). The results demonstrate the ability of a new fourth order propagation algorithm to accurately simulate the genuinely multidimensional wave dynamics of acoustic propagation in two space dimensions with the LEE. The results also show the ability of a new outflow boundary condition and fourth order algorithm to pass the evolving solution from the computational domain with no perceptible degradation of the solution remaining within the domain.
Pencil: Finite-difference Code for Compressible Hydrodynamic Flows
NASA Astrophysics Data System (ADS)
Brandenburg, Axel; Dobler, Wolfgang
2010-10-01
The Pencil code is a high-order finite-difference code for compressible hydrodynamic flows with magnetic fields. It is highly modular and can easily be adapted to different types of problems. The code runs efficiently under MPI on massively parallel shared- or distributed-memory computers, like e.g. large Beowulf clusters. The Pencil code is primarily designed to deal with weakly compressible turbulent flows. To achieve good parallelization, explicit (as opposed to compact) finite differences are used. Typical scientific targets include driven MHD turbulence in a periodic box, convection in a slab with non-periodic upper and lower boundaries, a convective star embedded in a fully nonperiodic box, accretion disc turbulence in the shearing sheet approximation, self-gravity, non-local radiation transfer, dust particle evolution with feedback on the gas, etc. A range of artificial viscosity and diffusion schemes can be invoked to deal with supersonic flows. For direct simulations regular viscosity and diffusion is being used. The code is written in well-commented Fortran90.
Sagiyama, Koki; Rudraraju, Shiva; Garikipati, Krishna
2016-09-13
Here, we consider solid state phase transformations that are caused by free energy densities with domains of non-convexity in strain-composition space; we refer to the non-convex domains as mechano-chemical spinodals. The non-convexity with respect to composition and strain causes segregation into phases with different crystal structures. We work on an existing model that couples the classical Cahn-Hilliard model with Toupin’s theory of gradient elasticity at finite strains. Both systems are represented by fourth-order, nonlinear, partial differential equations. The goal of this work is to develop unconditionally stable, second-order accurate time-integration schemes, motivated by the need to carry out large scalemore » computations of dynamically evolving microstructures in three dimensions. We also introduce reduced formulations naturally derived from these proposed schemes for faster computations that are still second-order accurate. Although our method is developed and analyzed here for a specific class of mechano-chemical problems, one can readily apply the same method to develop unconditionally stable, second-order accurate schemes for any problems for which free energy density functions are multivariate polynomials of solution components and component gradients. Apart from an analysis and construction of methods, we present a suite of numerical results that demonstrate the schemes in action.« less
NASA Astrophysics Data System (ADS)
Sagiyama, K.; Rudraraju, S.; Garikipati, K.
2016-11-01
We consider solid state phase transformations that are caused by free energy densities with domains of non-convexity in strain-composition space; we refer to the non-convex domains as mechano-chemical spinodals. The non-convexity with respect to composition and strain causes segregation into phases with different crystal structures. We work on an existing model that couples the classical Cahn-Hilliard model with Toupin's theory of gradient elasticity at finite strains. Both systems are represented by fourth-order, nonlinear, partial differential equations. The goal of this work is to develop unconditionally stable, second-order accurate time-integration schemes, motivated by the need to carry out large scale computations of dynamically evolving microstructures in three dimensions. We also introduce reduced formulations naturally derived from these proposed schemes for faster computations that are still second-order accurate. Although our method is developed and analyzed here for a specific class of mechano-chemical problems, one can readily apply the same method to develop unconditionally stable, second-order accurate schemes for any problems for which free energy density functions are multivariate polynomials of solution components and component gradients. Apart from an analysis and construction of methods, we present a suite of numerical results that demonstrate the schemes in action.
A New Time-Space Accurate Scheme for Hyperbolic Problems. 1; Quasi-Explicit Case
NASA Technical Reports Server (NTRS)
Sidilkover, David
1998-01-01
This paper presents a new discretization scheme for hyperbolic systems of conservations laws. It satisfies the TVD property and relies on the new high-resolution mechanism which is compatible with the genuinely multidimensional approach proposed recently. This work can be regarded as a first step towards extending the genuinely multidimensional approach to unsteady problems. Discontinuity capturing capabilities and accuracy of the scheme are verified by a set of numerical tests.
Finite difference modeling of Biot's poroelastic equations atseismic frequencies
Masson, Y.J.; Pride, S.R.; Nihei, K.T.
2006-02-24
Across the seismic band of frequencies (loosely defined as<10 kHz), a seismic wave propagating through a porous material willcreate flow in the pore space that is laminar; that is, in thislow-frequency "seismic limit," the development of viscous boundary layersin the pores need not be modeled. An explicit time steppingstaggered-grid finite difference scheme is presented for solving Biot'sequations of poroelasticity in this low-frequency limit. A key part ofthis work is the establishment of rigorous stability conditions. It isdemonstrated that over a wide range of porous material properties typicalof sedimentary rock and despite the presenceof fluid pressure diffusion(Biot slow waves), the usual Courant condition governs the stability asif the problem involved purely elastic waves. The accuracy of the methodis demonstrated by comparing to exact analytical solutions for both fastcompressional waves and slow waves. Additional numerical modelingexamples are also presented.
Numerical stability for finite difference approximations of Einstein's equations
Calabrese, G. . E-mail: G.Calabrese@soton.ac.uk; Hinder, I.; Husa, S.
2006-11-01
We extend the notion of numerical stability of finite difference approximations to include hyperbolic systems that are first order in time and second order in space, such as those that appear in numerical relativity and, more generally, in Hamiltonian formulations of field theories. By analyzing the symbol of the second order system, we obtain necessary and sufficient conditions for stability in a discrete norm containing one-sided difference operators. We prove stability for certain toy models and the linearized Nagy-Ortiz-Reula formulation of Einstein's equations. We also find that, unlike in the fully first order case, standard discretizations of some well-posed problems lead to unstable schemes and that the Courant limits are not always simply related to the characteristic speeds of the continuum problem. Finally, we propose methods for testing stability for second order in space hyperbolic systems.
Dong Hongjie Krylov, Nicolai V.
2007-06-15
We consider degenerate parabolic and elliptic fully nonlinear Bellman equations with Lipschitz coefficients in domains. Error bounds of order h{sup 1/2} in the sup norm for certain types of finite-difference schemes are obtained.
Viscoelastic Finite Difference Modeling Using Graphics Processing Units
NASA Astrophysics Data System (ADS)
Fabien-Ouellet, G.; Gloaguen, E.; Giroux, B.
2014-12-01
Full waveform seismic modeling requires a huge amount of computing power that still challenges today's technology. This limits the applicability of powerful processing approaches in seismic exploration like full-waveform inversion. This paper explores the use of Graphics Processing Units (GPU) to compute a time based finite-difference solution to the viscoelastic wave equation. The aim is to investigate whether the adoption of the GPU technology is susceptible to reduce significantly the computing time of simulations. The code presented herein is based on the freely accessible software of Bohlen (2002) in 2D provided under a General Public License (GNU) licence. This implementation is based on a second order centred differences scheme to approximate time differences and staggered grid schemes with centred difference of order 2, 4, 6, 8, and 12 for spatial derivatives. The code is fully parallel and is written using the Message Passing Interface (MPI), and it thus supports simulations of vast seismic models on a cluster of CPUs. To port the code from Bohlen (2002) on GPUs, the OpenCl framework was chosen for its ability to work on both CPUs and GPUs and its adoption by most of GPU manufacturers. In our implementation, OpenCL works in conjunction with MPI, which allows computations on a cluster of GPU for large-scale model simulations. We tested our code for model sizes between 1002 and 60002 elements. Comparison shows a decrease in computation time of more than two orders of magnitude between the GPU implementation run on a AMD Radeon HD 7950 and the CPU implementation run on a 2.26 GHz Intel Xeon Quad-Core. The speed-up varies depending on the order of the finite difference approximation and generally increases for higher orders. Increasing speed-ups are also obtained for increasing model size, which can be explained by kernel overheads and delays introduced by memory transfers to and from the GPU through the PCI-E bus. Those tests indicate that the GPU memory size
Erwin, Andrew; Sup, Frank C
2015-01-01
In this paper, a novel haptic feedback scheme, used for accurately positioning a 1DOF virtual wrist prosthesis through sensory substitution, is presented. The scheme employs a three-node tactor array and discretely and selectively modulates the stimulation frequency of each tactor to relay 11 discrete haptic stimuli to the user. Able-bodied participants were able to move the virtual wrist prosthesis via a surface electromyography based controller. The participants evaluated the feedback scheme without visual or audio feedback and relied solely on the haptic feedback alone to correctly position the hand. The scheme was evaluated through both normal (perpendicular) and shear (lateral) stimulations applied on the forearm. Normal stimulations were applied through a prototype device previously developed by the authors while shear stimulations were generated using an ubiquitous coin motor vibrotactor. Trials with no feedback served as a baseline to compare results within the study and to the literature. The results indicated that using normal and shear stimulations resulted in accurately positioning the virtual wrist, but were not significantly different. Using haptic feedback was substantially better than no feedback. The results found in this study are significant since the feedback scheme allows for using relatively few tactors to relay rich haptic information to the user and can be learned easily despite a relatively short amount of training. Additionally, the results are important for the haptic community since they contradict the common conception in the literature that normal stimulation is inferior to shear. From an ergonomic perspective normal stimulation has the potential to benefit upper limb amputees since it can operate at lower frequencies than shear-based vibrotactors while also generating less noise. Through further tuning of the novel haptic feedback scheme and normal stimulation device, a compact and comfortable sensory substitution device for upper
FWAVE V1.0 a framework for finite difference wave equation modeling
2002-07-01
FWAVE provides a computation framework for the rapid prototyping and efficient use of finite difference wave equation solutions. The user provides single grid Fortran solver components that are integrated using opaque handles to C++ distributed data structures. Permits the scientific researcher to make of clusters and parallel computers by concentrating only on the numerical schemes.
High Order Finite Difference Methods, Multidimensional Linear Problems and Curvilinear Coordinates
NASA Technical Reports Server (NTRS)
Nordstrom, Jan; Carpenter, Mark H.
1999-01-01
Boundary and interface conditions are derived for high order finite difference methods applied to multidimensional linear problems in curvilinear coordinates. The boundary and interface conditions lead to conservative schemes and strict and strong stability provided that certain metric conditions are met.
Finite difference computation of Casimir forces
NASA Astrophysics Data System (ADS)
Pinto, Fabrizio
2016-09-01
In this Invited paper, we begin by a historical introduction to provide a motivation for the classical problems of interatomic force computation and associated challenges. This analysis will lead us from early theoretical and experimental accomplishments to the integration of these fascinating interactions into the operation of realistic, next-generation micro- and nanodevices both for the advanced metrology of fundamental physical processes and in breakthrough industrial applications. Among several powerful strategies enabling vastly enhanced performance and entirely novel technological capabilities, we shall specifically consider Casimir force time-modulation and the adoption of non-trivial geometries. As to the former, the ability to alter the magnitude and sign of the Casimir force will be recognized as a crucial principle to implement thermodynamical nano-engines. As to the latter, we shall first briefly review various reported computational approaches. We shall then discuss the game-changing discovery, in the last decade, that standard methods of numerical classical electromagnetism can be retooled to formulate the problem of Casimir force computation in arbitrary geometries. This remarkable development will be practically illustrated by showing that such an apparently elementary method as standard finite-differencing can be successfully employed to numerically recover results known from the Lifshitz theory of dispersion forces in the case of interacting parallel-plane slabs. Other geometries will be also be explored and consideration given to the potential of non-standard finite-difference methods. Finally, we shall introduce problems at the computational frontier, such as those including membranes deformed by Casimir forces and the effects of anisotropic materials. Conclusions will highlight the dramatic transition from the enduring perception of this field as an exotic application of quantum electrodynamics to the recent demonstration of a human climbing
Stochastic finite-difference time-domain
NASA Astrophysics Data System (ADS)
Smith, Steven Michael
2011-12-01
This dissertation presents the derivation of an approximate method to determine the mean and the variance of electro-magnetic fields in the body using the Finite-Difference Time-Domain (FDTD) method. Unlike Monte Carlo analysis, which requires repeated FDTD simulations, this method directly computes the variance of the fields at every point in space at every sample of time in the simulation. This Stochastic FDTD simulation (S-FDTD) has at its root a new wave called the Variance wave, which is computed in the time domain along with the mean properties of the model space in the FDTD simulation. The Variance wave depends on the electro-magnetic fields, the reflections and transmission though the different dielectrics, and the variances of the electrical properties of the surrounding materials. Like the electro-magnetic fields, the Variance wave begins at zero (there is no variance before the source is turned on) and is computed in the time domain until all fields reach steady state. This process is performed in a fraction of the time of a Monte Carlo simulation and yields the first two statistical parameters (mean and variance). The mean of the field is computed using the traditional FDTD equations. Variance is computed by approximating the correlation coefficients between the constituitive properties and the use of the S-FDTD equations. The impetus for this work was the simulation time it takes to perform 3D Specific Absorption Rate (SAR) FDTD analysis of the human head model for cell phone power absorption in the human head due to the proximity of a cell phone being used. In many instances, Monte Carlo analysis is not performed due to the lengthy simulation times required. With the development of S-FDTD, these statistical analyses could be performed providing valuable statistical information with this information being provided in a small fraction of the time it would take to perform a Monte Carlo analysis.
Third-order accurate entropy-stable schemes for initial-boundary-value conservation laws
NASA Astrophysics Data System (ADS)
Svärd, Magnus
2012-08-01
We consider initial-boundary-value conservation laws with the objective to obtain high-order approximations. We study two different approaches to obtain third-order accuracy, local entropy stability and a global bound on the entropy. The results are applicable to, for example the Euler equations of gas dynamics, for which we present numerical results demonstrating the robustness and accuracy of the scheme.
NASA Technical Reports Server (NTRS)
Chen, G.; Zheng, Q.; Coleman, M.; Weerakoon, S.
1983-01-01
This paper briefly reviews convergent finite difference schemes for hyperbolic initial boundary value problems and their applications to boundary control systems of hyperbolic type which arise in the modelling of vibrations. These difference schemes are combined with the primal and the dual approaches to compute the optimal control in the unconstrained case, as well as the case when the control is subject to inequality constraints. Some of the preliminary numerical results are also presented.
Multi-Dimensional Asymptotically Stable 4th Order Accurate Schemes for the Diffusion Equation
NASA Technical Reports Server (NTRS)
Abarbanel, Saul; Ditkowski, Adi
1996-01-01
An algorithm is presented which solves the multi-dimensional diffusion equation on co mplex shapes to 4th-order accuracy and is asymptotically stable in time. This bounded-error result is achieved by constructing, on a rectangular grid, a differentiation matrix whose symmetric part is negative definite. The differentiation matrix accounts for the Dirichlet boundary condition by imposing penalty like terms. Numerical examples in 2-D show that the method is effective even where standard schemes, stable by traditional definitions fail.
Oyeyemi, Victor B.; Krisiloff, David B.; Keith, John A.; Libisch, Florian; Pavone, Michele; Carter, Emily A.
2014-01-28
Oxygenated hydrocarbons play important roles in combustion science as renewable fuels and additives, but many details about their combustion chemistry remain poorly understood. Although many methods exist for computing accurate electronic energies of molecules at equilibrium geometries, a consistent description of entire combustion reaction potential energy surfaces (PESs) requires multireference correlated wavefunction theories. Here we use bond dissociation energies (BDEs) as a foundational metric to benchmark methods based on multireference configuration interaction (MRCI) for several classes of oxygenated compounds (alcohols, aldehydes, carboxylic acids, and methyl esters). We compare results from multireference singles and doubles configuration interaction to those utilizing a posteriori and a priori size-extensivity corrections, benchmarked against experiment and coupled cluster theory. We demonstrate that size-extensivity corrections are necessary for chemically accurate BDE predictions even in relatively small molecules and furnish examples of unphysical BDE predictions resulting from using too-small orbital active spaces. We also outline the specific challenges in using MRCI methods for carbonyl-containing compounds. The resulting complete basis set extrapolated, size-extensivity-corrected MRCI scheme produces BDEs generally accurate to within 1 kcal/mol, laying the foundation for this scheme's use on larger molecules and for more complex regions of combustion PESs.
NASA Astrophysics Data System (ADS)
Oyeyemi, Victor B.; Krisiloff, David B.; Keith, John A.; Libisch, Florian; Pavone, Michele; Carter, Emily A.
2014-01-01
Oxygenated hydrocarbons play important roles in combustion science as renewable fuels and additives, but many details about their combustion chemistry remain poorly understood. Although many methods exist for computing accurate electronic energies of molecules at equilibrium geometries, a consistent description of entire combustion reaction potential energy surfaces (PESs) requires multireference correlated wavefunction theories. Here we use bond dissociation energies (BDEs) as a foundational metric to benchmark methods based on multireference configuration interaction (MRCI) for several classes of oxygenated compounds (alcohols, aldehydes, carboxylic acids, and methyl esters). We compare results from multireference singles and doubles configuration interaction to those utilizing a posteriori and a priori size-extensivity corrections, benchmarked against experiment and coupled cluster theory. We demonstrate that size-extensivity corrections are necessary for chemically accurate BDE predictions even in relatively small molecules and furnish examples of unphysical BDE predictions resulting from using too-small orbital active spaces. We also outline the specific challenges in using MRCI methods for carbonyl-containing compounds. The resulting complete basis set extrapolated, size-extensivity-corrected MRCI scheme produces BDEs generally accurate to within 1 kcal/mol, laying the foundation for this scheme's use on larger molecules and for more complex regions of combustion PESs.
SOME NEW FINITE DIFFERENCE METHODS FOR HELMHOLTZ EQUATIONS ON IRREGULAR DOMAINS OR WITH INTERFACES.
Wan, Xiaohai; Li, Zhilin
2012-06-01
Solving a Helmholtz equation Δu + λu = f efficiently is a challenge for many applications. For example, the core part of many efficient solvers for the incompressible Navier-Stokes equations is to solve one or several Helmholtz equations. In this paper, two new finite difference methods are proposed for solving Helmholtz equations on irregular domains, or with interfaces. For Helmholtz equations on irregular domains, the accuracy of the numerical solution obtained using the existing augmented immersed interface method (AIIM) may deteriorate when the magnitude of λ is large. In our new method, we use a level set function to extend the source term and the PDE to a larger domain before we apply the AIIM. For Helmholtz equations with interfaces, a new maximum principle preserving finite difference method is developed. The new method still uses the standard five-point stencil with modifications of the finite difference scheme at irregular grid points. The resulting coefficient matrix of the linear system of finite difference equations satisfies the sign property of the discrete maximum principle and can be solved efficiently using a multigrid solver. The finite difference method is also extended to handle temporal discretized equations where the solution coefficient λ is inversely proportional to the mesh size. PMID:22701346
Improving sub-grid scale accuracy of boundary features in regional finite-difference models
Panday, Sorab; Langevin, Christian D.
2012-01-01
As an alternative to grid refinement, the concept of a ghost node, which was developed for nested grid applications, has been extended towards improving sub-grid scale accuracy of flow to conduits, wells, rivers or other boundary features that interact with a finite-difference groundwater flow model. The formulation is presented for correcting the regular finite-difference groundwater flow equations for confined and unconfined cases, with or without Newton Raphson linearization of the nonlinearities, to include the Ghost Node Correction (GNC) for location displacement. The correction may be applied on the right-hand side vector for a symmetric finite-difference Picard implementation, or on the left-hand side matrix for an implicit but asymmetric implementation. The finite-difference matrix connectivity structure may be maintained for an implicit implementation by only selecting contributing nodes that are a part of the finite-difference connectivity. Proof of concept example problems are provided to demonstrate the improved accuracy that may be achieved through sub-grid scale corrections using the GNC schemes.
Semi-implicit finite difference methods for three-dimensional shallow water flow
Casulli, Vincenzo; Cheng, Ralph T.
1992-01-01
A semi-implicit finite difference method for the numerical solution of three-dimensional shallow water flows is presented and discussed. The governing equations are the primitive three-dimensional turbulent mean flow equations where the pressure distribution in the vertical has been assumed to be hydrostatic. In the method of solution a minimal degree of implicitness has been adopted in such a fashion that the resulting algorithm is stable and gives a maximal computational efficiency at a minimal computational cost. At each time step the numerical method requires the solution of one large linear system which can be formally decomposed into a set of small three-diagonal systems coupled with one five-diagonal system. All these linear systems are symmetric and positive definite. Thus the existence and uniquencess of the numerical solution are assured. When only one vertical layer is specified, this method reduces as a special case to a semi-implicit scheme for solving the corresponding two-dimensional shallow water equations. The resulting two- and three-dimensional algorithm has been shown to be fast, accurate and mass-conservative and can also be applied to simulate flooding and drying of tidal mud-flats in conjunction with three-dimensional flows. Furthermore, the resulting algorithm is fully vectorizable for an efficient implementation on modern vector computers.
Fracture flow simulation using a finite-difference lattice Boltzmann method.
Kim, I; Lindquist, W B; Durham, W B
2003-04-01
We present numerical computations for single phase flow through three-dimensional digitized rock fractures under varied simulated confining pressures appropriate to midcrustal depths. The computations are performed using a finite difference, lattice Boltzmann method and thus simulate Navier-Stokes flow. The digitized fracture data sets come from profiled elevations taken on tensile induced fractures in Harcourt granite. Numerical predictions of fracture permeability are compared with laboratory measurements performed on the same fractures. Use of the finite difference lattice Boltzmann method allows computation on nonuniform grid spacing, enabling accurate resolution across the aperture width without extensive refinement in the other two directions.
NASA Technical Reports Server (NTRS)
Jameson, A.
1976-01-01
A review is presented of some recently developed numerical methods for the solution of nonlinear equations of mixed type. The methods considered use finite difference approximations to the differential equation. Central difference formulas are employed in the subsonic zone and upwind difference formulas are used in the supersonic zone. The relaxation method for the small disturbance equation is discussed and a description is given of difference schemes for the potential flow equation in quasi-linear form. Attention is also given to difference schemes for the potential flow equation in conservation form, the analysis of relaxation schemes by the time dependent analogy, the accelerated iterative method, and three-dimensional calculations.
Fei, Tong; Larner, K.
1993-11-01
Finite-difference acoustic-wave modeling and reverse-time depth migration based on the full wave equation are general approaches that can take into account arbitary variations in velocity and density, and can handle turning waves well. However, conventional finite-difference methods for solving the acousticwave equation suffer from numerical dispersion when too few samples per wavelength are used. Here, we present two flux-corrected transport (FCT) algorithms, one based the second-order equation and the other based on first-order wave equations derived from the second-order one. Combining the FCT technique with conventional finite-difference modeling or reverse-time wave extrapolation can ensure finite-difference solutions without numerical dispersion even for shock waves and impulsive sources. Computed two-dimensional migration images show accurate positioning of reflectors with greater than 90-degree dip. Moreover, application to real data shows no indication of numerical dispersion. The FCT correction, which can be applied to finite-difference approximations of any order in space and time, is an efficient alternative to use of approximations of increasing order.
NASA Astrophysics Data System (ADS)
Hejranfar, Kazem; Ezzatneshan, Eslam
2014-06-01
In this work, the implementation of a high-order compact finite-difference lattice Boltzmann method (CFDLBM) is performed in the generalized curvilinear coordinates to improve the computational efficiency of the solution algorithm to handle curved geometries with non-uniform grids. The incompressible form of the discrete Boltzmann equation with the Bhatnagar-Gross-Krook (BGK) approximation with the pressure as the independent dynamic variable is transformed into the generalized curvilinear coordinates. Herein, the spatial derivatives in the resulting lattice Boltzmann (LB) equation in the computational plane are discretized by using the fourth-order compact finite-difference scheme and the temporal term is discretized with the fourth-order Runge-Kutta scheme to provide an accurate and efficient incompressible flow solver. A high-order spectral-type low-pass compact filter is used to regularize the numerical solution and remove spurious waves generated by boundary conditions, flow non-linearities and grid non-uniformity. All boundary conditions are implemented based on the solution of governing equations in the generalized curvilinear coordinates. The accuracy and efficiency of the solution methodology presented are demonstrated by computing different benchmark steady and unsteady incompressible flow problems. A sensitivity study is also conducted to evaluate the effects of grid size and filtering on the accuracy and convergence rate of the solution. Four test cases considered herein for validating the present computations and demonstrating the accuracy and robustness of the solution algorithm are: unsteady Couette flow and steady flow in a 2-D cavity with non-uniform grid and steady and unsteady flows over a circular cylinder and the NACA0012 hydrofoil at different flow conditions. Results obtained for the above test cases are in good agreement with the existing numerical and experimental results. The study shows the present solution methodology based on the
Finite difference time domain analysis of chirped dielectric gratings
NASA Technical Reports Server (NTRS)
Hochmuth, Diane H.; Johnson, Eric G.
1993-01-01
The finite difference time domain (FDTD) method for solving Maxwell's time-dependent curl equations is accurate, computationally efficient, and straight-forward to implement. Since both time and space derivatives are employed, the propagation of an electromagnetic wave can be treated as an initial-value problem. Second-order central-difference approximations are applied to the space and time derivatives of the electric and magnetic fields providing a discretization of the fields in a volume of space, for a period of time. The solution to this system of equations is stepped through time, thus, simulating the propagation of the incident wave. If the simulation is continued until a steady-state is reached, an appropriate far-field transformation can be applied to the time-domain scattered fields to obtain reflected and transmitted powers. From this information diffraction efficiencies can also be determined. In analyzing the chirped structure, a mesh is applied only to the area immediately around the grating. The size of the mesh is then proportional to the electric size of the grating. Doing this, however, imposes an artificial boundary around the area of interest. An absorbing boundary condition must be applied along the artificial boundary so that the outgoing waves are absorbed as if the boundary were absent. Many such boundary conditions have been developed that give near-perfect absorption. In this analysis, the Mur absorbing boundary conditions are employed. Several grating structures were analyzed using the FDTD method.
NASA Astrophysics Data System (ADS)
Li, Hong; Zhang, Wei; Zhang, Zhenguo; Chen, Xiaofei
2015-07-01
A discontinuous grid finite-difference (FD) method with non-uniform time step Runge-Kutta scheme on curvilinear collocated-grid is developed for seismic wave simulation. We introduce two transition zones: a spatial transition zone and a temporal transition zone, to exchange wavefield across the spatial and temporal discontinuous interfaces. A Gaussian filter is applied to suppress artificial numerical noise caused by down-sampling the wavefield from the finer grid to the coarser grid. We adapt the non-uniform time step Runge-Kutta scheme to a discontinuous grid FD method for further increasing the computational efficiency without losing the accuracy of time marching through the whole simulation region. When the topography is included in the modelling, we carry out the discontinuous grid method on a curvilinear collocated-grid to obtain a sufficiently accurate free-surface boundary condition implementation. Numerical tests show that the proposed method can sufficiently accurately simulate the seismic wave propagation on such grids and significantly reduce the computational resources consumption with respect to regular grids.
Finite-difference models of ordinary differential equations - Influence of denominator functions
NASA Technical Reports Server (NTRS)
Mickens, Ronald E.; Smith, Arthur
1990-01-01
This paper discusses the influence on the solutions of finite-difference schemes of using a variety of denominator functions in the discrete modeling of the derivative for any ordinary differential equation. The results obtained are a consequence of using a generalized definition of the first derivative. A particular example of the linear decay equation is used to illustrate in detail the various solution possibilities that can occur.
Hybrid finite element-finite difference method for thermal analysis of blood vessels.
Blanchard, C H; Gutierrez, G; White, J A; Roemer, R B
2000-01-01
A hybrid finite-difference/finite-element technique for the thermal analysis of blood vessels embedded in perfused tissue has been developed and evaluated. This method provides efficient and accurate solutions to the conjugated heat transfer problem of convection by blood coupled to conduction in the tissue. The technique uses a previously developed 3D automatic meshing method for creating a finite element mesh in the tissue surrounding the vessels, coupled iteratively with a 1-D marching finite difference method for the interior of the vessels. This hybrid technique retains the flexibility and ease of automated finite-element meshing techniques for modelling the complex geometry of blood vessels and irregularly shaped tissues, and speeds the solution time by using a simple finite-difference method to calculate the bulk mean temperatures within all blood vessels. The use of the 1D finite-difference technique in the blood vessels also eliminates the large computer memory requirements needed to accurately solve large vessel network problems when fine FE meshes are used in the interior of vessels. The accuracy of the hybrid technique has been verified against previously verified numerical solutions. In summary, the hybrid technique combines the accuracy and flexibility found in automated finite-element techniques, with the speed and reduction of computational memory requirements associated with the 1D finite-difference technique, something which has not been done before. This method, thus, has the potential to provide accurate, flexible and relatively fast solutions for the thermal analysis of coupled perfusion/blood vessel problems, and large vessel network problems.
Seismic-acoustic finite-difference wave propagation algorithm.
Preston, Leiph; Aldridge, David Franklin
2010-10-01
An efficient numerical algorithm for treating earth models composed of fluid and solid portions is obtained via straightforward modifications to a 3D time-domain finite-difference algorithm for simulating isotropic elastic wave propagation.
Finite Differences and Collocation Methods for the Solution of the Two Dimensional Heat Equation
NASA Technical Reports Server (NTRS)
Kouatchou, Jules
1999-01-01
In this paper we combine finite difference approximations (for spatial derivatives) and collocation techniques (for the time component) to numerically solve the two dimensional heat equation. We employ respectively a second-order and a fourth-order schemes for the spatial derivatives and the discretization method gives rise to a linear system of equations. We show that the matrix of the system is non-singular. Numerical experiments carried out on serial computers, show the unconditional stability of the proposed method and the high accuracy achieved by the fourth-order scheme.
NASA Astrophysics Data System (ADS)
Lin, M. C.; Nieter, C.; Stoltz, P. H.; Smithe, D. N.
2009-05-01
This work introduces a conformal finite difference time domain (CFDTD) method to accurately determine the dispersion relation of an A6 relativistic magnetron. The accuracy is measured by comparing with accurate SUPERFISH calculations based on finite element method. The results show that an accuracy of 99.4% can be achieved by using only 10,000 mesh points with Dey-Mittra algorithm. By comparison, a mesh number of 360,000 is needed to preserve 99% accuracy using conventional FDTD method. This suggests one can efficiently and accurately study the hot tests of microwave tubes using CFDTD particle-in-cell method instead of conventional FDTD one.
Properties of finite difference models of non-linear conservative oscillators
NASA Technical Reports Server (NTRS)
Mickens, R. E.
1988-01-01
Finite-difference (FD) approaches to the numerical solution of the differential equations describing the motion of a nonlinear conservative oscillator are investigated analytically. A generalized formulation of the Duffing and modified Duffing equations is derived and analyzed using several FD techniques, and it is concluded that, although it is always possible to contstruct FD models of conservative oscillators which are themselves conservative, caution is required to avoid numerical solutions which do not accurately reflect the properties of the original equation.
Double absorbing boundaries for finite-difference time-domain electromagnetics
NASA Astrophysics Data System (ADS)
LaGrone, John; Hagstrom, Thomas
2016-12-01
We describe the implementation of optimal local radiation boundary condition sequences for second order finite difference approximations to Maxwell's equations and the scalar wave equation using the double absorbing boundary formulation. Numerical experiments are presented which demonstrate that the design accuracy of the boundary conditions is achieved and, for comparable effort, exceeds that of a convolution perfectly matched layer with reasonably chosen parameters. An advantage of the proposed approach is that parameters can be chosen using an accurate a priori error bound.
NASA Astrophysics Data System (ADS)
Lao, Ka Un; Herbert, John M.
2013-06-01
An efficient, monomer-based electronic structure method is introduced for computing non-covalent interactions in molecular and ionic clusters. It builds upon our ``explicit polarization" (XPol) with pairwise-additive symmetry-adapted perturbation theory (SAPT) using the Kohn-Sham (KS) version of SAPT, but replaces the problematic and expensive sum-over-states dispersion terms with empirical potentials. This modification reduces the scaling from {O}(N^5) to {O}(N^3) and also facilitates the use of Kohn-Sham density functional theory (KS-DFT) as a low-cost means to capture intramolecular electron correlation. Accurate binding energies are obtained for benchmark databases of dimer binding energies, and potential energy curves are also captured accurately, for a variety of challenging systems. As compared to traditional DFT-SAPT or SAPT(DFT) methods, it removes the limitation to dimers and extends SAPT-based methodology to many-body systems. For many-body systems such as water clusters and halide-water cluster anions, the new method is superior to established density-functional methods for non-covalent interactions. We suggest that using different asymptotic corrections for different monomers is necessary to get good binding energies in general, as DFT-SAPT or SAPT(DFT), especially for hydrogen-bonded complexes. We also introduce a decomposition scheme for the interaction energy that extends traditional SAPT energy decomposition analysis to systems containing more than two monomers, and we find that the various energy components (electrostatic, exchange, induction, and dispersion) are in very good agreement with high-level SAPT benchmarks for dimers. For (H_2O)_6, the many-body contribution to the interaction energy agrees well with that obtained from traditional Kitaura-Morokuma energy decomposition analysis.
NASA Technical Reports Server (NTRS)
Handschuh, Robert F.
1987-01-01
An exponential finite difference algorithm, as first presented by Bhattacharya for one-dimensianal steady-state, heat conduction in Cartesian coordinates, has been extended. The finite difference algorithm developed was used to solve the diffusion equation in one-dimensional cylindrical coordinates and applied to two- and three-dimensional problems in Cartesian coordinates. The method was also used to solve nonlinear partial differential equations in one (Burger's equation) and two (Boundary Layer equations) dimensional Cartesian coordinates. Predicted results were compared to exact solutions where available, or to results obtained by other numerical methods. It was found that the exponential finite difference method produced results that were more accurate than those obtained by other numerical methods, especially during the initial transient portion of the solution. Other applications made using the exponential finite difference technique included unsteady one-dimensional heat transfer with temperature varying thermal conductivity and the development of the temperature field in a laminar Couette flow.
Stochastic finite difference lattice Boltzmann method for steady incompressible viscous flows
Fu, S.C.; So, R.M.C.; Leung, W.W.F.
2010-08-20
With the advent of state-of-the-art computers and their rapid availability, the time is ripe for the development of efficient uncertainty quantification (UQ) methods to reduce the complexity of numerical models used to simulate complicated systems with incomplete knowledge and data. The spectral stochastic finite element method (SSFEM) which is one of the widely used UQ methods, regards uncertainty as generating a new dimension and the solution as dependent on this dimension. A convergent expansion along the new dimension is then sought in terms of the polynomial chaos system, and the coefficients in this representation are determined through a Galerkin approach. This approach provides an accurate representation even when only a small number of terms are used in the spectral expansion; consequently, saving in computational resource can be realized compared to the Monte Carlo (MC) scheme. Recent development of a finite difference lattice Boltzmann method (FDLBM) that provides a convenient algorithm for setting the boundary condition allows the flow of Newtonian and non-Newtonian fluids, with and without external body forces to be simulated with ease. Also, the inherent compressibility effect in the conventional lattice Boltzmann method, which might produce significant errors in some incompressible flow simulations, is eliminated. As such, the FDLBM together with an efficient UQ method can be used to treat incompressible flows with built in uncertainty, such as blood flow in stenosed arteries. The objective of this paper is to develop a stochastic numerical solver for steady incompressible viscous flows by combining the FDLBM with a SSFEM. Validation against MC solutions of channel/Couette, driven cavity, and sudden expansion flows are carried out.
NASA Technical Reports Server (NTRS)
Macaraeg, M. G.
1985-01-01
A numerical study of the steady, axisymmetric flow in a heated, rotating spherical shell is conducted to model the Atmospheric General Circulation Experiment (AGCE) proposed to run aboard a later Shuttle mission. The AGCE will consist of concentric rotating spheres confining a dielectric fluid. By imposing a dielectric field across the fluid a radial body force will be created. The numerical solution technique is based on the incompressible Navier-Stokes equations. In the method a pseudospectral technique is used in the latitudinal direction, and a second-order accurate finite difference scheme discretizes time and radial derivatives. This paper discusses the development and performance of this numerical scheme for the AGCE which has been modeled in the past only by pure FD formulations. In addition, previous models have not investigated the effect of using a dielectric force to simulate terrestrial gravity. The effect of this dielectric force on the flow field is investigated as well as a parameter study of varying rotation rates and boundary temperatures. Among the effects noted are the production of larger velocities and enhanced reversals of radial temperature gradients for a body force generated by the electric field.
NASA Technical Reports Server (NTRS)
Macaraeg, M. G.
1985-01-01
A numerical study of the steady, axisymmetric flow in a heated, rotating spherical shell is conducted to model the Atmospheric General Circulation Experiment (AGCE) proposed to run aboard a later shuttle mission. The AGCE will consist of concentric rotating spheres confining a dielectric fluid. By imposing a dielectric field across the fluid a radial body force will be created. The numerical solution technique is based on the incompressible Navier-Stokes equations. In the method a pseudospectral technique is based on the incompressible Navier-Stokes equations. In the method a pseudospectral technique is used in the latitudinal direction, and a second-order accurate finite difference scheme discretizes time and radial derivatives. This paper discusses the development and performance of this numerical scheme for the AGCE which has been modelled in the past only by pure FD formulations. In addition, previous models have not investigated the effect of using a dielectric force to simulate terrestrial gravity. The effect of this dielectric force on the flow field is investigated as well as a parameter study of varying rotation rates and boundary temperatures. Among the effects noted are the production of larger velocities and enhanced reversals of radial temperature gradients for a body force generated by the electric field.
An improved finite-difference analysis of uncoupled vibrations of tapered cantilever beams
NASA Technical Reports Server (NTRS)
Subrahmanyam, K. B.; Kaza, K. R. V.
1983-01-01
An improved finite difference procedure for determining the natural frequencies and mode shapes of tapered cantilever beams undergoing uncoupled vibrations is presented. Boundary conditions are derived in the form of simple recursive relations involving the second order central differences. Results obtained by using the conventional first order central differences and the present second order central differences are compared, and it is observed that the present second order scheme is more efficient than the conventional approach. An important advantage offered by the present approach is that the results converge to exact values rapidly, and thus the extrapolation of the results is not necessary. Consequently, the basic handicap with the classical finite difference method of solution that requires the Richardson's extrapolation procedure is eliminated. Furthermore, for the cases considered herein, the present approach produces consistent lower bound solutions.
Convergence rates of finite difference stochastic approximation algorithms part I: general sampling
NASA Astrophysics Data System (ADS)
Dai, Liyi
2016-05-01
Stochastic optimization is a fundamental problem that finds applications in many areas including biological and cognitive sciences. The classical stochastic approximation algorithm for iterative stochastic optimization requires gradient information of the sample object function that is typically difficult to obtain in practice. Recently there has been renewed interests in derivative free approaches to stochastic optimization. In this paper, we examine the rates of convergence for the Kiefer-Wolfowitz algorithm and the mirror descent algorithm, under various updating schemes using finite differences as gradient approximations. The analysis is carried out under a general framework covering a wide range of updating scenarios. It is shown that the convergence of these algorithms can be accelerated by controlling the implementation of the finite differences.
Improved finite-difference vibration analysis of pretwisted, tapered beams
NASA Technical Reports Server (NTRS)
Subrahmanyam, K. B.; Kaza, K. R. V.
1984-01-01
An improved finite difference procedure based upon second order central differences is developed. Several difficulties encountered in earlier works with fictitious stations that arise in using second order central differences, are eliminated by developing certain recursive relations. The need for forward or backward differences at the beam boundaries or other similar procedures is eliminated in the present theory. By using this improved theory, the vibration characteristics of pretwisted and tapered blades are calculated. Results of the second order theory are compared with published theoretical and experimental results and are found to be in good agreement. The present method generally produces close lower bound solutions and shows fast convergence. Thus, extrapolation procedures that are customary with first order finite-difference methods are unnecessary. Furthermore, the computational time and effort needed for this improved method are almost the same as required for the conventional first order finite-difference approach.
3D Staggered-Grid Finite-Difference Simulation of Acoustic Waves in Turbulent Moving Media
NASA Astrophysics Data System (ADS)
Symons, N. P.; Aldridge, D. F.; Marlin, D.; Wilson, D. K.; Sullivan, P.; Ostashev, V.
2003-12-01
Acoustic wave propagation in a three-dimensional heterogeneous moving atmosphere is accurately simulated with a numerical algorithm recently developed under the DOD Common High Performance Computing Software Support Initiative (CHSSI). Sound waves within such a dynamic environment are mathematically described by a set of four, coupled, first-order partial differential equations governing small-amplitude fluctuations in pressure and particle velocity. The system is rigorously derived from fundamental principles of continuum mechanics, ideal-fluid constitutive relations, and reasonable assumptions that the ambient atmospheric motion is adiabatic and divergence-free. An explicit, time-domain, finite-difference (FD) numerical scheme is used to solve the system for both pressure and particle velocity wavefields. The atmosphere is characterized by 3D gridded models of sound speed, mass density, and the three components of the wind velocity vector. Dependent variables are stored on staggered spatial and temporal grids, and centered FD operators possess 2nd-order and 4th-order space/time accuracy. Accurate sound wave simulation is achieved provided grid intervals are chosen appropriately. The gridding must be fine enough to reduce numerical dispersion artifacts to an acceptable level and maintain stability. The algorithm is designed to execute on parallel computational platforms by utilizing a spatial domain-decomposition strategy. Currently, the algorithm has been validated on four different computational platforms, and parallel scalability of approximately 85% has been demonstrated. Comparisons with analytic solutions for uniform and vertically stratified wind models indicate that the FD algorithm generates accurate results with either a vanishing pressure or vanishing vertical-particle velocity boundary condition. Simulations are performed using a kinematic turbulence wind profile developed with the quasi-wavelet method. In addition, preliminary results are presented
Serpentine: Finite Difference Methods for Wave Propagation in Second Order Formulation
Petersson, N A; Sjogreen, B
2012-03-26
second order system is significantly smaller. Another issue with re-writing a second order system into first order form is that compatibility conditions often must be imposed on the first order form. These (Saint-Venant) conditions ensure that the solution of the first order system also satisfies the original second order system. However, such conditions can be difficult to enforce on the discretized equations, without introducing additional modeling errors. This project has previously developed robust and memory efficient algorithms for wave propagation including effects of curved boundaries, heterogeneous isotropic, and viscoelastic materials. Partially supported by internal funding from Lawrence Livermore National Laboratory, many of these methods have been implemented in the open source software WPP, which is geared towards 3-D seismic wave propagation applications. This code has shown excellent scaling on up to 32,768 processors and has enabled seismic wave calculations with up to 26 Billion grid points. TheWPP calculations have resulted in several publications in the field of computational seismology, e.g.. All of our current methods are second order accurate in both space and time. The benefits of higher order accurate schemes for wave propagation have been known for a long time, but have mostly been developed for first order hyperbolic systems. For second order hyperbolic systems, it has not been known how to make finite difference schemes stable with free surface boundary conditions, heterogeneous material properties, and curvilinear coordinates. The importance of higher order accurate methods is not necessarily to make the numerical solution more accurate, but to reduce the computational cost for obtaining a solution within an acceptable error tolerance. This is because the accuracy in the solution can always be improved by reducing the grid size h. However, in practice, the available computational resources might not be large enough to solve the problem with a
Finite-difference numerical simulations of underground explosion cavity decoupling
NASA Astrophysics Data System (ADS)
Aldridge, D. F.; Preston, L. A.; Jensen, R. P.
2012-12-01
Earth models containing a significant portion of ideal fluid (e.g., air and/or water) are of increasing interest in seismic wave propagation simulations. Examples include a marine model with a thick water layer, and a land model with air overlying a rugged topographic surface. The atmospheric infrasound community is currently interested in coupled seismic-acoustic propagation of low-frequency signals over long ranges (~tens to ~hundreds of kilometers). Also, accurate and efficient numerical treatment of models containing underground air-filled voids (caves, caverns, tunnels, subterranean man-made facilities) is essential. In support of the Source Physics Experiment (SPE) conducted at the Nevada National Security Site (NNSS), we are developing a numerical algorithm for simulating coupled seismic and acoustic wave propagation in mixed solid/fluid media. Solution methodology involves explicit, time-domain, finite-differencing of the elastodynamic velocity-stress partial differential system on a three-dimensional staggered spatial grid. Conditional logic is used to avoid shear stress updating within the fluid zones; this approach leads to computational efficiency gains for models containing a significant proportion of ideal fluid. Numerical stability and accuracy are maintained at air/rock interfaces (where the contrast in mass density is on the order of 1 to 2000) via a finite-difference operator "order switching" formalism. The fourth-order spatial FD operator used throughout the bulk of the earth model is reduced to second-order in the immediate vicinity of a high-contrast interface. Current modeling efforts are oriented toward quantifying the amount of atmospheric infrasound energy generated by various underground seismic sources (explosions and earthquakes). Source depth and orientation, and surface topography play obvious roles. The cavity decoupling problem, where an explosion is detonated within an air-filled void, is of special interest. A point explosion
Computer-Oriented Calculus Courses Using Finite Differences.
ERIC Educational Resources Information Center
Gordon, Sheldon P.
The so-called discrete approach in calculus instruction involves introducing topics from the calculus of finite differences and finite sums, both for motivation and as useful tools for applications of the calculus. In particular, it provides an ideal setting in which to incorporate computers into calculus courses. This approach has been…
Ackleh, Azmy S; Chellamuthu, Vinodh K; Ito, Kazufumi
2015-04-01
We study a quasilinear hierarchically size-structured population model presented in [4]. In this model the growth, mortality and reproduction rates are assumed to depend on a function of the population density. In [4] we showed that solutions to this model can become singular (measure-valued) in finite time even if all the individual parameters are smooth. Therefore, in this paper we develop a first order finite difference scheme to compute these measure-valued solutions. Convergence analysis for this method is provided. We also develop a high resolution second order scheme to compute the measure-valued solution of the model and perform a comparative study between the two schemes. PMID:25811433
Finite-difference time-domain analysis for the dynamics and diffraction of exciton-polaritons.
Chen, Minfeng; Chang, Yia-Chung; Hsieh, Wen-Feng
2015-10-01
We adopted a finite-difference time-domain (FDTD) scheme to simulate the dynamics and diffraction of exciton-polaritons, governed by the coupling of polarization waves with electromagnetic waves. The polarization wave, an approximate solution to the Schrödinger's equation at low frequencies, essentially captures the exciton behavior. Numerical stability of the scheme is analyzed and simple examples are provided to prove its validity. The system considered is both temporally and spatially dispersive, for which the FDTD analysis has attracted less attention in the literature. Here, we demonstrate that the FDTD scheme could be useful for studying the optical response of the exciton-polariton and its dynamics. The diffraction of a polariton wave from a polaritonic grating is also considered, and many sharp resonances are found, which manifest the interference effect of polariton waves. This illustrates that the measurement of transmittance or reflectance near polariton resonance can reveal subwavelength features in semiconductors, which are sensitive to polariton scattering.
Optimal implicit 2-D finite differences to model wave propagation in poroelastic media
NASA Astrophysics Data System (ADS)
Itzá, Reymundo; Iturrarán-Viveros, Ursula; Parra, Jorge O.
2016-05-01
Numerical modeling of seismic waves in heterogeneous porous reservoir rocks is an important tool for the interpretation of seismic surveys in reservoir engineering. We apply globally optimal implicit staggered-grid finite-differences to model 2-D wave propagation in heterogeneous poroelastic media at a low-frequency range (<10kHz). We validate the numerical solution by comparing it to an analytical-transient solution obtaining clear seismic wavefields including fast P, slow P and S waves (for a porous media saturated with fluid). The numerical dispersion and stability conditions are derived using von Neumann analysis, showing that over a wide range of porous materials the Courant condition governs the stability and this optimal implicit scheme improves the stability of explicit schemes. High order explicit finite-differences (FD) can be replaced by some lower order optimal implicit FD so computational cost will not be as expensive while maintaining the accuracy. Here we compute weights for the optimal implicit FD scheme to attain an accuracy of γ = 10-8. The implicit spatial differentiation involves solving tridiagonal linear systems of equations through Thomas' algorithm.
NASA Technical Reports Server (NTRS)
Osher, S.
1984-01-01
The construction of a reliable, shock capturing finite difference method to solve the Euler equations for inviscid, supersonic flow past fighter and missile type configurations is highly desirable. The numerical method must have a firm theoretical foundation and must be robust and efficient. It should be able to treat subsonic pockets in a predominantly supersonic flow. The method must also be easily applicable to the complex topologies of the aerodynamic configuration under consideration. The ongoing approach to this task is described and for steady supersonic flows is presented. This scheme is the basic numerical method. Results of work obtained during previous years are presented.
Two-dimensional finite difference program for thermal analysis of rocket thrust chambers
Naraghi, M.H.
1987-09-01
A two-dimensional finite difference computer model for thermal analysis of rocket thrust chambers has been developed. The model uses an iterative scheme for calculating the temperature distribution within the chamber wall and implements a successive overrelaxation formula for a quick convergence. The inputs of the models are the dimensions of the thrust chamber wall, types of materials used, heat transfer coefficients and temperatures of the hot gas and the coolant. The resulting output of the program consists of the nodal temperature distribution, heat transfer to the coolant and heat transfer from the hot gas.
Flux vector splitting of the inviscid equations with application to finite difference methods
NASA Technical Reports Server (NTRS)
Steger, J. L.; Warming, R. F.
1979-01-01
The conservation-law form of the inviscid gasdynamic equations has the remarkable property that the nonlinear flux vectors are homogeneous functions of degree one. This property readily permits the splitting of flux vectors into subvectors by similarity transformations so that each subvector has associated with it a specified eigenvalue spectrum. As a consequence of flux vector splitting, new explicit and implicit dissipative finite-difference schemes are developed for first-order hyperbolic systems of equations. Appropriate one-sided spatial differences for each split flux vector are used throughout the computational field even if the flow is locally subsonic. The results of some preliminary numerical computations are included.
A two-dimensional finite difference program for thermal analysis of rocket thrust chambers
NASA Technical Reports Server (NTRS)
Naraghi, Mohammad H.
1987-01-01
A two-dimensional finite difference computer model for thermal analysis of rocket thrust chambers has been developed. The model uses an iterative scheme for calculating the temperature distribution within the chamber wall and implements a successive overrelaxation formula for a quick convergence. The inputs of the models are the dimensions of the thrust chamber wall, types of materials used, heat transfer coefficients and temperatures of the hot gas and the coolant. The resulting output of the program consists of the nodal temperature distribution, heat transfer to the coolant and heat transfer from the hot gas.
NASA Astrophysics Data System (ADS)
Tan, Sirui; Huang, Lianjie
2014-05-01
For modelling large-scale 3-D scalar-wave propagation, the finite-difference (FD) method with high-order accuracy in space but second-order accuracy in time is widely used because of its relatively low requirements of computer memory. We develop a novel staggered-grid (SG) FD method with high-order accuracy not only in space, but also in time, for solving 2- and 3-D scalar-wave equations. We determine the coefficients of the FD operator in the joint time-space domain to achieve high-order accuracy in time while preserving high-order accuracy in space. Our new FD scheme is based on a stencil that contains a few more grid points than the standard stencil. It is 2M-th-order accurate in space and fourth-order accurate in time when using 2M grid points along each axis and wavefields at one time step as the standard SGFD method. We validate the accuracy and efficiency of our new FD scheme using dispersion analysis and numerical modelling of scalar-wave propagation in 2- and 3-D complex models with a wide range of velocity contrasts. For media with a velocity contrast up to five, our new FD scheme is approximately two times more computationally efficient than the standard SGFD scheme with almost the same computer-memory requirement as the latter. Further numerical experiments demonstrate that our new FD scheme loses its advantages over the standard SGFD scheme if the velocity contrast is 10. However, for most large-scale geophysical applications, the velocity contrasts often range approximately from 1 to 3. Our new method is thus particularly useful for large-scale 3-D scalar-wave modelling and full-waveform inversion.
NASA Astrophysics Data System (ADS)
Gladden, Herbert J.; Ko, Ching L.; Boddy, Douglas E.
1995-01-01
A higher-order finite-difference technique is developed to calculate the developing-flow field of steady incompressible laminar flows in the entrance regions of circular pipes. Navier-Stokes equations governing the motion of such a flow field are solved by using this new finite-difference scheme. This new technique can increase the accuracy of the finite-difference approximation, while also providing the option of using unevenly spaced clustered nodes for computation such that relatively fine grids can be adopted for regions with large velocity gradients. The velocity profile at the entrance of the pipe is assumed to be uniform for the computation. The velocity distribution and the surface pressure drop of the developing flow then are calculated and compared to existing experimental measurements reported in the literature. Computational results obtained are found to be in good agreement with existing experimental correlations and therefore, the reliability of the new technique has been successfully tested.
Finite-difference modeling with variable grid-size and adaptive time-step in porous media
NASA Astrophysics Data System (ADS)
Liu, Xinxin; Yin, Xingyao; Wu, Guochen
2014-04-01
Forward modeling of elastic wave propagation in porous media has great importance for understanding and interpreting the influences of rock properties on characteristics of seismic wavefield. However, the finite-difference forward-modeling method is usually implemented with global spatial grid-size and time-step; it consumes large amounts of computational cost when small-scaled oil/gas-bearing structures or large velocity-contrast exist underground. To overcome this handicap, combined with variable grid-size and time-step, this paper developed a staggered-grid finite-difference scheme for elastic wave modeling in porous media. Variable finite-difference coefficients and wavefield interpolation were used to realize the transition of wave propagation between regions of different grid-size. The accuracy and efficiency of the algorithm were shown by numerical examples. The proposed method is advanced with low computational cost in elastic wave simulation for heterogeneous oil/gas reservoirs.
Finite difference seismic modeling of axial magma chambers
Swift, S.A.; Dougherty, M.E.; Stephen, R.A. )
1990-11-01
The authors tested the feasibility of using finite difference methods to model seismic propagation at {approximately}10 Hx through a two-dimensional representation of an axial magma chamber with a thin, liquid lid. This technique produces time series of displacement or pressure at seafloor receivers to mimic a seismic refraction experiment and snapshots of P and S energy propagation. The results indicate that the implementation is stable for models with sharp velocity contrasts and complex geometries. The authors observe a high-energy, downward-traveling shear phase, observable only with borehole receivers, that would be useful in studying the nature and shape of magma chambers. The ability of finite difference methods to model high-order wave phenomena makes this method ideal for testing velocity models of spreading axes and for planning near-axis drilling of the East Pacific Rise in order to optimize the benefits from shear wave imaging of sub-axis structure.
Time dependent wave envelope finite difference analysis of sound propagation
NASA Technical Reports Server (NTRS)
Baumeister, K. J.
1984-01-01
A transient finite difference wave envelope formulation is presented for sound propagation, without steady flow. Before the finite difference equations are formulated, the governing wave equation is first transformed to a form whose solution tends not to oscillate along the propagation direction. This transformation reduces the required number of grid points by an order of magnitude. Physically, the transformed pressure represents the amplitude of the conventional sound wave. The derivation for the wave envelope transient wave equation and appropriate boundary conditions are presented as well as the difference equations and stability requirements. To illustrate the method, example solutions are presented for sound propagation in a straight hard wall duct and in a two dimensional straight soft wall duct. The numerical results are in good agreement with exact analytical results.
Semianalytical computation of path lines for finite-difference models
Pollock, D.W.
1988-01-01
A semianalytical particle tracking method was developed for use with velocities generated from block-centered finite-difference ground-water flow models. Based on the assumption that each directional velocity component varies linearly within a grid cell in its own coordinate directions, the method allows an analytical expression to be obtained describing the flow path within an individual grid cell. Given the intitial position of a particle anywhere in a cell, the coordinates of any other point along its path line within the cell, and the time of travel between them, can be computed directly. For steady-state systems, the exit point for a particle entering a cell at any arbitrary location can be computed in a single step. By following the particle as it moves from cell to cell, this method can be used to trace the path of a particle through any multidimensional flow field generated from a block-centered finite-difference flow model. -Author
Selecting step sizes in sensitivity analysis by finite differences
NASA Technical Reports Server (NTRS)
Iott, J.; Haftka, R. T.; Adelman, H. M.
1985-01-01
This paper deals with methods for obtaining near-optimum step sizes for finite difference approximations to first derivatives with particular application to sensitivity analysis. A technique denoted the finite difference (FD) algorithm, previously described in the literature and applicable to one derivative at a time, is extended to the calculation of several simultaneously. Both the original and extended FD algorithms are applied to sensitivity analysis for a data-fitting problem in which derivatives of the coefficients of an interpolation polynomial are calculated with respect to uncertainties in the data. The methods are also applied to sensitivity analysis of the structural response of a finite-element-modeled swept wing. In a previous study, this sensitivity analysis of the swept wing required a time-consuming trial-and-error effort to obtain a suitable step size, but it proved to be a routine application for the extended FD algorithm herein.
Asymptotic analysis of numerical wave propagation in finite difference equations
NASA Technical Reports Server (NTRS)
Giles, M.; Thompkins, W. T., Jr.
1983-01-01
An asymptotic technique is developed for analyzing the propagation and dissipation of wave-like solutions to finite difference equations. It is shown that for each fixed complex frequency there are usually several wave solutions with different wavenumbers and the slowly varying amplitude of each satisfies an asymptotic amplitude equation which includes the effects of smoothly varying coefficients in the finite difference equations. The local group velocity appears in this equation as the velocity of convection of the amplitude. Asymptotic boundary conditions coupling the amplitudes of the different wave solutions are also derived. A wavepacket theory is developed which predicts the motion, and interaction at boundaries, of wavepackets, wave-like disturbances of finite length. Comparison with numerical experiments demonstrates the success and limitations of the theory. Finally an asymptotic global stability analysis is developed.
Al-Rizzo, Hussain M; Tranquilla, Jim M; Feng, Ma
2005-01-01
In this paper, we present a versatile mathematical formulation of a newly developed 3-D locally conformal Finite Difference (FD) thermal algorithm developed specificallyfor coupled electromagnetic (EM) and heat diffusion simulations utilizing Overlapping Grids (OGFD) in the Cartesian and cylindrical coordinate systems. The motivation for this research arises from an attempt to characterize the dominant thermal transport phenomena typically encountered during the process cycle of a high-power, microwave-assisted material processing system employing a geometrically composite cylindrical multimode heating furnace. The cylindrical FD scheme is only applied to the outer shell of the housing cavity whereas the Cartesian FD scheme is used to advance the temperature elsewhere including top and bottom walls, and most of the inner region of the cavity volume. The temperature dependency of the EM constitutive and thermo-physical parameters of the material being processed is readily accommodated into the OGFD update equations. The time increment, which satisfies the stability constraint of the explicit OGFD time-marching scheme, is derived. In a departure from prior work, the salient features of the proposed algorithm are first, the locally conformal discretization scheme accurately describes the diffusion of heat and second, significant heat-loss mechanisms usually encountered in microwave heating problems at the interfacial boundary temperature nodes have been considered. These include convection and radiation between the surface of the workload and air inside the cavity, heat convection and radiation between the inner cavity walls and interior cavity volume, and free cooling of the outermost cavity walls.
NASA Astrophysics Data System (ADS)
Hejranfar, Kazem; Ezzatneshan, Eslam
2015-11-01
A high-order compact finite-difference lattice Boltzmann method (CFDLBM) is extended and applied to accurately simulate two-phase liquid-vapor flows with high density ratios. Herein, the He-Shan-Doolen-type lattice Boltzmann multiphase model is used and the spatial derivatives in the resulting equations are discretized by using the fourth-order compact finite-difference scheme and the temporal term is discretized with the fourth-order Runge-Kutta scheme to provide an accurate and efficient two-phase flow solver. A high-order spectral-type low-pass compact nonlinear filter is used to regularize the numerical solution and remove spurious waves generated by flow nonlinearities in smooth regions and at the same time to remove the numerical oscillations in the interfacial region between the two phases. Three discontinuity-detecting sensors for properly switching between a second-order and a higher-order filter are applied and assessed. It is shown that the filtering technique used can be conveniently adopted to reduce the spurious numerical effects and improve the numerical stability of the CFDLBM implemented. A sensitivity study is also conducted to evaluate the effects of grid size and the filtering procedure implemented on the accuracy and performance of the solution. The accuracy and efficiency of the proposed solution procedure based on the compact finite-difference LBM are examined by solving different two-phase systems. Five test cases considered herein for validating the results of the two-phase flows are an equilibrium state of a planar interface in a liquid-vapor system, a droplet suspended in the gaseous phase, a liquid droplet located between two parallel wettable surfaces, the coalescence of two droplets, and a phase separation in a liquid-vapor system at different conditions. Numerical results are also presented for the coexistence curve and the verification of the Laplace law. Results obtained are in good agreement with the analytical solutions and also
Finite difference discretisation of a model for biological nerve conduction
NASA Astrophysics Data System (ADS)
Aderogba, A. A.; Chapwanya, M.; Jejeniwa, O. A.
2016-06-01
A nonstandard finite difference method is proposed for the discretisation of the semilinear FitzHugh-Nagumo reaction diffusion equation. The equation has been useful in describing, for example, population models, biological models, heat and mass transfer models, and many other applications. The proposed approach involves splitting the equation into the space independent and the time independent sub equation. Numerical simulations for the full equation are presented.
Calculating rotordynamic coefficients of seals by finite-difference techniques
NASA Technical Reports Server (NTRS)
Dietzen, F. J.; Nordmann, R.
1987-01-01
For modelling the turbulent flow in a seal the Navier-Stokes equations in connection with a turbulence (kappa-epsilon) model are solved by a finite-difference method. A motion of the shaft round the centered position is assumed. After calculating the corresponding flow field and the pressure distribution, the rotor-dynamic coefficients of the seal can be determined. These coefficients are compared with results obtained by using the bulk flow theory of Childs and with experimental results.
Finite difference time domain grid generation from AMC helicopter models
NASA Technical Reports Server (NTRS)
Cravey, Robin L.
1992-01-01
A simple technique is presented which forms a cubic grid model of a helicopter from an Aircraft Modeling Code (AMC) input file. The AMC input file defines the helicopter fuselage as a series of polygonal cross sections. The cubic grid model is used as an input to a Finite Difference Time Domain (FDTD) code to obtain predictions of antenna performance on a generic helicopter model. The predictions compare reasonably well with measured data.
NASA Technical Reports Server (NTRS)
Yee, H. C.; Warming, R. F.; Harten, A.
1985-01-01
Highly accurate and yet stable shock-capturing finite difference schemes have been designed for the computation of the Euler equations of gas dynamics. Four different principles for the construction of high resolution total variation diminishing (TVD) schemes are available, including hybrid schemes, a second-order extension of Godunov's scheme by van Leer (1979), the modified flux approach of Harten (1983, 1984), and the numerical fluctuation approach of Roe (1985). The present paper has the objective to review the class of second-order TVD schemes via the modified flux approach. Attention is given to first-order TVD schemes, a second-order accurate explicit TVD scheme, the global order of accuracy of the second-order TVD scheme, extensions to systems and two-dimensional conservation laws, numerical experiments with a second-order explicit TVD scheme, implicit TVD schemes, and second-order implicit TVD schemes.
Modeling anisotropic flow and heat transport by using mimetic finite differences
NASA Astrophysics Data System (ADS)
Chen, Tao; Clauser, Christoph; Marquart, Gabriele; Willbrand, Karen; Büsing, Henrik
2016-08-01
Modeling anisotropic flow in porous or fractured rock often assumes that the permeability tensor is diagonal, which means that its principle directions are always aligned with the coordinate axes. However, the permeability of a heterogeneous anisotropic medium usually is a full tensor. For overcoming this shortcoming, we use the mimetic finite difference method (mFD) for discretizing the flow equation in a hydrothermal reservoir simulation code, SHEMAT-Suite, which couples this equation with the heat transport equation. We verify SHEMAT-Suite-mFD against analytical solutions of pumping tests, using both diagonal and full permeability tensors. We compare results from three benchmarks for testing the capability of SHEMAT-Suite-mFD to handle anisotropic flow in porous and fractured media. The benchmarks include coupled flow and heat transport problems, three-dimensional problems and flow through a fractured porous medium with full equivalent permeability tensor. It shows firstly that the mimetic finite difference method can model anisotropic flow both in porous and in fractured media accurately and its results are better than those obtained by the multi-point flux approximation method in highly anisotropic models, secondly that the asymmetric permeability tensor can be included and leads to improved results compared the symmetric permeability tensor in the equivalent fracture models, and thirdly that the method can be easily implemented in existing finite volume or finite difference codes, which has been demonstrated successfully for SHEMAT-Suite.
Introduction to finite-difference methods for numerical fluid dynamics
Scannapieco, E.; Harlow, F.H.
1995-09-01
This work is intended to be a beginner`s exercise book for the study of basic finite-difference techniques in computational fluid dynamics. It is written for a student level ranging from high-school senior to university senior. Equations are derived from basic principles using algebra. Some discussion of partial-differential equations is included, but knowledge of calculus is not essential. The student is expected, however, to have some familiarity with the FORTRAN computer language, as the syntax of the computer codes themselves is not discussed. Topics examined in this work include: one-dimensional heat flow, one-dimensional compressible fluid flow, two-dimensional compressible fluid flow, and two-dimensional incompressible fluid flow with additions of the equations of heat flow and the {Kappa}-{epsilon} model for turbulence transport. Emphasis is placed on numerical instabilities and methods by which they can be avoided, techniques that can be used to evaluate the accuracy of finite-difference approximations, and the writing of the finite-difference codes themselves. Concepts introduced in this work include: flux and conservation, implicit and explicit methods, Lagrangian and Eulerian methods, shocks and rarefactions, donor-cell and cell-centered advective fluxes, compressible and incompressible fluids, the Boussinesq approximation for heat flow, Cartesian tensor notation, the Boussinesq approximation for the Reynolds stress tensor, and the modeling of transport equations. A glossary is provided which defines these and other terms.
NASA Technical Reports Server (NTRS)
Baumeister, K. J.; Kreider, K. L.
1996-01-01
An explicit finite difference iteration scheme is developed to study harmonic sound propagation in ducts. To reduce storage requirements for large 3D problems, the time dependent potential form of the acoustic wave equation is used. To insure that the finite difference scheme is both explicit and stable, time is introduced into the Fourier transformed (steady-state) acoustic potential field as a parameter. Under a suitable transformation, the time dependent governing equation in frequency space is simplified to yield a parabolic partial differential equation, which is then marched through time to attain the steady-state solution. The input to the system is the amplitude of an incident harmonic sound source entering a quiescent duct at the input boundary, with standard impedance boundary conditions on the duct walls and duct exit. The introduction of the time parameter eliminates the large matrix storage requirements normally associated with frequency domain solutions, and time marching attains the steady-state quickly enough to make the method favorable when compared to frequency domain methods. For validation, this transient-frequency domain method is applied to sound propagation in a 2D hard wall duct with plug flow.
NASA Technical Reports Server (NTRS)
Baumeister, Kenneth J.; Kreider, Kevin L.
1996-01-01
An explicit finite difference iteration scheme is developed to study harmonic sound propagation in aircraft engine nacelles. To reduce storage requirements for large 3D problems, the time dependent potential form of the acoustic wave equation is used. To insure that the finite difference scheme is both explicit and stable, time is introduced into the Fourier transformed (steady-state) acoustic potential field as a parameter. Under a suitable transformation, the time dependent governing equation in frequency space is simplified to yield a parabolic partial differential equation, which is then marched through time to attain the steady-state solution. The input to the system is the amplitude of an incident harmonic sound source entering a quiescent duct at the input boundary, with standard impedance boundary conditions on the duct walls and duct exit. The introduction of the time parameter eliminates the large matrix storage requirements normally associated with frequency domain solutions, and time marching attains the steady-state quickly enough to make the method favorable when compared to frequency domain methods. For validation, this transient-frequency domain method is applied to sound propagation in a 2D hard wall duct with plug flow.
ATLAS: A real-space finite-difference implementation of orbital-free density functional theory
NASA Astrophysics Data System (ADS)
Mi, Wenhui; Shao, Xuecheng; Su, Chuanxun; Zhou, Yuanyuan; Zhang, Shoutao; Li, Quan; Wang, Hui; Zhang, Lijun; Miao, Maosheng; Wang, Yanchao; Ma, Yanming
2016-03-01
Orbital-free density functional theory (OF-DFT) is a promising method for large-scale quantum mechanics simulation as it provides a good balance of accuracy and computational cost. Its applicability to large-scale simulations has been aided by progress in constructing kinetic energy functionals and local pseudopotentials. However, the widespread adoption of OF-DFT requires further improvement in its efficiency and robustly implemented software. Here we develop a real-space finite-difference (FD) method for the numerical solution of OF-DFT in periodic systems. Instead of the traditional self-consistent method, a powerful scheme for energy minimization is introduced to solve the Euler-Lagrange equation. Our approach engages both the real-space finite-difference method and a direct energy-minimization scheme for the OF-DFT calculations. The method is coded into the ATLAS software package and benchmarked using periodic systems of solid Mg, Al, and Al3Mg. The test results show that our implementation can achieve high accuracy, efficiency, and numerical stability for large-scale simulations.
NASA Astrophysics Data System (ADS)
Yamamoto, Kaho; Iwai, Yosuke; Uchida, Yoshiaki; Nishiyama, Norikazu
2016-08-01
We numerically analyzed the light propagation in cholesteric liquid crystalline (CLC) droplet array by the finite-difference time-domain (FDTD) method. The FDTD method successfully reproduced the experimental light path observed in the complicated photonic structure of the CLC droplet array more accurately than the analysis of CLC droplets by geometric optics with Bragg condition, and this method help us understand the polarization of the propagating light waves. The FDTD method holds great promise for the design of various photonic devices composed of curved photonic materials like CLC droplets and microcapsules.
NASA Technical Reports Server (NTRS)
Shu, Chi-Wang
1998-01-01
This project is about the development of high order, non-oscillatory type schemes for computational fluid dynamics. Algorithm analysis, implementation, and applications are performed. Collaborations with NASA scientists have been carried out to ensure that the research is relevant to NASA objectives. The combination of ENO finite difference method with spectral method in two space dimension is considered, jointly with Cai [3]. The resulting scheme behaves nicely for the two dimensional test problems with or without shocks. Jointly with Cai and Gottlieb, we have also considered one-sided filters for spectral approximations to discontinuous functions [2]. We proved theoretically the existence of filters to recover spectral accuracy up to the discontinuity. We also constructed such filters for practical calculations.
NASA Astrophysics Data System (ADS)
Choi, S.-J.; Giraldo, F. X.; Kim, J.; Shin, S.
2014-11-01
The non-hydrostatic (NH) compressible Euler equations for dry atmosphere were solved in a simplified two-dimensional (2-D) slice framework employing a spectral element method (SEM) for the horizontal discretization and a finite difference method (FDM) for the vertical discretization. By using horizontal SEM, which decomposes the physical domain into smaller pieces with a small communication stencil, a high level of scalability can be achieved. By using vertical FDM, an easy method for coupling the dynamics and existing physics packages can be provided. The SEM uses high-order nodal basis functions associated with Lagrange polynomials based on Gauss-Lobatto-Legendre (GLL) quadrature points. The FDM employs a third-order upwind-biased scheme for the vertical flux terms and a centered finite difference scheme for the vertical derivative and integral terms. For temporal integration, a time-split, third-order Runge-Kutta (RK3) integration technique was applied. The Euler equations that were used here are in flux form based on the hydrostatic pressure vertical coordinate. The equations are the same as those used in the Weather Research and Forecasting (WRF) model, but a hybrid sigma-pressure vertical coordinate was implemented in this model. We validated the model by conducting the widely used standard tests: linear hydrostatic mountain wave, tracer advection, and gravity wave over the Schär-type mountain, as well as density current, inertia-gravity wave, and rising thermal bubble. The results from these tests demonstrated that the model using the horizontal SEM and the vertical FDM is accurate and robust provided sufficient diffusion is applied. The results with various horizontal resolutions also showed convergence of second-order accuracy due to the accuracy of the time integration scheme and that of the vertical direction, although high-order basis functions were used in the horizontal. By using the 2-D slice model, we effectively showed that the combined spatial
Finite difference time domain modeling of spiral antennas
NASA Technical Reports Server (NTRS)
Penney, Christopher W.; Beggs, John H.; Luebbers, Raymond J.
1992-01-01
The objectives outlined in the original proposal for this project were to create a well-documented computer analysis model based on the finite-difference, time-domain (FDTD) method that would be capable of computing antenna impedance, far-zone radiation patterns, and radar cross-section (RCS). The ability to model a variety of penetrable materials in addition to conductors is also desired. The spiral antennas under study by this project meet these requirements since they are constructed of slots cut into conducting surfaces which are backed by dielectric materials.
Macroscopic traffic modeling with the finite difference method
Mughabghab, S.; Azarm, A.; Stock, D.
1996-03-15
A traffic congestion forecasting model (ATOP), developed in the present investigation, is described briefly. Several macroscopic models, based on the solution of the partial differential equation of conservation of vehicles by the finite difference method, were tested using actual traffic data. The functional form, as well as the parameters, of the equation of state which describes the relation between traffic speed and traffic density, were determined for a section of the Long Island Expressway. The Lax method and the forward difference technique were applied. The results of extensive tests showed that the Lax method, in addition to giving very good agreement with the traffic data, produces stable solutions.
A finite difference approach to microstrip antenna design
Barth, M.J.; Bevensee, R.M.; Pennock, S.T.
1986-12-01
Microstrip antennas have received increased attention in recent years, due to their size and cost advantages. Analysis of the microstrip structure has proved difficult due to the presence of the dielectric substrate, particularly for complex geometries. One possible approach to a solution is the use of a finite difference computer code to model a proposed microstrip antenna design. The models are easily constructed and altered, and code versions are available which allow input impedance or far-field patterns to be calculated. Results for some simple antenna geometries will be presented.
FDIPS: Finite Difference Iterative Potential-field Solver
NASA Astrophysics Data System (ADS)
Toth, Gabor; van der Holst, Bartholomeus; Huang, Zhenguang
2016-06-01
FDIPS is a finite difference iterative potential-field solver that can generate the 3D potential magnetic field solution based on a magnetogram. It is offered as an alternative to the spherical harmonics approach, as when the number of spherical harmonics is increased, using the raw magnetogram data given on a grid that is uniform in the sine of the latitude coordinate can result in inaccurate and unreliable results, especially in the polar regions close to the Sun. FDIPS is written in Fortran 90 and uses the MPI library for parallel execution.
Finite difference program for calculating hydride bed wall temperature profiles
Klein, J.E.
1992-10-29
A QuickBASIC finite difference program was written for calculating one dimensional temperature profiles in up to two media with flat, cylindrical, or spherical geometries. The development of the program was motivated by the need to calculate maximum temperature differences across the walls of the Tritium metal hydrides beds for thermal fatigue analysis. The purpose of this report is to document the equations and the computer program used to calculate transient wall temperatures in stainless steel hydride vessels. The development of the computer code was motivated by the need to calculate maximum temperature differences across the walls of the hydrides beds in the Tritium Facility for thermal fatigue analysis.
NASA Astrophysics Data System (ADS)
Cassidy, Nigel J.; Millington, Tim M.
2009-04-01
Numerical modelling has recently established itself as an important, near-surface GPR interpretation tool with the finite-difference, time-domain (FDTD) method becoming one of the most popular techniques. Robust, flexible and accurate, the FDTD technique is capable of simulating GPR wave propagation in complex, three-dimensional, heterogeneous, lossy, subsurface environments to a high degree of realism. Unfortunately, many of the current FDTD methods still consider the subsurface materials as being 'non magnetic' and, as such, do not include the propagation and loss effects associated with magnetic materials (e.g., basic igneous rocks, iron-rich sands, corroded steel reinforced concrete, smelting wastes, etc). For magnetically lossy materials, the inclusion of a complex magnetic permeability into the FDTD scheme can result in smeared or 'fuzzy' interface problems, increased computational demand and equation-level coding changes. Therefore, it is prudent to describe the magnetically derived loss and propagation characteristics in a more generic manner where the 'electric' (e.g., permittivity and conductivity) properties of the material incorporate the magnetic loss effects explicitly. In this paper, we present a "generalised complex effective permittivity" approach to the FDTD material descriptors that allows for the true loss and propagation characteristics of the magnetic materials to modelled fully, regardless of their individual magnetic or electric field relaxation mechanisms. In doing so, we are able to incorporate the lossy, dispersive effects directly into existing FDTD schemes without modification, additional error or increased computational demand. To demonstrate its application, a three-dimensional, 450 MHz, near-surface model of GPR data simulation over a rusty pipe has been included that illustrates how the FDTD modelling can be used to evaluate subtle changes in the spectral nature of the reflected signals. The modelling results show that, for
NASA Astrophysics Data System (ADS)
Chaigne, Antoine; Askenfelt, Anders
1993-04-01
An improvement to the numerical approach and the underlying physical model made by Hiller and Ruiz in their attempt to generate musical sounds by solving the equations of vibrating strings by means of Finite Difference Methods (FDM) in order to simulate the notion of the piano string with a high degree of realism, is shown. Starting from the fundamental equations of a damped, stiff string interacting with a nonlinear hammer, a numerical finite difference scheme is derived, from which the time histories of string displacement and velocity for each point of the string are computed in the time domain. The interacting force between hammer and string, and the force acting on the bridge, are given by the same scheme. The performance of the model is illustrated by examples of simulated string waveforms. Aspects of numerical stability and dispersion are discussed with reference to the proper choice of sampling parameters.
Finite Difference Elastic Wave Field Simulation On GPU
NASA Astrophysics Data System (ADS)
Hu, Y.; Zhang, W.
2011-12-01
Numerical modeling of seismic wave propagation is considered as a basic and important aspect in investigation of the Earth's structure, and earthquake phenomenon. Among various numerical methods, the finite-difference method is considered one of the most efficient tools for the wave field simulation. However, with the increment of computing scale, the power of computing has becoming a bottleneck. With the development of hardware, in recent years, GPU shows powerful computational ability and bright application prospects in scientific computing. Many works using GPU demonstrate that GPU is powerful . Recently, GPU has not be used widely in the simulation of wave field. In this work, we present forward finite difference simulation of acoustic and elastic seismic wave propagation in heterogeneous media on NVIDIA graphics cards with the CUDA programming language. We also implement perfectly matched layers on the graphics cards to efficiently absorb outgoing waves on the fictitious edges of the grid Simulations compared with the results on CPU platform shows reliable accuracy and remarkable efficiency. This work proves that GPU can be an effective platform for wave field simulation, and it can also be used as a practical tool for real-time strong ground motion simulation.
Total Variation Diminishing (TVD) schemes of uniform accuracy
NASA Technical Reports Server (NTRS)
Hartwich, PETER-M.; Hsu, Chung-Hao; Liu, C. H.
1988-01-01
Explicit second-order accurate finite-difference schemes for the approximation of hyperbolic conservation laws are presented. These schemes are nonlinear even for the constant coefficient case. They are based on first-order upwind schemes. Their accuracy is enhanced by locally replacing the first-order one-sided differences with either second-order one-sided differences or central differences or a blend thereof. The appropriate local difference stencils are selected such that they give TVD schemes of uniform second-order accuracy in the scalar, or linear systems, case. Like conventional TVD schemes, the new schemes avoid a Gibbs phenomenon at discontinuities of the solution, but they do not switch back to first-order accuracy, in the sense of truncation error, at extrema of the solution. The performance of the new schemes is demonstrated in several numerical tests.
NASA Technical Reports Server (NTRS)
Garrett, L. B.
1971-01-01
An implicit finite difference scheme is developed for the fully coupled solution of the viscous radiating stagnation line equations, including strong blowing. Solutions are presented for both air injection and carbon phenolic ablation products injection into air at conditions near the peak radiative heating point in an earth entry trajectory from interplanetary return missions. A detailed radiative transport code that accounts for the important radiative exchange processes for gaseous mixtures in local thermodynamic and chemical equilibrium is utilized.
NASA Astrophysics Data System (ADS)
Rybynok, V. O.; Kyriacou, P. A.
2007-10-01
Diabetes is one of the biggest health challenges of the 21st century. The obesity epidemic, sedentary lifestyles and an ageing population mean prevalence of the condition is currently doubling every generation. Diabetes is associated with serious chronic ill health, disability and premature mortality. Long-term complications including heart disease, stroke, blindness, kidney disease and amputations, make the greatest contribution to the costs of diabetes care. Many of these long-term effects could be avoided with earlier, more effective monitoring and treatment. Currently, blood glucose can only be monitored through the use of invasive techniques. To date there is no widely accepted and readily available non-invasive monitoring technique to measure blood glucose despite the many attempts. This paper challenges one of the most difficult non-invasive monitoring techniques, that of blood glucose, and proposes a new novel approach that will enable the accurate, and calibration free estimation of glucose concentration in blood. This approach is based on spectroscopic techniques and a new adaptive modelling scheme. The theoretical implementation and the effectiveness of the adaptive modelling scheme for this application has been described and a detailed mathematical evaluation has been employed to prove that such a scheme has the capability of extracting accurately the concentration of glucose from a complex biological media.
NASA Astrophysics Data System (ADS)
Toyokuni, G.; Takenaka, H.
2007-12-01
We propose a method to obtain effective grid parameters for the finite-difference (FD) method with standard Earth models using analytical ways. In spite of the broad use of the heterogeneous FD formulation for seismic waveform modeling, accurate treatment of material discontinuities inside the grid cells has been a serious problem for many years. One possible way to solve this problem is to introduce effective grid elastic moduli and densities (effective parameters) calculated by the volume harmonic averaging of elastic moduli and volume arithmetic averaging of density in grid cells. This scheme enables us to put a material discontinuity into an arbitrary position in the spatial grids. Most of the methods used for synthetic seismogram calculation today receives the blessing of the standard Earth models, such as the PREM, IASP91, SP6, and AK135, represented as functions of normalized radius. For the FD computation of seismic waveform with such models, we first need accurate treatment of material discontinuities in radius. This study provides a numerical scheme for analytical calculations of the effective parameters for an arbitrary spatial grids in radial direction as to these major four standard Earth models making the best use of their functional features. This scheme can analytically obtain the integral volume averages through partial fraction decompositions (PFDs) and integral formulae. We have developed a FORTRAN subroutine to perform the computations, which is opened to utilization in a large variety of FD schemes ranging from 1-D to 3-D, with conventional- and staggered-grids. In the presentation, we show some numerical examples displaying the accuracy of the FD synthetics simulated with the analytical effective parameters.
A mimetic finite difference method for the Stokes problem with elected edge bubbles
Lipnikov, K; Berirao, L
2009-01-01
A new mimetic finite difference method for the Stokes problem is proposed and analyzed. The unstable P{sub 1}-P{sub 0} discretization is stabilized by adding a small number of bubble functions to selected mesh edges. A simple strategy for selecting such edges is proposed and verified with numerical experiments. The discretizations schemes for Stokes and Navier-Stokes equations must satisfy the celebrated inf-sup (or the LBB) stability condition. The stability condition implies a balance between discrete spaces for velocity and pressure. In finite elements, this balance is frequently achieved by adding bubble functions to the velocity space. The goal of this article is to show that the stabilizing edge bubble functions can be added only to a small set of mesh edges. This results in a smaller algebraic system and potentially in a faster calculations. We employ the mimetic finite difference (MFD) discretization technique that works for general polyhedral meshes and can accomodate non-uniform distribution of stabilizing bubbles.
Phase-field-based lattice Boltzmann finite-difference model for simulating thermocapillary flows
NASA Astrophysics Data System (ADS)
Liu, Haihu; Valocchi, Albert J.; Zhang, Yonghao; Kang, Qinjun
2013-01-01
A phase-field-based hybrid model that combines the lattice Boltzmann method with the finite difference method is proposed for simulating immiscible thermocapillary flows with variable fluid-property ratios. Using a phase field methodology, an interfacial force formula is analytically derived to model the interfacial tension force and the Marangoni stress. We present an improved lattice Boltzmann equation (LBE) method to capture the interface between different phases and solve the pressure and velocity fields, which can recover the correct Cahn-Hilliard equation (CHE) and Navier-Stokes equations. The LBE method allows not only use of variable mobility in the CHE, but also simulation of multiphase flows with high density ratio because a stable discretization scheme is used for calculating the derivative terms in forcing terms. An additional convection-diffusion equation is solved by the finite difference method for spatial discretization and the Runge-Kutta method for time marching to obtain the temperature field, which is coupled to the interfacial tension through an equation of state. The model is first validated against analytical solutions for the thermocapillary driven convection in two superimposed fluids at negligibly small Reynolds and Marangoni numbers. It is then used to simulate thermocapillary migration of a three-dimensional deformable droplet and bubble at various Marangoni numbers and density ratios, and satisfactory agreement is obtained between numerical results and theoretical predictions.
A Finite Difference-Based Modeling Approach for Prediction of Steel Hardenability
NASA Astrophysics Data System (ADS)
Sushanthi, Neethi; Maity, Joydeep
2014-06-01
In this research work an independent finite difference-based modeling approach was adopted for determination of the hardenability of steels. In this model, at first, cooling curves were generated by solving transient heat transfer equation through discretization with pure explicit finite difference scheme coupled with MATLAB-based programing in view of variable thermo-physical properties of 1080 steel. The cooling curves were solved against 50% transformation nose of TTT diagram in order to predict hardening behavior of 1080 steel in terms of hardenability parameters (Grossmann critical diameter, D C; and ideal critical diameter, D I) and the variation of the unhardened core diameter ( D u) to diameter of steel bar ( D) ratio with diameter of steel bar ( D). The experiments were also performed to determine actual D C value of 1080 steel for still water quenching. The D C value obtained by the developed model was found to match the experimental D C value with only 6% deviation. Therefore, the model developed in the present work can be used for direct determination of D I, D C, and D u without resorting to any rigorous experimentation.
Sheaffer, Jonathan; van Walstijn, Maarten; Fazenda, Bruno
2014-01-01
In finite difference time domain simulation of room acoustics, source functions are subject to various constraints. These depend on the way sources are injected into the grid and on the chosen parameters of the numerical scheme being used. This paper addresses the issue of selecting and designing sources for finite difference simulation, by first reviewing associated aims and constraints, and evaluating existing source models against these criteria. The process of exciting a model is generalized by introducing a system of three cascaded filters, respectively, characterizing the driving pulse, the source mechanics, and the injection of the resulting source function into the grid. It is shown that hard, soft, and transparent sources can be seen as special cases within this unified approach. Starting from the mechanics of a small pulsating sphere, a parametric source model is formulated by specifying suitable filters. This physically constrained source model is numerically consistent, does not scatter incoming waves, and is free from zero- and low-frequency artifacts. Simulation results are employed for comparison with existing source formulations in terms of meeting the spectral and temporal requirements on the outward propagating wave.
Phase-field-based lattice Boltzmann finite-difference model for simulating thermocapillary flows.
Liu, Haihu; Valocchi, Albert J; Zhang, Yonghao; Kang, Qinjun
2013-01-01
A phase-field-based hybrid model that combines the lattice Boltzmann method with the finite difference method is proposed for simulating immiscible thermocapillary flows with variable fluid-property ratios. Using a phase field methodology, an interfacial force formula is analytically derived to model the interfacial tension force and the Marangoni stress. We present an improved lattice Boltzmann equation (LBE) method to capture the interface between different phases and solve the pressure and velocity fields, which can recover the correct Cahn-Hilliard equation (CHE) and Navier-Stokes equations. The LBE method allows not only use of variable mobility in the CHE, but also simulation of multiphase flows with high density ratio because a stable discretization scheme is used for calculating the derivative terms in forcing terms. An additional convection-diffusion equation is solved by the finite difference method for spatial discretization and the Runge-Kutta method for time marching to obtain the temperature field, which is coupled to the interfacial tension through an equation of state. The model is first validated against analytical solutions for the thermocapillary driven convection in two superimposed fluids at negligibly small Reynolds and Marangoni numbers. It is then used to simulate thermocapillary migration of a three-dimensional deformable droplet and bubble at various Marangoni numbers and density ratios, and satisfactory agreement is obtained between numerical results and theoretical predictions. PMID:23410429
Wang, Wei; Shu, Chi-Wang; Yee, H.C.; Sjögreen, Björn
2012-01-01
A new high order finite-difference method utilizing the idea of Harten ENO subcell resolution method is proposed for chemical reactive flows and combustion. In reaction problems, when the reaction time scale is very small, e.g., orders of magnitude smaller than the fluid dynamics time scales, the governing equations will become very stiff. Wrong propagation speed of discontinuity may occur due to the underresolved numerical solution in both space and time. The present proposed method is a modified fractional step method which solves the convection step and reaction step separately. In the convection step, any high order shock-capturing method can be used. In the reaction step, an ODE solver is applied but with the computed flow variables in the shock region modified by the Harten subcell resolution idea. For numerical experiments, a fifth-order finite-difference WENO scheme and its anti-diffusion WENO variant are considered. A wide range of 1D and 2D scalar and Euler system test cases are investigated. Studies indicate that for the considered test cases, the new method maintains high order accuracy in space for smooth flows, and for stiff source terms with discontinuities, it can capture the correct propagation speed of discontinuities in very coarse meshes with reasonable CFL numbers.
Two-dimensional time-domain finite-difference modeling for viscoelastic seismic wave propagation
NASA Astrophysics Data System (ADS)
Fan, Na; Zhao, Lian-Feng; Xie, Xiao-Bi; Ge, Zengxi; Yao, Zhen-Xing
2016-09-01
Real Earth media are not perfectly elastic. Instead, they attenuate propagating mechanical waves. This anelastic phenomenon in wave propagation can be modeled by a viscoelastic mechanical model consisting of several standard linear solids. Using this viscoelastic model, we approximate a constant Q over a frequency band of interest. We use a four-element viscoelastic model with a trade-off between accuracy and computational costs to incorporate Q into 2-D time-domain first-order velocity-stress wave equations. To improve the computational efficiency, we limit the Q in the model to a list of discrete values between 2 and 1000. The related stress and strain relaxation times that characterize the viscoelastic model are pre-calculated and stored in a database for use by the finite-difference calculation. A viscoelastic finite-difference scheme that is second order in time and fourth order in space is developed based on the MacCormack algorithm. The new method is validated by comparing the numerical result with analytical solutions that are calculated using the generalized reflection/transmission coefficient method. The synthetic seismograms exhibit greater than 95 per cent consistency in a two-layer viscoelastic model. The dispersion generated from the simulation is consistent with the Kolsky-Futterman dispersion relationship.
2D time-domain finite-difference modeling for viscoelastic seismic wave propagation
NASA Astrophysics Data System (ADS)
Fan, Na; Zhao, Lian-Feng; Xie, Xiao-Bi; Ge, Zengxi; Yao, Zhen-Xing
2016-07-01
Real Earth media are not perfectly elastic. Instead, they attenuate propagating mechanical waves. This anelastic phenomenon in wave propagation can be modeled by a viscoelastic mechanical model consisting of several standard linear solids. Using this viscoelastic model, we approximate a constant Q over a frequency band of interest. We use a four-element viscoelastic model with a tradeoff between accuracy and computational costs to incorporate Q into 2D time-domain first-order velocity-stress wave equations. To improve the computational efficiency, we limit the Q in the model to a list of discrete values between 2 and 1000. The related stress and strain relaxation times that characterize the viscoelastic model are pre-calculated and stored in a database for use by the finite-difference calculation. A viscoelastic finite-difference scheme that is second-order in time and fourth-order in space is developed based on the MacCormack algorithm. The new method is validated by comparing the numerical result with analytical solutions that are calculated using the generalized reflection/transmission coefficient method. The synthetic seismograms exhibit greater than 95 per cent consistency in a two-layer viscoelastic model. The dispersion generated from the simulation is consistent with the Kolsky-Futterman dispersion relationship.
Finite difference grid generation by multivariate blending function interpolation
NASA Technical Reports Server (NTRS)
Anderson, P. G.; Spradley, L. W.
1980-01-01
The General Interpolants Method (GIM) code which solves the multidimensional Navier-Stokes equations for arbitrary geometric domains is described. The geometry module in the GIM code generates two and three dimensional grids over specified flow regimes, establishes boundary condition information and computes finite difference analogs for use in the GIM code numerical solution module. The technique can be classified as an algebraic equation approach. The geometry package uses multivariate blending function interpolation of vector-values functions which define the shapes of the edges and surfaces bounding the flow domain. By employing blending functions which conform to the cardinality conditions the flow domain may be mapped onto a unit square (2-D) or unit cube (3-D), thus producing an intrinsic coordinate system for the region of interest. The intrinsic coordinate system facilitates grid spacing control to allow for optimum distribution of nodes in the flow domain.
Effects of sources on time-domain finite difference models.
Botts, Jonathan; Savioja, Lauri
2014-07-01
Recent work on excitation mechanisms in acoustic finite difference models focuses primarily on physical interpretations of observed phenomena. This paper offers an alternative view by examining the properties of models from the perspectives of linear algebra and signal processing. Interpretation of a simulation as matrix exponentiation clarifies the separate roles of sources as boundaries and signals. Boundary conditions modify the matrix and thus its modal structure, and initial conditions or source signals shape the solution, but not the modal structure. Low-frequency artifacts are shown to follow from eigenvalues and eigenvectors of the matrix, and previously reported artifacts are predicted from eigenvalue estimates. The role of source signals is also briefly discussed. PMID:24993210
Finite-difference modeling of commercial aircraft using TSAR
Pennock, S.T.; Poggio, A.J.
1994-11-15
Future aircraft may have systems controlled by fiber optic cables, to reduce susceptibility to electromagnetic interference. However, the digital systems associated with the fiber optic network could still experience upset due to powerful radio stations, radars, and other electromagnetic sources, with potentially serious consequences. We are modeling the electromagnetic behavior of commercial transport aircraft in support of the NASA Fly-by-Light/Power-by-Wire program, using the TSAR finite-difference time-domain code initially developed for the military. By comparing results obtained from TSAR with data taken on a Boeing 757 at the Air Force Phillips Lab., we hope to show that FDTD codes can serve as an important tool in the design and certification of U.S. commercial aircraft, helping American companies to produce safe, reliable air transportation.
Visualization of elastic wavefields computed with a finite difference code
Larsen, S.; Harris, D.
1994-11-15
The authors have developed a finite difference elastic propagation model to simulate seismic wave propagation through geophysically complex regions. To facilitate debugging and to assist seismologists in interpreting the seismograms generated by the code, they have developed an X Windows interface that permits viewing of successive temporal snapshots of the (2D) wavefield as they are calculated. The authors present a brief video displaying the generation of seismic waves by an explosive source on a continent, which propagate to the edge of the continent then convert to two types of acoustic waves. This sample calculation was part of an effort to study the potential of offshore hydroacoustic systems to monitor seismic events occurring onshore.
Modeling pulse driven antenna systems with finite differences
Barth, M.; Pennock, S.; Ziolkowski, R.; McLeod, R.
1990-03-01
We have developed a capability of modeling the performance of general, pulse driven, antenna systems. Our approach is to use TSAR, a three dimensional finite difference time domain (FDTD) code, to model the antenna structure and the surrounding near field environment. We then use a far field projection algorithm to obtain its far field response. Specifically, this algorithm utilizes the tangential electric and magnetic fields at a specified surface of the TSAR FDTD computational volume and calculates the resulting fields far from the equivalent magnetic and electric sources. This approach will be illustrated by considering the TEB antenna system. The system is modeled with the code and the results are compared with anechoic chamber data. 10 figs., 2 tabs.
Finite difference time domain analysis of microwave ferrite devices and mobile antenna systems
NASA Astrophysics Data System (ADS)
Yildirim, Bahadir Suleyman
This dissertation presents analysis and design of shielded mobile antenna systems and microwave ferrite devices using a finite-difference time-domain method. Novel shielded antenna structures suitable for cellular communications have been analyzed and designed with emphasize on reducing excessive radiated energy absorbed in user's head and hand, while keeping the antenna performance at its peak in the presence of user. These novel antennas include a magnetically shielded antenna, a dual-resonance shielded antenna and, a shorted and truncated microstrip antenna. The effect of magnetic coating on the performance of a shielded monopole antenna is studied extensively. A parametric study is performed to analyze the dual-resonance phenomenon observed in the dual-resonance shielded antenna, optimize the antenna design within the cellular communications band, and improve the antenna performance. Input impedance, near and far fields of the dual-resonance shielded antenna are calculated using the finite-difference time-domain method. Experimental validation is also presented. In addition, performance of a shorted and truncated microstrip antenna has been investigated over a wide range of substrate parameters and dimensions. Objectives of the research work also include development of a finite-difference time-domain technique to accurately model magnetically anisotropic media, including the effect of non-uniform magnetization within the finite-size ferrite material due to demagnetizing fields. A slow wave thin film isolator and a stripline disc junction circulator are analyzed. An extensive parametric study calculates wide-band frequency-dependent parameters of these devices for various device dimensions and material parameters. Finally, a ferrite-filled stripline configuration is analyzed to study the non- linear behaviour of ferrite by introducing a modified damping factor.
Experiments with explicit filtering for LES using a finite-difference method
NASA Technical Reports Server (NTRS)
Lund, T. S.; Kaltenbach, H. J.
1995-01-01
The equations for large-eddy simulation (LES) are derived formally by applying a spatial filter to the Navier-Stokes equations. The filter width as well as the details of the filter shape are free parameters in LES, and these can be used both to control the effective resolution of the simulation and to establish the relative importance of different portions of the resolved spectrum. An analogous, but less well justified, approach to filtering is more or less universally used in conjunction with LES using finite-difference methods. In this approach, the finite support provided by the computational mesh as well as the wavenumber-dependent truncation errors associated with the finite-difference operators are assumed to define the filter operation. This approach has the advantage that it is also 'automatic' in the sense that no explicit filtering: operations need to be performed. While it is certainly convenient to avoid the explicit filtering operation, there are some practical considerations associated with finite-difference methods that favor the use of an explicit filter. Foremost among these considerations is the issue of truncation error. All finite-difference approximations have an associated truncation error that increases with increasing wavenumber. These errors can be quite severe for the smallest resolved scales, and these errors will interfere with the dynamics of the small eddies if no corrective action is taken. Years of experience at CTR with a second-order finite-difference scheme for high Reynolds number LES has repeatedly indicated that truncation errors must be minimized in order to obtain acceptable simulation results. While the potential advantages of explicit filtering are rather clear, there is a significant cost associated with its implementation. In particular, explicit filtering reduces the effective resolution of the simulation compared with that afforded by the mesh. The resolution requirements for LES are usually set by the need to capture
Samak, M. Mosleh E. Abu; Bakar, A. Ashrif A.; Kashif, Muhammad; Zan, Mohd Saiful Dzulkifly
2016-01-01
This paper discusses numerical analysis methods for different geometrical features that have limited interval values for typically used sensor wavelengths. Compared with existing Finite Difference Time Domain (FDTD) methods, the alternating direction implicit (ADI)-FDTD method reduces the number of sub-steps by a factor of two to three, which represents a 33% time savings in each single run. The local one-dimensional (LOD)-FDTD method has similar numerical equation properties, which should be calculated as in the previous method. Generally, a small number of arithmetic processes, which result in a shorter simulation time, are desired. The alternating direction implicit technique can be considered a significant step forward for improving the efficiency of unconditionally stable FDTD schemes. This comparative study shows that the local one-dimensional method had minimum relative error ranges of less than 40% for analytical frequencies above 42.85 GHz, and the same accuracy was generated by both methods.
An Eigenvalue Analysis of finite-difference approximations for hyperbolic IBVPs
NASA Technical Reports Server (NTRS)
Warming, Robert F.; Beam, Richard M.
1989-01-01
The eigenvalue spectrum associated with a linear finite-difference approximation plays a crucial role in the stability analysis and in the actual computational performance of the discrete approximation. The eigenvalue spectrum associated with the Lax-Wendroff scheme applied to a model hyperbolic equation was investigated. For an initial-boundary-value problem (IBVP) on a finite domain, the eigenvalue or normal mode analysis is analytically intractable. A study of auxiliary problems (Dirichlet and quarter-plane) leads to asymptotic estimates of the eigenvalue spectrum and to an identification of individual modes as either benign or unstable. The asymptotic analysis establishes an intuitive as well as quantitative connection between the algebraic tests in the theory of Gustafsson, Kreiss, and Sundstrom and Lax-Richtmyer L(sub 2) stability on a finite domain.
An eigenvalue analysis of finite-difference approximations for hyperbolic IBVPs
NASA Technical Reports Server (NTRS)
Warming, Robert F.; Beam, Richard M.
1990-01-01
The eigenvalue spectrum associated with a linear finite-difference approximation plays a crucial role in the stability analysis and in the actual computational performance of the discrete approximation. The eigenvalue spectrum associated with the Lax-Wendroff scheme applied to a model hyperbolic equation was investigated. For an initial-boundary-value problem (IBVP) on a finite domain, the eigenvalue or normal mode analysis is analytically intractable. A study of auxiliary problems (Dirichlet and quarter-plane) leads to asymptotic estimates of the eigenvalue spectrum and to an identification of individual modes as either benign or unstable. The asymptotic analysis establishes an intuitive as well as quantitative connection between the algebraic tests in the theory of Gustafsson, Kreiss, and Sundstrom and Lax-Richtmyer L (sub 2) stability on a finite domain.
Robust and efficient upwind finite-difference traveltime calculations in three dimensions
Schneider, W.A. Jr.
1995-07-01
First-arrival traveltimes in complicated 3-D geologic media may be computed robustly and efficiently using an upwind finite-difference solution of the 3-D eikonal equation. An important application of this technique is computing traveltimes for imaging seismic data with 3-D prestack Kirchhoff depth migration. The method performs radial extrapolation of the three components of the slowness vector in spherical coordinates. Traveltimes are computed by numerically integrating the radial component of the slowness vector. The original finite-difference equations are recast into unitless forms that are more stable to numerical errors. A stability condition adaptively determines the radial steps that are used to extrapolate. Computations are done in a rotated spherical coordinate system that places the small arc-length regions of the spherical grid at the earth`s surface (z = 0 plane). This improves efficiency by placing large grid cells in the central regions of the grid where wavefields are complicated, thereby maximizing the radial steps. Adaptive gridding allows the angular grid spacings to vary with radius. The computation grid is also adaptively truncated so that it does not extend beyond the predefined Cartesian traveltime grid. This grid handling improves efficiency. The method cannot compute traveltimes corresponding to wavefronts that have ``turned`` so that they propagate in the negative radial direction. Such wavefronts usually represent headwaves and are not needed to image seismic data. An adaptive angular normalization prevent this turning, while allowing lower-angle wavefront components to accurately propagate.
Optimal implicit 2-D finite differences to model wave propagation in poroelastic media
NASA Astrophysics Data System (ADS)
Itzá, Reymundo; Iturrarán-Viveros, Ursula; Parra, Jorge O.
2016-08-01
Numerical modeling of seismic waves in heterogeneous porous reservoir rocks is an important tool for the interpretation of seismic surveys in reservoir engineering. We apply globally optimal implicit staggered-grid finite differences (FD) to model 2-D wave propagation in heterogeneous poroelastic media at a low-frequency range (<10 kHz). We validate the numerical solution by comparing it to an analytical-transient solution obtaining clear seismic wavefields including fast P and slow P and S waves (for a porous media saturated with fluid). The numerical dispersion and stability conditions are derived using von Neumann analysis, showing that over a wide range of porous materials the Courant condition governs the stability and this optimal implicit scheme improves the stability of explicit schemes. High-order explicit FD can be replaced by some lower order optimal implicit FD so computational cost will not be as expensive while maintaining the accuracy. Here, we compute weights for the optimal implicit FD scheme to attain an accuracy of γ = 10-8. The implicit spatial differentiation involves solving tridiagonal linear systems of equations through Thomas' algorithm.
Holden, H.; Hu, Yaozhong
1996-12-31
In modelling the pressure p(x,w) at x {element_of} D {contained_in} R{sup d} of an incompressible fluid in a heterogeneous, isotropic medium with a stochastic permeability k(x,w) {ge} 0, Holden, Lindstrom, Oksendal, Uboe and Zhang [HLOUZ95] studied the following stochastic differential equation div(k(x,w){diamond}{del}p(x,w))=-{line_integral}(x), x{element_of} D/p(x,w)=0, x{element_of}{partial_derivative}D, where {integral} is a given source and {diamond} denotes the Wick product. They proved existence of an explicit solution in (S){sup -1}. In this paper we define a finite difference scheme to approximate the above equation and prove that this scheme converges in (S){sup -1}. To solve the obtained finite difference equation, we prove that an adapted Jacobi iterative method is convergent. The paper is organized as follows: (1) Introduction. (2) The finite difference equation. (3) Uniform integrability of the stopping times. (4) Proof of Theorem 1.2. (5) Continuity of some functionals on D(0,{infinity}). (6) Jacobi`s iterative method. (A) The space (S){sup -1}. (B) Weak convergence. 13 refs.
Luo, Y.; Xia, J.; Xu, Y.; Zeng, C.; Liu, J.
2010-01-01
Love-wave propagation has been a topic of interest to crustal, earthquake, and engineering seismologists for many years because it is independent of Poisson's ratio and more sensitive to shear (S)-wave velocity changes and layer thickness changes than are Rayleigh waves. It is well known that Love-wave generation requires the existence of a low S-wave velocity layer in a multilayered earth model. In order to study numerically the propagation of Love waves in a layered earth model and dispersion characteristics for near-surface applications, we simulate high-frequency (>5 Hz) Love waves by the staggered-grid finite-difference (FD) method. The air-earth boundary (the shear stress above the free surface) is treated using the stress-imaging technique. We use a two-layer model to demonstrate the accuracy of the staggered-grid modeling scheme. We also simulate four-layer models including a low-velocity layer (LVL) or a high-velocity layer (HVL) to analyze dispersive energy characteristics for near-surface applications. Results demonstrate that: (1) the staggered-grid FD code and stress-imaging technique are suitable for treating the free-surface boundary conditions for Love-wave modeling, (2) Love-wave inversion should be treated with extra care when a LVL exists because of a lack of LVL information in dispersions aggravating uncertainties in the inversion procedure, and (3) energy of high modes in a low-frequency range is very weak, so that it is difficult to estimate the cutoff frequency accurately, and "mode-crossing" occurs between the second higher and third higher modes when a HVL exists. ?? 2010 Birkh??user / Springer Basel AG.
Gidas-Ni-Nirenberg results for finite difference equations
NASA Astrophysics Data System (ADS)
McKenna, P. J.; Reichel, W.
2007-10-01
Are positive solutions of finite difference boundary value problems [Delta]hu=f(u) in [Omega]h, u=0 on [not partial differential][Omega]h as symmetric as the domain? To answer this question we first show by examples that almost arbitrary non-symmetric solutions can be constructed. This is in striking difference to the continuous case, where by the famous Gidas-Ni-Nirenberg theorem [B. Gidas, Wei-Ming Ni, L. Nirenberg, Symmetry and related problems via the maximum principle, Comm. Math. Phys. 68 (1979) 209-243] positive solutions inherit the symmetry of the underlying domain. Then we prove approximate symmetry theorems for solutions on equidistantly meshed n-dimensional cubes: explicit estimates depending on the data are given which show that the solutions become more symmetric as the discretization gets finer. The quality of the estimates depends on whether or not f(0)<0. The one-dimensional case stands out in two ways: the proofs are elementary and the estimates for the defect of symmetry are O(h) compared to O(1/log(h)) in the higher-dimensional case.
Elastic finite-difference method for irregular grids
Oprsal, I.; Zahradnik, J.
1999-01-01
Finite-difference (FD) modeling of complicated structures requires simple algorithms. This paper presents a new elastic FD method for spatially irregular grids that is simple and, at the same time, saves considerable memory and computing time. Features like faults, low-velocity layers, cavities, and/or nonplanar surfaces are treated on a fine grid, while the remaining parts of the model are, with equal accuracy, represented on a coarse grid. No interpolation is needed between the fine and coarse parts due to the rectangular grid cells. Relatively abrupt transitions between the small and large grid steps produce no numerical artifacts in the present method. Planar or nonplanar free surfaces, including underground cavities, are treated in a way similar to internal grid points but with consideration of the zero-valued elastic parameters and density outside the free surface (vacuum formalism). A theoretical proof that vacuum formalism fulfills the free-surface conditions is given. Numerical validation is performed through comparison with independent methods, comparing FD with explicitly prescribed boundary conditions and finite elements. Memory and computing time needed in the studied models was only about 10 to 40% of that employing regular square grids of equal accuracy. A practical example of a synthetic seismic section, showing clear signatures of a coal seam and cavity, is presented. The method can be extended to three dimensions.
A finite difference model for free surface gravity drainage
Couri, F.R.; Ramey, H.J. Jr.
1993-09-01
The unconfined gravity flow of liquid with a free surface into a well is a classical well test problem which has not been well understood by either hydrologists or petroleum engineers. Paradigms have led many authors to treat an incompressible flow as compressible flow to justify the delayed yield behavior of a time-drawdown test. A finite-difference model has been developed to simulate the free surface gravity flow of an unconfined single phase, infinitely large reservoir into a well. The model was verified with experimental results in sandbox models in the literature and with classical methods applied to observation wells in the Groundwater literature. The simulator response was also compared with analytical Theis (1935) and Ramey et al. (1989) approaches for wellbore pressure at late producing times. The seepage face in the sandface and the delayed yield behavior were reproduced by the model considering a small liquid compressibility and incompressible porous medium. The potential buildup (recovery) simulated by the model evidenced a different- phenomenon from the drawdown, contrary to statements found in the Groundwater literature. Graphs of buildup potential vs time, buildup seepage face length vs time, and free surface head and sand bottom head radial profiles evidenced that the liquid refills the desaturating cone as a flat moving surface. The late time pseudo radial behavior was only approached after exaggerated long times.
Contraction pre-conditioner in finite-difference electromagnetic modelling
NASA Astrophysics Data System (ADS)
Yavich, Nikolay; Zhdanov, Michael S.
2016-09-01
This paper introduces a novel approach to constructing an effective pre-conditioner for finite-difference (FD) electromagnetic modelling in geophysical applications. This approach is based on introducing an FD contraction operator, similar to one developed for integral equation formulation of Maxwell's equation. The properties of the FD contraction operator were established using an FD analogue of the energy equality for the anomalous electromagnetic field. A new pre-conditioner uses a discrete Green's function of a 1-D layered background conductivity. We also developed the formulae for an estimation of the condition number of the system of FD equations pre-conditioned with the introduced FD contraction operator. Based on this estimation, we have established that the condition number is bounded by the maximum conductivity contrast between the background conductivity and actual conductivity. When there are both resistive and conductive anomalies relative to the background, the new pre-conditioner is advantageous over using the 1-D discrete Green's function directly. In our numerical experiments with both resistive and conductive anomalies, for a land geoelectrical model with 1:10 contrast, the method accelerates convergence of an iterative method (BiCGStab) by factors of 2-2.5, and in a marine example with 1:50 contrast, by a factor of 4.6, compared to direct use of the discrete 1-D Green's function as a pre-conditioner.
QED multi-dimensional vacuum polarization finite-difference solver
NASA Astrophysics Data System (ADS)
Carneiro, Pedro; Grismayer, Thomas; Silva, Luís; Fonseca, Ricardo
2015-11-01
The Extreme Light Infrastructure (ELI) is expected to deliver peak intensities of 1023 - 1024 W/cm2 allowing to probe nonlinear Quantum Electrodynamics (QED) phenomena in an unprecedented regime. Within the framework of QED, the second order process of photon-photon scattering leads to a set of extended Maxwell's equations [W. Heisenberg and H. Euler, Z. Physik 98, 714] effectively creating nonlinear polarization and magnetization terms that account for the nonlinear response of the vacuum. To model this in a self-consistent way, we present a multi dimensional generalized Maxwell equation finite difference solver with significantly enhanced dispersive properties, which was implemented in the OSIRIS particle-in-cell code [R.A. Fonseca et al. LNCS 2331, pp. 342-351, 2002]. We present a detailed numerical analysis of this electromagnetic solver. As an illustration of the properties of the solver, we explore several examples in extreme conditions. We confirm the theoretical prediction of vacuum birefringence of a pulse propagating in the presence of an intense static background field [arXiv:1301.4918 [quant-ph
Nonlinear triggered lightning models for use in finite difference calculations
NASA Technical Reports Server (NTRS)
Rudolph, Terence; Perala, Rodney A.; Ng, Poh H.
1989-01-01
Two nonlinear triggered lightning models have been developed for use in finite difference calculations. Both are based on three species of air chemistry physics and couple nonlinearly calculated air conductivity to Maxwell's equations. The first model is suitable for use in three-dimensional modeling and has been applied to the analysis of triggered lightning on the NASA F106B Thunderstorm Research Aircraft. The model calculates number densities of positive ions, negative ions, and electrons as a function of time and space through continuity equations, including convective derivative terms. The set of equations is closed by using experimentally determined mobilities, and the mobilities are also used to determine the air conductivity. Results from the model's application to the F106B are shown. The second model is two-dimensional and incorporates an enhanced air chemistry formulation. Momentum conservation equations replace the mobility assumption of the first model. Energy conservation equations for neutrals, heavy ions, and electrons are also used. Energy transfer into molecular vibrational modes is accounted for. The purpose for the enhanced model is to include the effects of temperature into the air breakdown, a necessary step if the model is to simulate more than the very earliest stages of breakdown. Therefore, the model also incorporates a temperature-dependent electron avalanche rate. Results from the model's application to breakdown around a conducting ellipsoid placed in an electric field are shown.
Thermo-mechanically coupled deformation with the finite difference method
NASA Astrophysics Data System (ADS)
Duretz, Thibault; Raess, Ludovic; Podladchikov, Yury; Schmalholz, Stefan
2016-04-01
Numerous geological observations are the result of thermo-mechanical processes. In particular, tectonic processes such as ductile shear localization can be induced by the intrinsic coupling that exists between deformation, energy and rheology. In order to study these processes, we have designed two-dimensional implicit and explicit finite difference models. These models take into account a temperature-dependent power-law rheology as well as diffusion, advection, and conversion of mechanical work into heat. For implicit models, different non-linear solving strategies were implemented (implicit/explicit thermo-mechanical coupling, Picard/Newton linearisations). We model thermo-mechanically activated shear localization in lower crustal conditions using these different numerical methods. We show that all methods capture the thermo-mechanical instability and exhibit similar temporal evolution. We perform quantitative comparisons with specifically designed tests (conservation of energy, analytical solution, scaling law). For implicit approaches, we discuss the treatment of thermo-mechanical coupling (implicit/explicit) and the impact of the imposed accuracy (tolerance) of the non-linear solvers. We compare the accuracy of the explicit method with the one of the implicit methods. Numerical algorithms based on explicit methods to study thermo-mechanical shear localisation are attractive because they are easy to program and very comprehensible.
Finite difference algorithm in real-time optical CD applications
NASA Astrophysics Data System (ADS)
Opsal, Jon L.; Chu, Hanyou; Leng, Jingmin
2004-05-01
In real-time optical CD applications of shallow trench isolation (STI), shallow trench removal (STR), deep trench isolation (DTI), and deep trench removal (DTR), a single recipe is required for each type of application to accommodate wide ranges of process windows by monitoring parameters such as bottom CD (BCD), middle CD (MCD), top CD (TCD) and side wall angle (SWA). The modeling of the grating profiles of silicon trenches with nitride caps requires a large number of slices (> 10) to generate smooth shapes for top rounding of the nitride, curvature of the silicon trench waist, and the silicon trench footing or undercut. The number of orders for Fourier expansion is also high (larger than 13 in the best case). With these requirements we found that the rigorous coupled wave analysis (RCWA) algorithm is generally too slow to calculate the CD profiles from the raw scatterometry spectra. In this paper we present a finite difference (FD) algorithm and its applications to real-time CD scatterometry. The mathematical analysis of the FD algorithm was published elsewhere. We demonstrate that the FD algorithm has an advantage over RCWA in terms of calculation speed (up to a factor of 10 improvement), better capture of profile shapes in comparison with cross sectional SEM (X-SEM) and more robust in terms of numerical stability. Details of comparisons between FD and RCWA will be shown for the applications of STR and DTR.
Contraction preconditioner in finite-difference electromagnetic modeling
NASA Astrophysics Data System (ADS)
Yavich, Nikolay; Zhdanov, Michael S.
2016-06-01
This paper introduces a novel approach to constructing an effective preconditioner for finite-difference (FD) electromagnetic modeling in geophysical applications. This approach is based on introducing an FD contraction operator, similar to one developed for integral equation formulation of Maxwell's equation. The properties of the FD contraction operator were established using an FD analog of the energy equality for the anomalous electromagnetic field. A new preconditioner uses a discrete Green's function of a 1D layered background conductivity. We also developed the formulas for an estimation of the condition number of the system of FD equations preconditioned with the introduced FD contraction operator. Based on this estimation, we have established that for high contrasts, the condition number is bounded by the maximum conductivity contrast between the background conductivity and actual conductivity. When there are both resistive and conductive anomalies relative to the background, the new preconditioner is advantageous over using the 1D discrete Green's function directly. In our numerical experiments with both resistive and conductive anomalies, for a land geoelectrical model with 1:10 contrast, the method accelerates convergence of an iterative method (BiCGStab) by factors of 2 to 2.5, and in a marine example with 1:50 contrast, by a factor of 4.6, compared to direct use of the discrete 1D Green's function as a preconditioner.
An improved finite-difference calculation of downhole dynamometer cards for sucker-rod pumps
Everitt, T.A. )
1992-02-01
Sucker-rod pumping is the most widely used means of artificial lift. About 85% to 90% of all producing wells in the U.S. are rod-pumped. Thus, a reliable method of analyzing these pumping system is a necessity. For many years, the surface dynamometer has been used to analyze sucker-rod systems. Interpretation of actual pump conditions from surface dynamometer cards is often difficult, if not impossible. Results obtained from surface cards are strictly qualitative and are dependent on the analyzer's expertise. The ideal analysis procedure would be to measure the actual pump conditions with a downhole dynamometer. However, this situation is not economically feasible. Therefore, an accurate method of calculating downhole pump cards form measured surface cards is needed. This paper presents a method for calculating these downhole cards that uses a finite-difference representation of the wave equation. First, a brief description of previous calculation techniques is given.
The arbitrary order mixed mimetic finite difference method for the diffusion equation
Gyrya, Vitaliy; Lipnikov, Konstantin; Manzini, Gianmarco
2016-05-01
Here, we propose an arbitrary-order accurate mimetic finite difference (MFD) method for the approximation of diffusion problems in mixed form on unstructured polygonal and polyhedral meshes. As usual in the mimetic numerical technology, the method satisfies local consistency and stability conditions, which determines the accuracy and the well-posedness of the resulting approximation. The method also requires the definition of a high-order discrete divergence operator that is the discrete analog of the divergence operator and is acting on the degrees of freedom. The new family of mimetic methods is proved theoretically to be convergent and optimal error estimates for flux andmore » scalar variable are derived from the convergence analysis. A numerical experiment confirms the high-order accuracy of the method in solving diffusion problems with variable diffusion tensor. It is worth mentioning that the approximation of the scalar variable presents a superconvergence effect.« less
Hierarchical Parallelism in Finite Difference Analysis of Heat Conduction
NASA Technical Reports Server (NTRS)
Padovan, Joseph; Krishna, Lala; Gute, Douglas
1997-01-01
Based on the concept of hierarchical parallelism, this research effort resulted in highly efficient parallel solution strategies for very large scale heat conduction problems. Overall, the method of hierarchical parallelism involves the partitioning of thermal models into several substructured levels wherein an optimal balance into various associated bandwidths is achieved. The details are described in this report. Overall, the report is organized into two parts. Part 1 describes the parallel modelling methodology and associated multilevel direct, iterative and mixed solution schemes. Part 2 establishes both the formal and computational properties of the scheme.
On a finite-difference method for solving transient viscous flow problems
NASA Technical Reports Server (NTRS)
Li, C. P.
1983-01-01
A method has been developed to solve the unsteady, compressible Navier-Stokes equation with the property of consistency and the ability of minimizing the equation stiffness. It relies on innovative extensions of the state-of-the-art finite-difference techniques and is composed of: (1) the upwind scheme for split-flux and the central scheme for conventional flux terms in the inviscid and viscous regions, respectively; (2) the characteristic treatment of both inviscid and viscous boundaries; (3) an ADI procedure compatible with interior and boundary points; and (4) a scalar matrix coefficient including viscous terms. The performance of this method is assessed with four sample problems; namely, a standing shock in the Laval duct, a shock reflected from the wall, the shock-induced boundary-layer separation, and a transient internal nozzle flow. The results from the present method, an existing hybrid block method, and a well-known two-step explicit method are compared and discussed. It is concluded that this method has an optimal trade-off between the solution accuracy and computational economy, and other desirable properties for analyzing transient viscous flow problems.
NASA Astrophysics Data System (ADS)
Lefrancois, Daniel; Rehn, Dirk R.; Dreuw, Andreas
2016-08-01
For the calculation of adiabatic singlet-triplet gaps (STG) in diradicaloid systems the spin-flip (SF) variant of the algebraic diagrammatic construction (ADC) scheme for the polarization propagator in third order perturbation theory (SF-ADC(3)) has been applied. Due to the methodology of the SF approach the singlet and triplet states are treated on an equal footing since they are part of the same determinant subspace. This leads to a systematically more accurate description of, e.g., diradicaloid systems than with the corresponding non-SF single-reference methods. Furthermore, using analytical excited state gradients at ADC(3) level, geometry optimizations of the singlet and triplet states were performed leading to a fully consistent description of the systems, leading to only small errors in the calculated STGs ranging between 0.6 and 2.4 kcal/mol with respect to experimental references.
Lefrancois, Daniel; Rehn, Dirk R; Dreuw, Andreas
2016-08-28
For the calculation of adiabatic singlet-triplet gaps (STG) in diradicaloid systems the spin-flip (SF) variant of the algebraic diagrammatic construction (ADC) scheme for the polarization propagator in third order perturbation theory (SF-ADC(3)) has been applied. Due to the methodology of the SF approach the singlet and triplet states are treated on an equal footing since they are part of the same determinant subspace. This leads to a systematically more accurate description of, e.g., diradicaloid systems than with the corresponding non-SF single-reference methods. Furthermore, using analytical excited state gradients at ADC(3) level, geometry optimizations of the singlet and triplet states were performed leading to a fully consistent description of the systems, leading to only small errors in the calculated STGs ranging between 0.6 and 2.4 kcal/mol with respect to experimental references. PMID:27586899
Dispersion-relation-preserving schemes for computational aeroacoustics
NASA Technical Reports Server (NTRS)
Tam, Christopher K. W.; Webb, Jay C.
1992-01-01
Finite difference schemes that have the same dispersion relations as the original partial differential equations are referred to as dispersion-relation-preserving (DRP) schemes. A method to construct time marching DRP schemes by optimizing the finite difference approximations of the space and time derivatives in the wave number and frequency space is presented. A sequence of numerical simulations is then performed.
The construction of high-accuracy schemes for acoustic equations
NASA Technical Reports Server (NTRS)
Tang, Lei; Baeder, James D.
1995-01-01
An accuracy analysis of various high order schemes is performed from an interpolation point of view. The analysis indicates that classical high order finite difference schemes, which use polynomial interpolation, hold high accuracy only at nodes and are therefore not suitable for time-dependent problems. Thus, some schemes improve their numerical accuracy within grid cells by the near-minimax approximation method, but their practical significance is degraded by maintaining the same stencil as classical schemes. One-step methods in space discretization, which use piecewise polynomial interpolation and involve data at only two points, can generate a uniform accuracy over the whole grid cell and avoid spurious roots. As a result, they are more accurate and efficient than multistep methods. In particular, the Cubic-Interpolated Psuedoparticle (CIP) scheme is recommended for computational acoustics.
NASA Astrophysics Data System (ADS)
Qiang, Rui; Chen, Ji; Yang, Fan
2010-10-01
A novel three-dimensional time domain method is developed to study interactions between finite-sized electromagnetic sources and infinite periodic structures. The method is based on a periodic finite difference time domain method combined with the spectral expansion of electromagnetic sources. Using this method, only a single periodic cell needs to be modeled in finite difference time domain simulations. The convergence, guidelines on using the algorithm, and the acceleration scheme for the algorithm are discussed. Several periodic structures are simulated by this proposed method. It is shown that this method can significantly reduce the required computer memory and computational time.
NASA Astrophysics Data System (ADS)
Moczo, P.; Kristek, J.; Galis, M.; Pazak, P.
2009-12-01
Numerical prediction of earthquake ground motion in sedimentary basins and valleys often has to account for P-wave to S-wave speed ratios (Vp/Vs) as large as 5 and even larger, mainly in sediments below groundwater level. The ratio can attain values larger than 10 in unconsolidated sediments (e.g. in Ciudad de México). In a process of developing 3D optimally-accurate finite-difference schemes we encountered a serious problem with accuracy in media with large Vp/Vs ratio. This led us to investigate the very fundamental reasons for the inaccuracy. In order to identify the very basic inherent aspects of the numerical schemes responsible for their behavior with varying Vp/Vs ratio, we restricted to the most basic 2nd-order 2D numerical schemes on a uniform grid in a homogeneous medium. Although basic in the specified sense, the schemes comprise the decisive features for accuracy of wide class of numerical schemes. We investigated 6 numerical schemes: finite-difference_displacement_conventional grid (FD_D_CG) finite-element_Lobatto integration (FE_L) finite-element_Gauss integration (FE_G) finite-difference_displacement-stress_partly-staggered grid (FD_DS_PSG) finite-difference_displacement-stress_staggered grid (FD_DS_SG) finite-difference_velocity-stress_staggered grid (FD_VS_SG) We defined and calculated local errors of the schemes in amplitude and polarization. Because different schemes use different time steps, they need different numbers of time levels to calculate solution for a desired time window. Therefore, we normalized errors for a unit time. The normalization allowed for a direct comparison of errors of different schemes. Extensive numerical calculations for wide ranges of values of the Vp/Vs ratio, spatial sampling ratio, stability ratio, and entire range of directions of propagation with respect to the spatial grid led to interesting and surprising findings. Accuracy of FD_D_CG, FE_L and FE_G strongly depends on Vp/Vs ratio. The schemes are not
NASA Astrophysics Data System (ADS)
Ouyang, Chaojun; He, Siming; Xu, Qiang; Luo, Yu; Zhang, Wencheng
2013-03-01
A two-dimensional mountainous mass flow dynamic procedure solver (Massflow-2D) using the MacCormack-TVD finite difference scheme is proposed. The solver is implemented in Matlab on structured meshes with variable computational domain. To verify the model, a variety of numerical test scenarios, namely, the classical one-dimensional and two-dimensional dam break, the landslide in Hong Kong in 1993 and the Nora debris flow in the Italian Alps in 2000, are executed, and the model outputs are compared with published results. It is established that the model predictions agree well with both the analytical solution as well as the field observations.
NASA Astrophysics Data System (ADS)
Chen, Hanming; Zhou, Hui; Sheng, Shanbo
2016-10-01
We develop the general rectangular grid discretization based time-space domain high-order staggered-grid finite-difference (SGFD) methods for modeling three-dimension (3D) scalar wave propagation. The proposed two high-order SGFD schemes can achieve the arbitrary even-order accuracy in space, and the fourth- and sixth-order accuracies in time, respectively. We derive the analytical expression of the high-order FD coefficients based on a general rectangular grid discretization with different grid spacing in all axial directions. The general rectangular grid discretization makes our time-space domain SGFD schemes more flexible than the existing ones developed on the cubic grid with the same grid spacing in the axial directions. Theoretical analysis indicates that our time-space domain SGFD schemes have a better stability and a higher accuracy than the traditional temporal second-order SGFD scheme. Our time-space domain SGFD schemes allow larger time steps than the traditional SGFD scheme for attaining a similar accuracy, and thus are more efficient. Numerical example further confirms the superior accuracy, stability and efficiency of our time-space domain SGFD schemes.
Dynamic Rupture Simulation of Bent Faults with a New Finite Difference Approach
NASA Astrophysics Data System (ADS)
Cruz-Atienza, V. M.; Virieux, J.; Operto, S.
2003-04-01
Many questions about physical parameters governing the rupture propagation of earthquakes find their answers within realistic dynamic considerations. For instance, sophisticated constitutive relations based on laboratory experiments have led to a better understanding of rupture evolution from its very beginning to its arrest. In fact, large amount of field observations as well as recent simulations have shown the importance of considering more reasonable geological settings (e.g., bent and step-over fault geometries; heterogeneous surrounding media). However, despite the development of powerful numerical tools, one important question remains unanswered. How important is in rupture process the feedback coming from a heterogeneous structure if the fault geometry is complex? To start answering this question, we propose a numerical approach based in a new staggered-grid finite-difference technique. In this work, we use a recently proposed four-order staggered-grid finite-difference scheme to dynamically model in-plane (mode II) shear fracturing propagation in faults with any pre-established geometry. In contrast with the classical 2-D staggered grid elementary cell in which all the elastic fields are defined in different positions (except the normal stresses), the stencil used here considers the velocity and stress fields separately in only two staggered grids. Such an elementary structure is a straightforward consequence of a new definition of the four-order spatial differential operators: they are decoupled into two 45-degree rotated operators. This approach permits efficient treatment of boundary conditions to impose the shear stress drop in the nodes where the entire stress tensor is located. Furthermore, this procedure reduces numerical anisotropy along preferred directions and provides stable solutions for any fault orientation. The fault is defined as a set of point sources placed in the middle of the grid without using any ad hoc numerical ghost plane often
NASA Technical Reports Server (NTRS)
Barnwell, R. W.; Dejarnette, F. R.; Wahls, R. A.
1987-01-01
A new turbulent boundary-layer method is developed which models the inner region with the law of the wall while the outer region uses Clauser's eddy viscosity in Matsuno's finite-difference method. The match point between the inner and outer regions as well as the wall shear stress are determined at each marching step during the computation. Results obtained for incompressible, two-dimensional flow over flat plates and ellipses are compared with solutions from a baseline method which uses a finite-difference method for the entire boundary layer. Since the present method used the finite-difference method in the outer region only, the number of grid points required was about half that needed for the baseline method. Accurate displacement and momentum thicknesses were predicted for all cases. Skin friction was predicted well for the flat plate, but the accuracy decreased significantly for the ellipses. Adding a wake functions to the law of the wall allows some of the pressure gradient effect to be taken into account thereby increasing the accuracy of the method.
NASA Technical Reports Server (NTRS)
Weatherill, W. H.; Ehlers, F. E.; Yip, E.; Sebastian, J. D.
1980-01-01
Analytical and empirical studies of a finite difference method for the solution of the transonic flow about harmonically oscillating wings and airfoils are presented. The procedure is based on separating the velocity potential into steady and unsteady parts and linearizing the resulting unsteady equations for small disturbances. The steady velocity potential is obtained first from the well-known nonlinear equation for steady transonic flow. The unsteady velocity potential is then obtained from a linear differential equation in complex form with spatially varying coefficients. Since sinusoidal motion is assumed, the unsteady equation is independent of time. An out-of-core direct solution procedure was developed and applied to two-dimensional sections. Results are presented for a section of vanishing thickness in subsonic flow and an NACA 64A006 airfoil in supersonic flow. Good correlation is obtained in the first case at values of Mach number and reduced frequency of direct interest in flutter analyses. Reasonable results are obtained in the second case. Comparisons of two-dimensional finite difference solutions with exact analytic solutions indicate that the accuracy of the difference solution is dependent on the boundary conditions used on the outer boundaries. Homogeneous boundary conditions on the mesh edges that yield complex eigenvalues give the most accurate finite difference solutions. The plane outgoing wave boundary conditions meet these requirements.
Harte, Philip T.
1994-01-01
Proper discretization of a ground-water-flow field is necessary for the accurate simulation of ground-water flow by models. Although discretiza- tion guidelines are available to ensure numerical stability, current guidelines arc flexible enough (particularly in vertical discretization) to allow for some ambiguity of model results. Testing of two common types of vertical-discretization schemes (horizontal and nonhorizontal-model-layer approach) were done to simulate sloping hydrogeologic units characteristic of New England. Differences of results of model simulations using these two approaches are small. Numerical errors associated with use of nonhorizontal model layers are small (4 percent). even though this discretization technique does not adhere to the strict formulation of the finite-difference method. It was concluded that vertical discretization by means of the nonhorizontal layer approach has advantages in representing the hydrogeologic units tested and in simplicity of model-data input. In addition, vertical distortion of model cells by this approach may improve the representation of shallow flow processes.
NASA Astrophysics Data System (ADS)
Yang, Qingjie; Mao, Weijian
2016-10-01
The poroelastodynamic equations are used to describe the dynamic solid-fluid interaction in the reservoir. To obtain the intrinsic properties of reservoir rocks from geophysical data measured in both laboratory and field, we need an accurate solution of the wave propagation in porous media. At present, the poroelastic wave equations are mostly solved in the time domain, which involves a difficult and complicated time convolution. In order to avoid the issues caused by the time convolution, we propose a frequency-space domain method. The poroelastic wave equations are composed of a linear system in the frequency domain, which easily takes into account the effects of all frequencies on the dispersion and attenuation of seismic wave. A 25-point weighted-averaging finite different scheme is proposed to discretize the equations. For the finite model, the perfectly matched layer technique is applied at the model boundaries. We validated the proposed algorithm by testing three numerical examples of poroelastic models, which are homogenous, two-layered and heterogeneous with different fluids, respectively. The testing results are encouraging in the aspects of both computational accuracy and efficiency.
Development of an explicit non-staggered scheme for solving three-dimensional Maxwell's equations
NASA Astrophysics Data System (ADS)
Sheu, Tony W. H.; Chung, Y. W.; Li, J. H.; Wang, Y. C.
2016-10-01
An explicit finite-difference scheme for solving the three-dimensional Maxwell's equations in non-staggered grids is presented. We aspire to obtain time-dependent solutions of the Faraday's and Ampère's equations and predict the electric and magnetic fields within the discrete zero-divergence context (or Gauss's law). The local conservation laws in Maxwell's equations are numerically preserved using the explicit second-order accurate symplectic partitioned Runge-Kutta temporal scheme. Following the method of lines, the spatial derivative terms in the semi-discretized Faraday's and Ampère's equations are approximated theoretically to obtain a highly accurate numerical phase velocity. The proposed fourth-order accurate space-centered finite difference scheme minimizes the discrepancy between the exact and numerical phase velocities. This minimization process considerably reduces the dispersion and anisotropy errors normally associated with finite difference time-domain methods. The computational efficiency of getting the same level of accuracy at less computing time and the ability of preserving the symplectic property have been numerically demonstrated through several test problems.
NASA Astrophysics Data System (ADS)
Nikkar, Samira; Nordström, Jan
2015-06-01
A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered. By applying the energy method, well-posed boundary conditions for the continuous problem are derived. Summation-by-Parts (SBP) operators for the space and time discretization, together with a weak imposition of boundary and initial conditions using Simultaneously Approximation Terms (SATs) lead to a provable fully-discrete energy-stable conservative finite difference scheme. We show how to construct a time-dependent SAT formulation that automatically imposes boundary conditions, when and where they are required. We also prove that a uniform flow field is preserved, i.e. the Numerical Geometric Conservation Law (NGCL) holds automatically by using SBP-SAT in time and space. The developed technique is illustrated by considering an application using the linearized Euler equations: the sound generated by moving boundaries. Numerical calculations corroborate the stability and accuracy of the new fully discrete approximations.
Solution to PDEs using radial basis function finite-differences (RBF-FD) on multiple GPUs
Bollig, Evan F.; Flyer, Natasha; Erlebacher, Gordon
2012-08-30
This paper presents parallelization strategies for the radial basis function-finite difference (RBF-FD) method. As a generalized finite differencing scheme, the RBF-FD method functions without the need for underlying meshes to structure nodes. It offers high-order accuracy approximation and scales as O(N) per time step, with N being with the total number of nodes. To our knowledge, this is the first implementation of the RBF-FD method to leverage GPU accelerators for the solution of PDEs. Additionally, this implementation is the first to span both multiple CPUs and multiple GPUs. OpenCL kernels target the GPUs and inter-processor communication and synchronization is managed by the Message Passing Interface (MPI). We verify our implementation of the RBF-FD method with two hyperbolic PDEs on the sphere, and demonstrate up to 9x speedup on a commodity GPU with unoptimized kernel implementations. On a high performance cluster, the method achieves up to 7x speedup for the maximum problem size of 27,556 nodes.
Goode, D.J.; Appel, C.A.
1992-01-01
More accurate alternatives to the widely used harmonic mean interblock transmissivity are proposed for block-centered finite-difference models of ground-water flow in unconfined aquifers and in aquifers having smoothly varying transmissivity. The harmonic mean is the exact interblock transmissivity for steady-state one-dimensional flow with no recharge if the transmissivity is assumed to be spatially uniform over each finite-difference block, changing abruptly at the block interface. However, the harmonic mean may be inferior to other means if transmissivity varies in a continuous or smooth manner between nodes. Alternative interblock transmissivity functions are analytically derived for the case of steady-state one-dimensional flow with no recharge. The second author has previously derived the exact interblock transmissivity, the logarithmic mean, for one-dimensional flow when transmissivity is a linear function of distance in the direction of flow. We show that the logarithmic mean transmissivity is also exact for uniform flow parallel to the direction of changing transmissivity in a two- or three-dimensional model, regardless of grid orientation relative to the flow vector. For the case of horizontal flow in a homogeneous unconfined or water-table aquifer with a horizontal bottom and with areally distributed recharge, the exact interblock transmissivity is the unweighted arithmetic mean of transmissivity at the nodes. This mean also exhibits no grid-orientation effect for unidirectional flow in a two-dimensional model. For horizontal flow in an unconfined aquifer with no recharge where hydraulic conductivity is a linear function of distance in the direction of flow the exact interblock transmissivity is the product of the arithmetic mean saturated thickness and the logarithmic mean hydraulic conductivity. For several hypothetical two- and three-dimensional cases with smoothly varying transmissivity or hydraulic conductivity, the harmonic mean is shown to yield
NASA Technical Reports Server (NTRS)
Kreider, Kevin L.; Baumeister, Kenneth J.
1996-01-01
An explicit finite difference real time iteration scheme is developed to study harmonic sound propagation in aircraft engine nacelles. To reduce storage requirements for future large 3D problems, the time dependent potential form of the acoustic wave equation is used. To insure that the finite difference scheme is both explicit and stable for a harmonic monochromatic sound field, a parabolic (in time) approximation is introduced to reduce the order of the governing equation. The analysis begins with a harmonic sound source radiating into a quiescent duct. This fully explicit iteration method then calculates stepwise in time to obtain the 'steady state' harmonic solutions of the acoustic field. For stability, applications of conventional impedance boundary conditions requires coupling to explicit hyperbolic difference equations at the boundary. The introduction of the time parameter eliminates the large matrix storage requirements normally associated with frequency domain solutions, and time marching attains the steady-state quickly enough to make the method favorable when compared to frequency domain methods. For validation, this transient-frequency domain method is applied to sound propagation in a 2D hard wall duct with plug flow.
Validation of NSWING, a multi-core finite difference code for tsunami propagation and run-up
NASA Astrophysics Data System (ADS)
Miranda, J. M. A.; Luis, J. M. F.; Reis, C.; Omira, R.; Baptista, M. A.
2014-12-01
We present the finite difference tsunami code NSWING (Non-linear Shallow Water model With Nested Grids), that solves the non-linear shallow water equations using the discretization and explicit leap-frog finite difference scheme, in a Cartesian or Spherical frame, as developed by Liu et al. (1998). An open boundary condition is used on the outward limit of the grid, whenever it does not correspond to land. The model also incorporates Coriolis acceleration, bottom friction and a moving boundary scheme to model run-up. Multiple levels of nesting are possible. NSWING runs on MS windows operating system using more than one core. The code is applied to classical benchmark tests (Synolakis et al., 2007) and to a test case in SW Portugal. It is shown that the code reproduces well the numerical benchmarks, improves its accuracy with increasing resolution and ensures mass conservation. It is also shown that NSWING can efficiently provide inundation modelling for high resolution studies. This work is a contribution to GEONUM project FCT-ANR/MAT-NAN/0122/2012
Numerical computation of transonic flows by finite-element and finite-difference methods
NASA Technical Reports Server (NTRS)
Hafez, M. M.; Wellford, L. C.; Merkle, C. L.; Murman, E. M.
1978-01-01
Studies on applications of the finite element approach to transonic flow calculations are reported. Different discretization techniques of the differential equations and boundary conditions are compared. Finite element analogs of Murman's mixed type finite difference operators for small disturbance formulations were constructed and the time dependent approach (using finite differences in time and finite elements in space) was examined.
NASA Technical Reports Server (NTRS)
Batina, John T.
1992-01-01
A time-accurate approximate-factorization (AF) algorithm is described for solution of the three-dimensional unsteady transonic small-disturbance equation. The AF algorithm consists of a time-linearization procedure coupled with a subiteration technique. The algorithm is the basis for the Computational Aeroelasticity Program-Transonic Small Disturbance (CAP-TSD) computer code, which was developed for the analysis of unsteady aerodynamics and aeroelasticity of realistic aircraft configurations. The paper describes details on the governing flow equations and boundary conditions, with an emphasis on documenting the finite-difference formulas of the AF algorithm.
NASA Technical Reports Server (NTRS)
Vinh, Hoang; Dwyer, Harry A.; Van Dam, C. P.
1992-01-01
The applications of two CFD-based finite-difference methods to computational electromagnetics are investigated. In the first method, the time-domain Maxwell's equations are solved using the explicit Lax-Wendroff scheme and in the second method, the second-order wave equations satisfying the Maxwell's equations are solved using the implicit Crank-Nicolson scheme. The governing equations are transformed to a generalized curvilinear coordinate system and solved on a body-conforming mesh using the scattered-field formulation. The induced surface current and the bistatic radar cross section are computed and the results are validated for several two-dimensional test cases involving perfectly-conducting scatterers submerged in transverse-magnetic plane waves.
2-D Finite Difference Modeling of the D'' Structure Beneath the Eastern Cocos Plate: Part I
NASA Astrophysics Data System (ADS)
Helmberger, D. V.; Song, T. A.; Sun, D.
2005-12-01
The discovery of phase transition from Perovskite (Pv) to Post-Perovskite (PPv) at depth nears the lowermost mantle has revealed a new view of the earth's D'' layer (Oganov et al. 2004; Murakami et al. 2004). Hernlund et al. (2004) recently pusposed that, depending on the geotherm at the core-mantle boundary (CMB), a double-crossing of the phase boundary by the geotherm at two different depths may also occur. To explore these new findings, we adopt 2-D finite difference scheme (Helmberger and Vidale, 1988) to model wave propagation in rapidly varying structure. We collect broadband waveform data recorded by several Passcal experiments, such as La Ristra transect and CDROM transect in the southwest US to constrain the lateral variations in D'' structure. These data provide fairly dense sampling (~ 20 km) in the lowermost mantle beneath the eastern Cocos plate. Since the source-receiver paths are mostly in the same azimuth, we make 2-D cross-sections from global tomography model (Grand, 2002) and compute finite difference synthetics. We modify the lowermost mantle below 2500 km with constraints from transverse-component waveform data at epicentral distances of 70-82 degrees in the time window between S and ScS, essentially foward modeling waveforms. Assuming a velocity jump of 3 % at D'', our preferred model shows that the D'' topography deepens from the north to the south by about 120 km over a lateral distance of 300 km. Such large topography jumps have been proposed by Thomas et al. (2004) using data recorded by TriNet. In addition, there is a negative velocity jump (-3 %) 100 km above the CMB in the south. This simple model compare favorably with results from a study by Sun, Song and Helmberger (2005), who follow Sidorin et al. (1999) approach and produce a thermodynamically consistent velocity model with Pv-PPv phase boundary. It appears that much of this complexity exists in Grand's tomographic maps with rapid variation in velocities just above the D''. We also
Finite-difference time-domain studies of the optical properties of nanoshell dimers.
Oubre, C; Nordlander, P
2005-05-26
The optical properties of metallic nanoshell dimers are investigated using the finite difference time domain (FDTD) method. We discuss issues of numerical convergence specific for the dimer system. We present results for both homodimers and heterodimers. The results show that retardation effects must be taken into account for an accurate description of realistic size nanoparticle dimers. The optical properties of the nanoshell dimer are found to be strongly polarization dependent. Maximal coupling between the nanoshells in a dimer occurs when the electric field of the incident pulse is aligned parallel to the dimer axis. The wavelengths of the peaks in the extinction cross section of the dimer are shown to vary by more than 100 nm, depending on the incident electric field polarization. The calculations show that electric field enhancements in the dimer junctions depend strongly on dimer separation. The maximum field enhancements occur in the dimer junction and at the expense of a reduced electric field enhancement in other regions of space. We investigate the usefulness of nanoshell dimers substrates for SERS by integrating the fourth power of the electric field enhancements around the surfaces of the nanoparticles as a function of dimer separation and wavelength. The SERS efficiency is shown to depend strongly on dimer separation but much weaker than the fourth power of the maximum electric field enhancement at a particular point. The SERS efficiency is also found to depend strongly on the wavelength of the incident light. Maximum SERS efficiency occurs for resonant excitation of the dimer plasmons. PMID:16852215
Forensic GPR: finite-difference simulations of responses from buried human remains
NASA Astrophysics Data System (ADS)
Hammon, William S.; McMechan, George A.; Zeng, Xiaoxian
2000-10-01
Time domain 2.5-D finite-difference simulations of ground-penetrating radar (GPR) responses from models of buried human remains suggest the potential of GPR for detailed non-destructive forensic site investigation. Extraction of information beyond simple detection of cadavers in forensic investigations should be possible with current GPR technology. GPR responses are simulated for various body cross-sections with different depths of burial, soil types, soil moisture contents, survey frequencies and antenna separations. Biological tissues have high electrical conductivity so diagnostic features for the imaging of human bodies are restricted to the soil/skin interface and shallow tissue interfaces. A low amplitude reflection shadow zone occurs beneath a body because of high GPR attenuation within the body. Resolution of diagnostic features of a human target requires a survey frequency of 900 MHz or greater and an increment between recording stations of 10 cm or less. Depth migration focuses field GPR data into an image that reveals accurate information on the number, dimensions, locations and orientations of body elements. The main limitation on image quality is attenuation in the surrounding soil and within the body. 3-D imaging is also feasible.
Nonlinear waves in deforming porous media with the finite difference method
NASA Astrophysics Data System (ADS)
Räss, Ludovic; Duretz, Thibault; Podladchikov, Yuri
2016-04-01
The actual trend in computational geodynamics tends toward the development of coupled multi-physics models. The resulting models involve various types of nonlinear processes such as non-Newtonian rheologies and multi-physics coupling. One of the main challenge is to treat these nonlinearities in order to preserve the predictive power of these forward models. In this framework, we designed two dimensional (2D) finite difference models using both implicit and explicit discretisations. We apply the models to study the initiation and the propagation of nonlinear waves in deforming porous media (porosity waves). A strong coupling of a Stokes solver to a nonlinear Darcy flow is required to describe the complex evolution of permeability and porosity in space and in time. In our models, we also take into account porosity-dependant permeability and rheologies that are representative of major reservoir rock type (i.e. tight shales). We conduct numerical simulations and show that both implicit and explicit approaches capture the channeling instabilities and the development of focused flow. We perform quantitative comparison of the two methods and discuss the treatment of multi-physics coupling and rheological nonlinearities. We show that implicit and explicit discretisations converge to similar solutions only if the nonlinearities are accurately resolved. Explicit numerical algorithms can therefore be attractive because they are very comprehensible and well suited for HPC while implicit models are performing well on desktop computations.
NASA Astrophysics Data System (ADS)
Weinert, C. M.; Agrawal, N.
1994-12-01
A self-consistent finite difference method for the simulation of quantum well electron transfer structures is developed and applied to optimize InGaAsP/InP/InAlAs structures for fast optical switching devices. Simultaneous solution of Poisson's equation, continuity equation, and Schroedinger's equation on a discretized mesh yields a fast and accurate simulation method which may be applied to arbitrary layer structures and needs no artificial assumptions like abrupt space charge layers. Because of the exact treatment of charge distribution and leakage current the simulation gives new insight into the performance of barrier, reservoir, and quantum well electron transfer structrues, which could not be found by previous approximate theories. With this method we calculate the important physical parameters of these devices, namely, the shift of the optical absorption edge, band filling, leakage current, and capacitance. In addition, each layer is investigated separately with respect to its influence on device performance and fabrication tolerances; the results are used for optimization. Moreover, the exact numerical simulation is used to derive simplified relations for the dependence of band filling, capacitance, and high speed behavior on the heterostructure design.
Zou, Han; Jiang, Hao; Luo, Yiwen; Zhu, Jianjie; Lu, Xiaoxuan; Xie, Lihua
2016-01-01
The location and contextual status (indoor or outdoor) is fundamental and critical information for upper-layer applications, such as activity recognition and location-based services (LBS) for individuals. In addition, optimizations of building management systems (BMS), such as the pre-cooling or heating process of the air-conditioning system according to the human traffic entering or exiting a building, can utilize the information, as well. The emerging mobile devices, which are equipped with various sensors, become a feasible and flexible platform to perform indoor-outdoor (IO) detection. However, power-hungry sensors, such as GPS and WiFi, should be used with caution due to the constrained battery storage on mobile device. We propose BlueDetect: an accurate, fast response and energy-efficient scheme for IO detection and seamless LBS running on the mobile device based on the emerging low-power iBeacon technology. By leveraging the on-broad Bluetooth module and our proposed algorithms, BlueDetect provides a precise IO detection service that can turn on/off on-board power-hungry sensors smartly and automatically, optimize their performances and reduce the power consumption of mobile devices simultaneously. Moreover, seamless positioning and navigation services can be realized by it, especially in a semi-outdoor environment, which cannot be achieved by GPS or an indoor positioning system (IPS) easily. We prototype BlueDetect on Android mobile devices and evaluate its performance comprehensively. The experimental results have validated the superiority of BlueDetect in terms of IO detection accuracy, localization accuracy and energy consumption. PMID:26907295
Zou, Han; Jiang, Hao; Luo, Yiwen; Zhu, Jianjie; Lu, Xiaoxuan; Xie, Lihua
2016-01-01
The location and contextual status (indoor or outdoor) is fundamental and critical information for upper-layer applications, such as activity recognition and location-based services (LBS) for individuals. In addition, optimizations of building management systems (BMS), such as the pre-cooling or heating process of the air-conditioning system according to the human traffic entering or exiting a building, can utilize the information, as well. The emerging mobile devices, which are equipped with various sensors, become a feasible and flexible platform to perform indoor-outdoor (IO) detection. However, power-hungry sensors, such as GPS and WiFi, should be used with caution due to the constrained battery storage on mobile device. We propose BlueDetect: an accurate, fast response and energy-efficient scheme for IO detection and seamless LBS running on the mobile device based on the emerging low-power iBeacon technology. By leveraging the on-broad Bluetooth module and our proposed algorithms, BlueDetect provides a precise IO detection service that can turn on/off on-board power-hungry sensors smartly and automatically, optimize their performances and reduce the power consumption of mobile devices simultaneously. Moreover, seamless positioning and navigation services can be realized by it, especially in a semi-outdoor environment, which cannot be achieved by GPS or an indoor positioning system (IPS) easily. We prototype BlueDetect on Android mobile devices and evaluate its performance comprehensively. The experimental results have validated the superiority of BlueDetect in terms of IO detection accuracy, localization accuracy and energy consumption. PMID:26907295
Zou, Han; Jiang, Hao; Luo, Yiwen; Zhu, Jianjie; Lu, Xiaoxuan; Xie, Lihua
2016-02-22
The location and contextual status (indoor or outdoor) is fundamental and critical information for upper-layer applications, such as activity recognition and location-based services (LBS) for individuals. In addition, optimizations of building management systems (BMS), such as the pre-cooling or heating process of the air-conditioning system according to the human traffic entering or exiting a building, can utilize the information, as well. The emerging mobile devices, which are equipped with various sensors, become a feasible and flexible platform to perform indoor-outdoor (IO) detection. However, power-hungry sensors, such as GPS and WiFi, should be used with caution due to the constrained battery storage on mobile device. We propose BlueDetect: an accurate, fast response and energy-efficient scheme for IO detection and seamless LBS running on the mobile device based on the emerging low-power iBeacon technology. By leveraging the on-broad Bluetooth module and our proposed algorithms, BlueDetect provides a precise IO detection service that can turn on/off on-board power-hungry sensors smartly and automatically, optimize their performances and reduce the power consumption of mobile devices simultaneously. Moreover, seamless positioning and navigation services can be realized by it, especially in a semi-outdoor environment, which cannot be achieved by GPS or an indoor positioning system (IPS) easily. We prototype BlueDetect on Android mobile devices and evaluate its performance comprehensively. The experimental results have validated the superiority of BlueDetect in terms of IO detection accuracy, localization accuracy and energy consumption.
Saarelma, Jukka; Botts, Jonathan; Hamilton, Brian; Savioja, Lauri
2016-04-01
Finite-difference time-domain (FDTD) simulation has been a popular area of research in room acoustics due to its capability to simulate wave phenomena in a wide bandwidth directly in the time-domain. A downside of the method is that it introduces a direction and frequency dependent error to the simulated sound field due to the non-linear dispersion relation of the discrete system. In this study, the perceptual threshold of the dispersion error is measured in three-dimensional FDTD schemes as a function of simulation distance. Dispersion error is evaluated for three different explicit, non-staggered FDTD schemes using the numerical wavenumber in the direction of the worst-case error of each scheme. It is found that the thresholds for the different schemes do not vary significantly when the phase velocity error level is fixed. The thresholds are found to vary significantly between the different sound samples. The measured threshold for the audibility of dispersion error at the probability level of 82% correct discrimination for three-alternative forced choice is found to be 9.1 m of propagation in a free field, that leads to a maximum group delay error of 1.8 ms at 20 kHz with the chosen phase velocity error level of 2%. PMID:27106330
Saarelma, Jukka; Botts, Jonathan; Hamilton, Brian; Savioja, Lauri
2016-04-01
Finite-difference time-domain (FDTD) simulation has been a popular area of research in room acoustics due to its capability to simulate wave phenomena in a wide bandwidth directly in the time-domain. A downside of the method is that it introduces a direction and frequency dependent error to the simulated sound field due to the non-linear dispersion relation of the discrete system. In this study, the perceptual threshold of the dispersion error is measured in three-dimensional FDTD schemes as a function of simulation distance. Dispersion error is evaluated for three different explicit, non-staggered FDTD schemes using the numerical wavenumber in the direction of the worst-case error of each scheme. It is found that the thresholds for the different schemes do not vary significantly when the phase velocity error level is fixed. The thresholds are found to vary significantly between the different sound samples. The measured threshold for the audibility of dispersion error at the probability level of 82% correct discrimination for three-alternative forced choice is found to be 9.1 m of propagation in a free field, that leads to a maximum group delay error of 1.8 ms at 20 kHz with the chosen phase velocity error level of 2%.
NASA Technical Reports Server (NTRS)
Fix, G. J.; Rose, M. E.
1983-01-01
A least squares formulation of the system divu = rho, curlu = zeta is surveyed from the viewpoint of both finite element and finite difference methods. Closely related arguments are shown to establish convergence estimates.
NASA Technical Reports Server (NTRS)
Bauld, N. R., Jr.; Goree, J. G.
1983-01-01
The accuracy of the finite difference method in the solution of linear elasticity problems that involve either a stress discontinuity or a stress singularity is considered. Solutions to three elasticity problems are discussed in detail: a semi-infinite plane subjected to a uniform load over a portion of its boundary; a bimetallic plate under uniform tensile stress; and a long, midplane symmetric, fiber reinforced laminate subjected to uniform axial strain. Finite difference solutions to the three problems are compared with finite element solutions to corresponding problems. For the first problem a comparison with the exact solution is also made. The finite difference formulations for the three problems are based on second order finite difference formulas that provide for variable spacings in two perpendicular directions. Forward and backward difference formulas are used near boundaries where their use eliminates the need for fictitious grid points.
Construction of weighted upwind compact scheme
NASA Astrophysics Data System (ADS)
Wang, Zhengjie
Enormous endeavor has been devoted in spatial high order high resolution schemes in more than twenty five years previously, like total variation diminishing (TVD), essentially non-oscillatory scheme, weighted essentially non-oscillatory scheme for finite difference, and Discontinuous Galerkin methods for finite element and the finite volume. In this dissertation, a high order finite difference Weighted Upwind Compact Scheme has been constructed by dissipation and dispersion analysis. Secondly, a new method to construct global weights has been tested. Thirdly, a methodology to compromise dissipation and dispersion in constructing Weighted Upwind Compact Scheme has been derived. Finally, several numerical test cases have been shown.
NASA Technical Reports Server (NTRS)
Garrett, L. B.; Smith, G. L.; Perkins, J. N.
1972-01-01
An implicit finite-difference scheme is developed for the fully coupled solution of the viscous, radiating stagnation-streamline equations, including strong blowing. Solutions are presented for both air injection and injection of carbon-phenolic ablation products into air at conditions near the peak radiative heating point in an earth entry trajectory from interplanetary return missions. A detailed radiative-transport code that accounts for the important radiative exchange processes for gaseous mixtures in local thermodynamic and chemical equilibrium is utilized in the study. With minimum number of assumptions for the initially unknown parameters and profile distributions, convergent solutions to the full stagnation-line equations are rapidly obtained by a method of successive approximations. Damping of selected profiles is required to aid convergence of the solutions for massive blowing. It is shown that certain finite-difference approximations to the governing differential equations stabilize and improve the solutions. Detailed comparisons are made with the numerical results of previous investigations. Results of the present study indicate lower radiative heat fluxes at the wall for carbonphenolic ablation than previously predicted.
NASA Technical Reports Server (NTRS)
Desideri, J. A.; Steger, J. L.; Tannehill, J. C.
1978-01-01
The iterative convergence properties of an approximate-factorization implicit finite-difference algorithm are analyzed both theoretically and numerically. Modifications to the base algorithm were made to remove the inconsistency in the original implementation of artificial dissipation. In this way, the steady-state solution became independent of the time-step, and much larger time-steps can be used stably. To accelerate the iterative convergence, large time-steps and a cyclic sequence of time-steps were used. For a model transonic flow problem governed by the Euler equations, convergence was achieved with 10 times fewer time-steps using the modified differencing scheme. A particular form of instability due to variable coefficients is also analyzed.
NASA Astrophysics Data System (ADS)
Kaya, Adem; Sendur, Ali
2015-11-01
A numerical scheme for the convection-diffusion-reaction (CDR) problems is studied herein. We propose a finite difference method on a special grid for solving CDR problems particularly designed to treat the most interesting case of small diffusion. We use the subgrid nodes in the Link-cutting bubble (LCB) strategy [5] to construct a numerical algorithm that can easily be extended to the higher dimensions. The method adapts very well to all regimes with continuous transitions from one regime to another. We also compare the performance of the present method with the Streamline-upwind Petrov-Galerkin (SUPG) and the Residual-Free Bubbles (RFB) methods on several benchmark problems. The numerical experiments confirm the good performance of the proposed method.
NASA Technical Reports Server (NTRS)
Byun, Chansup; Guruswamy, Guru P.
1993-01-01
This paper presents a procedure for computing the aeroelasticity of wing-body configurations on multiple-instruction, multiple-data (MIMD) parallel computers. In this procedure, fluids are modeled using Euler equations discretized by a finite difference method, and structures are modeled using finite element equations. The procedure is designed in such a way that each discipline can be developed and maintained independently by using a domain decomposition approach. A parallel integration scheme is used to compute aeroelastic responses by solving the coupled fluid and structural equations concurrently while keeping modularity of each discipline. The present procedure is validated by computing the aeroelastic response of a wing and comparing with experiment. Aeroelastic computations are illustrated for a High Speed Civil Transport type wing-body configuration.
A Technique of Treating Negative Weights in WENO Schemes
NASA Technical Reports Server (NTRS)
Shi, Jing; Hu, Changqing; Shu, Chi-Wang
2000-01-01
High order accurate weighted essentially non-oscillatory (WENO) schemes have recently been developed for finite difference and finite volume methods both in structural and in unstructured meshes. A key idea in WENO scheme is a linear combination of lower order fluxes or reconstructions to obtain a high order approximation. The combination coefficients, also called linear weights, are determined by local geometry of the mesh and order of accuracy and may become negative. WENO procedures cannot be applied directly to obtain a stable scheme if negative linear weights are present. Previous strategy for handling this difficulty is by either regrouping of stencils or reducing the order of accuracy to get rid of the negative linear weights. In this paper we present a simple and effective technique for handling negative linear weights without a need to get rid of them.
TE/TM scheme for computation of electromagnetic fields in accelerators
Zagorodnov, Igor . E-mail: zagor@temf.de; Weiland, Thomas . E-mail: thomas.weiland@temf.de
2005-07-20
We propose a new two-level economical conservative scheme for short-range wake field calculation in three dimensions. The scheme does not have dispersion in the longitudinal direction and is staircase free (second order convergent). Unlike the finite-difference time domain method (FDTD), it is based on a TE/TM like splitting of the field components in time. Additionally, it uses an enhanced alternating direction splitting of the transverse space operator that makes the scheme computationally as effective as the conventional FDTD method. Unlike the FDTD ADI and low-order Strang methods, the splitting error in our scheme is only of fourth order. As numerical examples show, the new scheme is much more accurate on the long-time scale than the conventional FDTD approach.
NASA Astrophysics Data System (ADS)
Petersson, N. Anders; Sjögreen, Björn
2015-10-01
We develop a fourth order accurate finite difference method for solving the three-dimensional elastic wave equation in general heterogeneous anisotropic materials on curvilinear grids. The proposed method is an extension of the method for isotropic materials, previously described in the paper by Sjögreen and Petersson (2012) [11]. The proposed method discretizes the anisotropic elastic wave equation in second order formulation, using a node centered finite difference method that satisfies the principle of summation by parts. The summation by parts technique results in a provably stable numerical method that is energy conserving. We also generalize and evaluate the super-grid far-field technique for truncating unbounded domains. Unlike the commonly used perfectly matched layers (PML), the super-grid technique is stable for general anisotropic material, because it is based on a coordinate stretching combined with an artificial dissipation. As a result, the discretization satisfies an energy estimate, proving that the numerical approximation is stable. We demonstrate by numerical experiments that sufficiently wide super-grid layers result in very small artificial reflections. Applications of the proposed method are demonstrated by three-dimensional simulations of anisotropic wave propagation in crystals.
NASA Astrophysics Data System (ADS)
He, Qinglong; Han, Bo; Chen, Yong; Li, Yang
2016-01-01
In this paper, we present an efficient inversion method to reconstruct the velocity and density model based on the acoustic wave equation. The inversion is performed in the frequency domain using the finite-difference contrast source inversion (FD-CSI) method. The full forward problem is required to be solved only once at the beginning of the inversion process, which makes the method very computationally efficient. Furthermore, the flexibility of the finite-difference operator ensures FD-CSI the capability of handling complex geophysical applications with inhomogeneous background media. Moreover, different parameters are automatically normalized, avoiding the numerical difficulty arising from the different magnitudes of the parameters. A variant of total variation regularization called multiplicative constraint is incorporated to resolve the sharp discontinuities of the parameters. We employ a two-phase inversion strategy to carry out the FD-CSI method. After simultaneously reconstructing the bulk modulus and density, we obtain a relatively reliable bulk modulus, which is used as the background in the next phase to retrieve more accurate bulk modulus and density. A simple experiment is carried out to present the capability of the FD-CSI method in dealing with the cross talk effect between different parameters. The application on the Marmousi model further emphasizes the performance of the method for more complex geophysical problems.
Petersson, N. Anders; Sjogreen, Bjorn
2015-07-20
We develop a fourth order accurate finite difference method for solving the three-dimensional elastic wave equation in general heterogeneous anisotropic materials on curvilinear grids. The proposed method is an extension of the method for isotropic materials, previously described in the paper by Sjögreen and Petersson (2012) [11]. The method we proposed discretizes the anisotropic elastic wave equation in second order formulation, using a node centered finite difference method that satisfies the principle of summation by parts. The summation by parts technique results in a provably stable numerical method that is energy conserving. Also, we generalize and evaluate the super-grid far-fieldmore » technique for truncating unbounded domains. Unlike the commonly used perfectly matched layers (PML), the super-grid technique is stable for general anisotropic material, because it is based on a coordinate stretching combined with an artificial dissipation. Moreover, the discretization satisfies an energy estimate, proving that the numerical approximation is stable. We demonstrate by numerical experiments that sufficiently wide super-grid layers result in very small artificial reflections. Applications of the proposed method are demonstrated by three-dimensional simulations of anisotropic wave propagation in crystals.« less
Petersson, N. Anders; Sjogreen, Bjorn
2015-07-20
We develop a fourth order accurate finite difference method for solving the three-dimensional elastic wave equation in general heterogeneous anisotropic materials on curvilinear grids. The proposed method is an extension of the method for isotropic materials, previously described in the paper by Sjögreen and Petersson (2012) [11]. The method we proposed discretizes the anisotropic elastic wave equation in second order formulation, using a node centered finite difference method that satisfies the principle of summation by parts. The summation by parts technique results in a provably stable numerical method that is energy conserving. Also, we generalize and evaluate the super-grid far-field technique for truncating unbounded domains. Unlike the commonly used perfectly matched layers (PML), the super-grid technique is stable for general anisotropic material, because it is based on a coordinate stretching combined with an artificial dissipation. Moreover, the discretization satisfies an energy estimate, proving that the numerical approximation is stable. We demonstrate by numerical experiments that sufficiently wide super-grid layers result in very small artificial reflections. Applications of the proposed method are demonstrated by three-dimensional simulations of anisotropic wave propagation in crystals.
NASA Technical Reports Server (NTRS)
LeCroy, Stuart R.; Whitlock, Charles H.; Suttles, John T.
1997-01-01
A finite difference radiative transfer program was developed to handle most anisotropic scattering and reflectance problems encountered in the Earth's atmospheric system. The model has been used to reproduce the radiance received by both satellite and ground based radiation measuring instruments. It accurately replicates the radiance measured by both narrow and wide field-of-view instruments with either narrow or broadband wavelength ranges located on the surface and at satellite altitudes. The output of the finite difference code is compared to the measurements by surface pyranometers and a spectroradiometer aboard a high flying aircraft. The program output is also compared to ERBE measurements aboard the ERBS and NOAA-9 satellites as well as the visible bands aboard the GOES-6 and GOES-7 satellites and AVHRR bands 1 and 2 of the NOAA-9 and NOAA-1 1 satellites. The model is within 0.2 % of the radiance received by pyranometers, within 0.6 % of the ERBE radiances, and within 3 % of the radiances measured by the visible bands of the GOES and NOAA AVHRR radiometers.
Dynamic Rupture Simulation of Bending Faults With a Finite Difference Approach
NASA Astrophysics Data System (ADS)
Cruz-Atienza, V. M.; Virieux, J.; Operto, S.
2002-12-01
Many questions about physical parameters governing the rupture propagation of earthquakes seem to find their answers within realistic dynamic considerations. Sophisticated constitutive relations based in laboratory experiments have lead to a better understanding of rupture evolution from its very beginning to its arrest. On the other hand, large amount of field observations as well as recent numerical simulations have also demonstrated the importance, in rupture growing, of considering more reasonable geological settings (e.g., bending and step-over fault geometries; heterogeneous surrounding media). So far, despite the development of powerful numerical tools, there still exist some numerical considerations that overstep their possibilities. Authors have solved the dynamic problem by applying the boundary integral equations method (BIEM) in order to explore the influence of fault geometry. This can be possible because of the fact that only the rupture path must be discretized, reducing the impact of numerical discretization. However, the BIEM needs the analytical solution of Green functions that can only be computed for a homogeneous space. Up to date, no interaction with heterogeneous structures can be taken in to account. In contrast, finite difference (FD) approaches have been widely used. In this case, due to the specific discretization of the elastodynamic equations through the entire domain, and the azimuthal anisotropy intrinsic to differential operators, only planar faults have been considered and numerical artefacts have to be carefully checked. In this work, we have used a recently proposed four-order staggered grid finite difference scheme to model in-plane (mode II) dynamic shear fracturing propagation with any pre-established geometry. In contrast with the classical 2-D staggered grid elementary cell in which all the elastic fields are defined in different positions (except the normal stresses), the stencil used here consider the velocity and stress
NASA Astrophysics Data System (ADS)
Takenaka, H.; Komatsu, M.; Toyokuni, G.; Nakamura, T.; Okamoto, T.
2015-12-01
A simple and efficient finite-difference scheme is developed to compute seismic wave propagation for a partial spherical shell model of a three-dimensionally (3-D) heterogeneous global earth structure. This new scheme solves the elastodynamic equations in the "quasi-Cartesian" coordinate system similar to a local Cartesian one, instead of the spherical coordinate system, with a staggered-grid finite-difference method in time domain (FDTD) which is one of the most popular numerical methods in seismic motion simulations for local to regional scale models. The proposed scheme may be useful for modeling seismic wave propagation in a very large region of sub-global scale beyond regional and less than global ones, where the effects of roundness of earth cannot be ignored. In "quasi-Cartesian" coordinates, x, y, and z are set to be locally in directions of latitude, longitude and depth, respectively. The stencil for each of the x-derivatives then depends on the depth coordinate at the evaluation point, while the stencil for each of the y-derivatives varies with both coordinates of the depth and latitude. In order to reduce lateral variations of the horizontal finite-difference stencils over the computational domain, we move the target area to a location around the equator of the computational spherical coordinate system using a way similar to the conversion from equatorial coordinates to ecliptic coordinates. The developed scheme can be easily implemented in 3-D Cartesian FDTD codes for local to regional scale modeling by changing a very small part of the codes. Our scheme may be able to open a window for multi-scale modeling of seismic wave propagation in scales from sub-global to local one.
Minimum divergence viscous flow simulation through finite difference and regularization techniques
NASA Astrophysics Data System (ADS)
Victor, Rodolfo A.; Mirabolghasemi, Maryam; Bryant, Steven L.; Prodanović, Maša
2016-09-01
We develop a new algorithm to simulate single- and two-phase viscous flow through a three-dimensional Cartesian representation of the porous space, such as those available through X-ray microtomography. We use the finite difference method to discretize the governing equations and also propose a new method to enforce the incompressible flow constraint under zero Neumann boundary conditions for the velocity components. Finite difference formulation leads to fast parallel implementation through linear solvers for sparse matrices, allowing relatively fast simulations, while regularization techniques used on solving inverse problems lead to the desired incompressible fluid flow. Tests performed using benchmark samples show good agreement with experimental/theoretical values. Additional tests are run on Bentheimer and Buff Berea sandstone samples with available laboratory measurements. We compare the results from our new method, based on finite differences, with an open source finite volume implementation as well as experimental results, specifically to evaluate the benefits and drawbacks of each method. Finally, we calculate relative permeability by using this modified finite difference technique together with a level set based algorithm for multi-phase fluid distribution in the pore space. To our knowledge this is the first time regularization techniques are used in combination with finite difference fluid flow simulations.
NASA Astrophysics Data System (ADS)
Zhang, Xi; Liu, Yang; Cai, Xiaohui; Ren, Zhiming
2015-12-01
The reverse-time migration (RTM) crosscorrelation imaging condition requires that the forward-propagated source wavefield and the backward-propagated receiver wavefield must be obtained at the same time. The easiest way to get the source wavefield is to save the entire time history of the full wavefield into computer memory. However, this strategy requires huge amount of data storage. It is impossible for large-scale 3D RTM. To reduce the computer memory cost, the back-propagated source wavefield is reconstructed by using the stored boundary wavefield. Its computer memory is proportional to the saved boundary grid points. For high order of spatial finite-difference (FD) schemes, more boundary grid points are needed to be stored, which consumes a large amount of the computer memory required for RTM. To further reduce the computer memory cost, we adopt the hybrid absorbing boundary condition (ABC) combined with the arbitrarily wide-angle wave equations (AWWEs). In our method, three boundary grid points can obtain good absorption. The source wavefield can be accurately reconstructed by using these points and the mirror-image symmetry method. Numerical experiments demonstrate the correctness and effectiveness of the proposed method. We compared our method with the conventional hybrid ABC method based on the 15°one way wave equations (OWWEs). Comparisons show that our method with three boundary grid points can achieve the same absorption as the conventional method with ten boundary grid points. For twentieth order of accuracy in space, our method uses only about 30% of memory requirement and about 59% of computation time required by the conventional method.
Accuracy study of the IDO scheme by Fourier analysis
NASA Astrophysics Data System (ADS)
Imai, Yohsuke; Aoki, Takayuki
2006-09-01
The numerical accuracy of the Interpolated Differential Operator (IDO) scheme is studied with Fourier analysis for the solutions of Partial Differential Equations (PDEs): advection, diffusion, and Poisson equations. The IDO scheme solves governing equations not only for physical variable but also for first-order spatial derivative. Spatial discretizations are based on Hermite interpolation functions with both of them. In the Fourier analysis for the IDO scheme, the Fourier coefficients of the physical variable and the first-order derivative are coupled by the equations derived from the governing equations. The analysis shows the IDO scheme resolves all the wavenumbers with higher accuracy than the fourth-order Finite Difference (FD) and Compact Difference (CD) schemes for advection equation. In particular, for high wavenumbers, the accuracy is superior to that of the sixth-order Combined Compact Difference (CCD) scheme. The diffusion and Poisson equations are also more accurately solved in comparison with the FD and CD schemes. These results show that the IDO scheme guarantees highly resolved solutions for all the terms of fluid flow equations.
Finite-difference evolution of a scattered laser pulse in ocean water
NASA Astrophysics Data System (ADS)
Tessendorf, J.; Piotrowski, C.; Kelly, R. L.
1988-01-01
The effects of absorption and scattering on the propagation of a finite-size laser pulse through ocean water are investigated theoretically, applying a finite-difference model based on the time-dependent radiative-transfer equation. The derivation of the finite-difference evolution algorithm is outlined; its FORTRAN numerical implementation is explained; and simulation results for simple test problems are presented in graphs. The method is shown to provide unconditional stability and physically correct propagation velocities in all directions. The need to eliminate or compensate for ray effects is indicated.
Finite difference solutions of the Euler equations in the vicinity of sharp edges
NASA Technical Reports Server (NTRS)
Hartwich, P.-M.
1985-01-01
Attempts have been made to explain why finite difference solutions of the Euler equations can describe flows with large vortical structures around sharp-edged bodies. The present paper is concerned with the influence of a singular sharp edge on the truncation error for a set of discretized Euler equations. An analysis is conducted of the distribution of the truncation error of one finite difference approximation of the Euler equations near a sharp edge of a thin plate. The analysis leads to a determination of the size of the region of the neighborhood of such a singularity. Attention is given to the consistency of a discretization of the Euler equations, and numerical experiments.
Application of a novel finite difference method to dynamic crack problems
NASA Technical Reports Server (NTRS)
Chen, Y. M.; Wilkins, M. L.
1976-01-01
A versatile finite difference method (HEMP and HEMP 3D computer programs) was developed originally for solving dynamic problems in continuum mechanics. It was extended to analyze the stress field around cracks in a solid with finite geometry subjected to dynamic loads and to simulate numerically the dynamic fracture phenomena with success. This method is an explicit finite difference method applied to the Lagrangian formulation of the equations of continuum mechanics in two and three space dimensions and time. The calculational grid moves with the material and in this way it gives a more detailed description of the physics of the problem than the Eulerian formulation.
NASA Astrophysics Data System (ADS)
Sanmiguel-Rojas, Enrique; Ortega-Casanova, Joaquin; del Pino, Carlos; Fernandez-Feria, Ramon
2004-11-01
A method for generating a non-uniform cartesian grid for irregular two-dimensional (2D) geometries such that all the boundary points are regular mesh points is given. The resulting non-uniform grid is used to discretize the Navier-Stokes equations for 2D incompressible viscous flows using finite difference approximations. To that end, finite-difference approximations of the derivatives on a non-uniform mesh are given. We test the method with two different examples: the shallow water flow on a lake with irregular contour, and the pressure driven flow through an irregular array of circular cylinders.
Locally conformal finite-difference time-domain techniques for particle-in-cell plasma simulation
NASA Astrophysics Data System (ADS)
Clark, R. E.; Welch, D. R.; Zimmerman, W. R.; Miller, C. L.; Genoni, T. C.; Rose, D. V.; Price, D. W.; Martin, P. N.; Short, D. J.; Jones, A. W. P.; Threadgold, J. R.
2011-02-01
The Dey-Mittra [S. Dey, R. Mitra, A locally conformal finite-difference time-domain (FDTD) algorithm for modeling three-dimensional perfectly conducting objects, IEEE Microwave Guided Wave Lett. 7 (273) 1997] finite-difference time-domain partial cell method enables the modeling of irregularly shaped conducting surfaces while retaining second-order accuracy. We present an algorithm to extend this method to include charged particle emission and absorption in particle-in-cell codes. Several examples are presented that illustrate the possible improvements that can be realized using the new algorithm for problems relevant to plasma simulation.
NASA Astrophysics Data System (ADS)
Vincenti, H.; Vay, J.-L.
2016-03-01
Very high order or pseudo-spectral Maxwell solvers are the method of choice to reduce discretization effects (e.g. numerical dispersion) that are inherent to low order Finite-Difference Time-Domain (FDTD) schemes. However, due to their large stencils, these solvers are often subject to truncation errors in many electromagnetic simulations. These truncation errors come from non-physical modifications of Maxwell's equations in space that may generate spurious signals affecting the overall accuracy of the simulation results. Such modifications for instance occur when Perfectly Matched Layers (PMLs) are used at simulation domain boundaries to simulate open media. Another example is the use of arbitrary order Maxwell solver with domain decomposition technique that may under some condition involve stencil truncations at subdomain boundaries, resulting in small spurious errors that do eventually build up. In each case, a careful evaluation of the characteristics and magnitude of the errors resulting from these approximations, and their impact at any frequency and angle, requires detailed analytical and numerical studies. To this end, we present a general analytical approach that enables the evaluation of numerical errors of fully three-dimensional arbitrary order finite-difference Maxwell solver, with arbitrary modification of the local stencil in the simulation domain. The analytical model is validated against simulations of domain decomposition technique and PMLs, when these are used with very high-order Maxwell solver, as well as in the infinite order limit of pseudo-spectral solvers. Results confirm that the new analytical approach enables exact predictions in each case. It also confirms that the domain decomposition technique can be used with very high-order Maxwell solvers and a reasonably low number of guard cells with negligible effects on the whole accuracy of the simulation.
Rayleigh Wave Numerical Dispersion in a 3D Finite-Difference Algorithm
NASA Astrophysics Data System (ADS)
Preston, L. A.; Aldridge, D. F.
2010-12-01
A Rayleigh wave propagates laterally without dispersion in the vicinity of the plane stress-free surface of a homogeneous and isotropic elastic halfspace. The phase speed is independent of frequency and depends only on the Poisson ratio of the medium. However, after temporal and spatial discretization, a Rayleigh wave simulated by a 3D staggered-grid finite-difference (FD) seismic wave propagation algorithm suffers from frequency- and direction-dependent numerical dispersion. The magnitude of this dispersion depends critically on FD algorithm implementation details. Nevertheless, proper gridding can control numerical dispersion to within an acceptable level, leading to accurate Rayleigh wave simulations. Many investigators have derived dispersion relations appropriate for body wave propagation by various FD algorithms. However, the situation for surface waves is less well-studied. We have devised a numerical search procedure to estimate Rayleigh phase speed and group speed curves for 3D O(2,2) and O(2,4) staggered-grid FD algorithms. In contrast with the continuous time-space situation (where phase speed is obtained by extracting the appropriate root of the Rayleigh cubic), we cannot develop a closed-form mathematical formula governing the phase speed. Rather, we numerically seek the particular phase speed that leads to a solution of the discrete wave propagation equations, while holding medium properties, frequency, horizontal propagation direction, and gridding intervals fixed. Group speed is then obtained by numerically differentiating the phase speed with respect to frequency. The problem is formulated for an explicit stress-free surface positioned at two different levels within the staggered spatial grid. Additionally, an interesting variant involving zero-valued medium properties above the surface is addressed. We refer to the latter as an implicit free surface. Our preliminary conclusion is that an explicit free surface, implemented with O(4) spatial FD
Finite difference analysis of rotordynamic seal coefficients for an eccentric shaft position
NASA Technical Reports Server (NTRS)
Nordmann, R.; Dietzen, F. J.
1989-01-01
The dynamic coefficients of seals are calculated for shaft movements around an eccentric position. The turbulent flow is described by the Navier-Stokes equations in connection with a turbulence model. The equations are solved by a finite-difference procedure.
Finite-difference, spectral and Galerkin methods for time-dependent problems
NASA Technical Reports Server (NTRS)
Tadmor, E.
1983-01-01
Finite difference, spectral and Galerkin methods for the approximate solution of time dependent problems are surveyed. A unified discussion on their accuracy, stability and convergence is given. In particular, the dilemma of high accuracy versus stability is studied in some detail.
NASA Technical Reports Server (NTRS)
Iida, H. T.
1966-01-01
Computational procedure reduces the numerical effort whenever the method of finite differences is used to solve ablation problems for which the surface recession is large relative to the initial slab thickness. The number of numerical operations required for a given maximum space mesh size is reduced.
Rupture Dynamics Simulation for Non-Planar fault by a Curved Grid Finite Difference Method
NASA Astrophysics Data System (ADS)
Zhang, Z.; Zhu, G.; Chen, X.
2011-12-01
We first implement the non-staggered finite difference method to solve the dynamic rupture problem, with split-node, for non-planar fault. Split-node method for dynamic simulation has been used widely, because of that it's more precise to represent the fault plane than other methods, for example, thick fault, stress glut and so on. The finite difference method is also a popular numeric method to solve kinematic and dynamic problem in seismology. However, previous works focus most of theirs eyes on the staggered-grid method, because of its simplicity and computational efficiency. However this method has its own disadvantage comparing to non-staggered finite difference method at some fact for example describing the boundary condition, especially the irregular boundary, or non-planar fault. Zhang and Chen (2006) proposed the MacCormack high order non-staggered finite difference method based on curved grids to precisely solve irregular boundary problem. Based upon on this non-staggered grid method, we make success of simulating the spontaneous rupture problem. The fault plane is a kind of boundary condition, which could be irregular of course. So it's convinced that we could simulate rupture process in the case of any kind of bending fault plane. We will prove this method is valid in the case of Cartesian coordinate first. In the case of bending fault, the curvilinear grids will be used.
Relative and Absolute Error Control in a Finite-Difference Method Solution of Poisson's Equation
ERIC Educational Resources Information Center
Prentice, J. S. C.
2012-01-01
An algorithm for error control (absolute and relative) in the five-point finite-difference method applied to Poisson's equation is described. The algorithm is based on discretization of the domain of the problem by means of three rectilinear grids, each of different resolution. We discuss some hardware limitations associated with the algorithm,…
Finite difference micromagnetic simulation with self-consistent currents and smooth surfaces
Cerjan, C; Gibbons, M R; Hewett, D W; Parker, G
1999-05-27
A micromagnetic algorithm has been developed using the finite difference method (FDM). Elliptic field equations are solved on the mesh using the efficient Dynamic Alternating Direction Implicit method. Smooth surfaces have been included in the FDM formulation so structures of irregular shape can be modeled. The current distribution and temperature of devices are also calculated. Keywords: Micromagnetic simulation, Magnetic dots, Read heads, Thermal Effects
A FINITE-DIFFERENCE, DISCRETE-WAVENUMBER METHOD FOR CALCULATING RADAR TRACES
A hybrid of the finite-difference method and the discrete-wavenumber method is developed to calculate radar traces. The method is based on a three-dimensional model defined in the Cartesian coordinate system; the electromagnetic properties of the model are symmetric with respect ...
A FINITE-DIFFERENCE, DISCRETE-WAVENUMBER METHOD FOR CALCULATING RADAR TRACES
A hybrid of the finite-difference method and the discrete-wavenumber method is developed to calculate radar traces. The method is based on a three-dimensional model defined in the Cartesian coordinate system; the electromag-netic properties of the model are symmetric with respect...
NASA Astrophysics Data System (ADS)
Wang, Enjiang; Liu, Yang; Sen, Mrinal K.
2016-09-01
The 2-D acoustic wave equation is commonly solved numerically by finite-difference (FD) methods in which the accuracy of solution is significantly affected by the FD stencils. The commonly used cross stencil can reach either only second-order accuracy for space domain dispersion-relation-based FD method or (2M)th-order accuracy along eight specific propagation directions for time-space domain dispersion-relation-based FD method, if the conventional (2M)th-order spatial FD and second-order temporal FD are used to discretize the equation. One other newly developed rhombus stencil can reach arbitrary even-order accuracy. However, this stencil adds significantly to computational cost when the operator length is large. To achieve a balance between the solution accuracy and efficiency, we develop a new FD stencil to solve the 2-D acoustic wave equation. This stencil is a combination of the cross stencil and rhombus stencil. A cross stencil with an operator length parameter M is used to approximate the spatial partial derivatives while a rhombus stencil with an operator length parameter N together with the conventional second-order temporal FD is employed in approximating the temporal partial derivatives. Using this stencil, a new FD scheme is developed; we demonstrate that this scheme can reach (2M)th-order accuracy in space and (2N)th-order accuracy in time when spatial FD coefficients and temporal FD coefficients are derived from respective dispersion relation using Taylor-series expansion (TE) method. To further increase the accuracy, we derive the FD coefficients by employing the time-space domain dispersion relation of this FD scheme using TE. We also use least-squares (LS) optimization method to reduce dispersion at high wavenumbers. Dispersion analysis, stability analysis and modelling examples demonstrate that our new scheme has greater accuracy and better stability than conventional FD schemes, and thus can adopt large time steps. To reduce the extra
NASA Astrophysics Data System (ADS)
Wang, Enjiang; Liu, Yang; Sen, Mrinal K.
2016-07-01
The 2D acoustic wave equation is commonly solved numerically by finite-difference (FD) methods in which the accuracy of solution is significantly affected by the FD stencils. The commonly used cross stencil can reach either only second-order accuracy for space domain dispersion-relation-based FD method or (2 M)th-order accuracy along eight specific propagation directions for time-space domain dispersion-relation-based FD method, if the conventional (2 M)th-order spatial FD and second-order temporal FD are used to discretize the equation. One other newly developed rhombus stencil can reach arbitrary even-order accuracy. However, this stencil adds significantly computational cost when the operator length is large. To achieve a balance between the solution accuracy and efficiency, we develop a new FD stencil to solve the 2D acoustic wave equation. This stencil is a combination of the cross stencil and rhombus stencil. A cross stencil with an operator length parameter M is used to approximate the spatial partial derivatives while a rhombus stencil with an operator length parameter N together with the conventional 2nd-order temporal FD is employed in approximating the temporal partial derivatives. Using this stencil, a new FD scheme is developed; we demonstrate that this scheme can reach (2 M)th-order accuracy in space and (2 N)th-order accuracy in time when spatial FD coefficients and temporal FD coefficients are derived from respective dispersion relation using Taylor-series expansion (TE) method. To further increase the accuracy, we derive the FD coefficients by employing the time-space domain dispersion relation of this FD scheme using TE. We also use least-squares (LS) optimization method to reduce dispersion at high wavenumbers. Dispersion analysis, stability analysis and modelling examples demonstrate that our new scheme has greater accuracy and better stability than conventional FD schemes, and thus can adopt large time steps. To reduce the extra computational
Appelo, D; Petersson, N A
2007-12-17
The isotropic elastic wave equation governs the propagation of seismic waves caused by earthquakes and other seismic events. It also governs the propagation of waves in solid material structures and devices, such as gas pipes, wave guides, railroad rails and disc brakes. In the vast majority of wave propagation problems arising in seismology and solid mechanics there are free surfaces. These free surfaces have, in general, complicated shapes and are rarely flat. Another feature, characterizing problems arising in these areas, is the strong heterogeneity of the media, in which the problems are posed. For example, on the characteristic length scales of seismological problems, the geological structures of the earth can be considered piecewise constant, leading to models where the values of the elastic properties are also piecewise constant. Large spatial contrasts are also found in solid mechanics devices composed of different materials welded together. The presence of curved free surfaces, together with the typical strong material heterogeneity, makes the design of stable, efficient and accurate numerical methods for the elastic wave equation challenging. Today, many different classes of numerical methods are used for the simulation of elastic waves. Early on, most of the methods were based on finite difference approximations of space and time derivatives of the equations in second order differential form (displacement formulation), see for example [1, 2]. The main problem with these early discretizations were their inability to approximate free surface boundary conditions in a stable and fully explicit manner, see e.g. [10, 11, 18, 20]. The instabilities of these early methods were especially bad for problems with materials with high ratios between the P-wave (C{sub p}) and S-wave (C{sub s}) velocities. For rectangular domains, a stable and explicit discretization of the free surface boundary conditions is presented in the paper [17] by Nilsson et al. In summary
NASA Technical Reports Server (NTRS)
Shu, Chi-Wang
1997-01-01
In these lecture notes we describe the construction, analysis, and application of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws and related Hamilton- Jacobi equations. ENO and WENO schemes are high order accurate finite difference schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics. These lecture notes are basically self-contained. It is our hope that with these notes and with the help of the quoted references, the reader can understand the algorithms and code them up for applications.
Accurate numerical solution of compressible, linear stability equations
NASA Technical Reports Server (NTRS)
Malik, M. R.; Chuang, S.; Hussaini, M. Y.
1982-01-01
The present investigation is concerned with a fourth order accurate finite difference method and its application to the study of the temporal and spatial stability of the three-dimensional compressible boundary layer flow on a swept wing. This method belongs to the class of compact two-point difference schemes discussed by White (1974) and Keller (1974). The method was apparently first used for solving the two-dimensional boundary layer equations. Attention is given to the governing equations, the solution technique, and the search for eigenvalues. A general purpose subroutine is employed for solving a block tridiagonal system of equations. The computer time can be reduced significantly by exploiting the special structure of two matrices.
NASA Technical Reports Server (NTRS)
DeBonis, James R.
2013-01-01
A computational fluid dynamics code that solves the compressible Navier-Stokes equations was applied to the Taylor-Green vortex problem to examine the code s ability to accurately simulate the vortex decay and subsequent turbulence. The code, WRLES (Wave Resolving Large-Eddy Simulation), uses explicit central-differencing to compute the spatial derivatives and explicit Low Dispersion Runge-Kutta methods for the temporal discretization. The flow was first studied and characterized using Bogey & Bailley s 13-point dispersion relation preserving (DRP) scheme. The kinetic energy dissipation rate, computed both directly and from the enstrophy field, vorticity contours, and the energy spectra are examined. Results are in excellent agreement with a reference solution obtained using a spectral method and provide insight into computations of turbulent flows. In addition the following studies were performed: a comparison of 4th-, 8th-, 12th- and DRP spatial differencing schemes, the effect of the solution filtering on the results, the effect of large-eddy simulation sub-grid scale models, and the effect of high-order discretization of the viscous terms.
Time-accurate unsteady aerodynamic and aeroelastic calculations for wings using Euler equations
NASA Technical Reports Server (NTRS)
Guruswamy, Guru P.
1988-01-01
A time-accurate approach to simultaneously solve the Euler flow equations and modal structural equations of motion is presented for computing aeroelastic responses of wings. The Euler flow eauations are solved by a time-accurate finite difference scheme with dynamic grids. The coupled aeroelastic equations of motion are solved using the linear acceleration method. The aeroelastic configuration adaptive dynamic grids are time accurately generated using the aeroelastically deformed shape of the wing. The unsteady flow calculations are validated wih experiment, both for a semi-infinite wing and a wall-mounted cantilever rectangular wings. Aeroelastic responses are computed for a rectangular wing using the modal data generated by the finite-element method. The robustness of the present approach in computing unsteady flows and aeroelastic responses that are beyond the capability of earlier approaches using the potential equations are demonstrated.
Jiménez-Osés, Gonzalo; García, José I; Corzana, Francisco; Elguero, José
2011-05-20
A new protocol combining classical MD simulations and DFT calculations is presented to accurately estimate the (1)H NMR chemical shifts of highly mobile guest-host systems and their thermal dependence. This strategy has been successfully applied for the hydrogen molecule trapped into C(60) fullerene, an unresolved and challenging prototypical case for which experimental values have never been reproduced. The dependence of the final values on the theoretical method and their implications to avoid over interpretation of the obtained results are carefully described.
Chai, Zhenhua; Zhao, T S
2014-07-01
In this paper, we propose a local nonequilibrium scheme for computing the flux of the convection-diffusion equation with a source term in the framework of the multiple-relaxation-time (MRT) lattice Boltzmann method (LBM). Both the Chapman-Enskog analysis and the numerical results show that, at the diffusive scaling, the present nonequilibrium scheme has a second-order convergence rate in space. A comparison between the nonequilibrium scheme and the conventional second-order central-difference scheme indicates that, although both schemes have a second-order convergence rate in space, the present nonequilibrium scheme is more accurate than the central-difference scheme. In addition, the flux computation rendered by the present scheme also preserves the parallel computation feature of the LBM, making the scheme more efficient than conventional finite-difference schemes in the study of large-scale problems. Finally, a comparison between the single-relaxation-time model and the MRT model is also conducted, and the results show that the MRT model is more accurate than the single-relaxation-time model, both in solving the convection-diffusion equation and in computing the flux.
NASA Technical Reports Server (NTRS)
Subrahmanyam, K. B.; Kaza, K. R. V.
1985-01-01
Theoretical natural frequencies of the first three modes of torsional vibration of pretwisted, rotating cantilever beams are determined for various thickness and aspect ratios. Conclusions concerning individual and collective effects of warping, pretwist, tension-torsion coupling and tennis racket effect (twist-rotational coupling) terms on the natural frequencies are drawn from numerical results obtained by using a finite difference procedure with first order central differences. The relative importance of structural warping, inertial warping, pretwist, tension-torsion and twist-rotational coupling terms is discussed for various rotational speeds. The accuracy of results obtained by using the finite difference approach is verified by a comparison with the exact solution for specialized simple cases of the equation of motion used in this paper.
NASA Astrophysics Data System (ADS)
Dai, Liyi
2016-05-01
Stochastic optimization is a fundamental problem that finds applications in many areas including biological and cognitive sciences. The classical stochastic approximation algorithm for iterative stochastic optimization requires gradient information of the sample object function that is typically difficult to obtain in practice. Recently there has been renewed interests in derivative free approaches to stochastic optimization. In this paper, we examine the rates of convergence for the Kiefer-Wolfowitz algorithm and the mirror descent algorithm, by approximating gradient using finite differences generated through common random numbers. It is shown that the convergence of these algorithms can be accelerated by controlling the implementation of the finite differences. Particularly, it is shown that the rate can be increased to n-2/5 in general and to n-1/2, the best possible rate of stochastic approximation, in Monte Carlo optimization for a broad class of problems, in the iteration number n.
On One-Dimensional Stretching Functions for Finite-Difference Calculations
NASA Technical Reports Server (NTRS)
Vinokur, M.
1980-01-01
The class of one dimensional stretching function used in finite difference calculations is studied. For solutions containing a highly localized region of rapid variation, simple criteria for a stretching function are derived using a truncation error analysis. These criteria are used to investigate two types of stretching functions. One is an interior stretching function, for which the location and slope of an interior clustering region are specified. The simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine. The other type of function is a two sided stretching function, for which the arbitrary slopes at the two ends of the one dimensional interval are specified. The simplest such general function is found to be one based on the inverse tangent. The general two sided function has many applications in the construction of finite difference grids.
Finite difference methods for transient signal propagation in stratified dispersive media
NASA Technical Reports Server (NTRS)
Lam, D. H.
1975-01-01
Explicit difference equations are presented for the solution of a signal of arbitrary waveform propagating in an ohmic dielectric, a cold plasma, a Debye model dielectric, and a Lorentz model dielectric. These difference equations are derived from the governing time-dependent integro-differential equations for the electric fields by a finite difference method. A special difference equation is derived for the grid point at the boundary of two different media. Employing this difference equation, transient signal propagation in an inhomogeneous media can be solved provided that the medium is approximated in a step-wise fashion. The solutions are generated simply by marching on in time. It is concluded that while the classical transform methods will remain useful in certain cases, with the development of the finite difference methods described, an extensive class of problems of transient signal propagating in stratified dispersive media can be effectively solved by numerical methods.
NASA Astrophysics Data System (ADS)
Bohlen, Thomas; Wittkamp, Florian
2016-03-01
We analyse the performance of a higher order accurate staggered viscoelastic time-domain finite-difference method, in which the staggered Adams-Bashforth (ABS) third-order and fourth-order accurate time integrators are used for temporal discretization. ABS is a multistep method that uses previously calculated wavefields to increase the order of accuracy in time. The analysis shows that the numerical dispersion is much lower than that of the widely used second-order leapfrog method. Numerical dissipation is introduced by the ABS method which is significantly smaller for fourth-order than third-order accuracy. In 1-D and 3-D simulation experiments, we verify the convincing improvements of simulation accuracy of the fourth-order ABS method. In a realistic elastic 3-D scenario, the computing time reduces by a factor of approximately 2.4, whereas the memory requirements increase by approximately a factor of 2.2. The ABS method thus provides an alternative strategy to increase the simulation accuracy in time by investing computer memory instead of computing time.
Coomar, Arunima; Arntsen, Christopher; Lopata, Kenneth A; Pistinner, Shlomi; Neuhauser, Daniel
2011-08-28
We develop near-field (NF), a very efficient finite-difference time-dependent (FDTD) approach for simulating electromagnetic systems in the near-field regime. NF is essentially a time-dependent version of the quasistatic frequency-dependent Poisson algorithm. We assume that the electric field is longitudinal, and hence propagates only a set of time-dependent polarizations and currents. For near-field scales, the time step (dt) is much larger than in the usual Maxwell FDTD approach, as it is not related to the velocity of light; rather, it is determined by the rate of damping and plasma oscillations in the material, so dt = 2.5 a.u. was well converged in our simulations. The propagation in time is done via a leapfrog algorithm much like Yee's method, and only a single spatial convolution is needed per time step. In conjunction, we also develop a new and very accurate 8 and 9 Drude-oscillators fit to the permittivity of gold and silver, desired here because we use a large time step. We show that NF agrees with Mie-theory in the limit of small spheres and that it also accurately describes the evolution of the spectral shape as a function of the separation between two gold or silver spheres. The NF algorithm is especially efficient for systems with small scale dynamics and makes it very simple to introduce additional effects such as embedding.
Zhao, Huawei; Crozier, Stuart; Liu, Feng
2002-12-01
Numerical modeling of the eddy currents induced in the human body by the pulsed field gradients in MRI presents a difficult computational problem. It requires an efficient and accurate computational method for high spatial resolution analyses with a relatively low input frequency. In this article, a new technique is described which allows the finite difference time domain (FDTD) method to be efficiently applied over a very large frequency range, including low frequencies. This is not the case in conventional FDTD-based methods. A method of implementing streamline gradients in FDTD is presented, as well as comparative analyses which show that the correct source injection in the FDTD simulation plays a crucial rule in obtaining accurate solutions. In particular, making use of the derivative of the input source waveform is shown to provide distinct benefits in accuracy over direct source injection. In the method, no alterations to the properties of either the source or the transmission media are required. The method is essentially frequency independent and the source injection method has been verified against examples with analytical solutions. Results are presented showing the spatial distribution of gradient-induced electric fields and eddy currents in a complete body model.
NASA Technical Reports Server (NTRS)
Steger, J. L.; Caradonna, F. X.
1980-01-01
An implicit finite difference procedure is developed to solve the unsteady full potential equation in conservation law form. Computational efficiency is maintained by use of approximate factorization techniques. The numerical algorithm is first order in time and second order in space. A circulation model and difference equations are developed for lifting airfoils in unsteady flow; however, thin airfoil body boundary conditions have been used with stretching functions to simplify the development of the numerical algorithm.
Transport and dispersion of pollutants in surface impoundments: a finite difference model
Yeh, G.T.
1980-07-01
A surface impoundment model by finite-difference (SIMFD) has been developed. SIMFD computes the flow rate, velocity field, and the concentration distribution of pollutants in surface impoundments with any number of islands located within the region of interest. Theoretical derivations and numerical algorithm are described in detail. Instructions for the application of SIMFD and listings of the FORTRAN IV source program are provided. Two sample problems are given to illustrate the application and validity of the model.
Simulation of realistic rotor blade-vortex interactions using a finite-difference technique
NASA Technical Reports Server (NTRS)
Hassan, Ahmed A.; Charles, Bruce D.
1989-01-01
A numerical finite-difference code has been used to predict helicopter blade loads during realistic self-generated three-dimensional blade-vortex interactions. The velocity field is determined via a nonlinear superposition of the rotor flowfield. Data obtained from a lifting-line helicopter/rotor trim code are used to determine the instantaneous position of the interaction vortex elements with respect to the blade. Data obtained for three rotor advance ratios show a reasonable correlation with wind tunnel data.
Finite-difference, time-domain analysis of a folded acoustic transmission line.
Jackson, Charles M
2005-03-01
Recently designed, modern versions of renais sance woodwind instruments such as the recorder and serpent use square cross sections and a folded acoustic transmission line. Conventional microwave techniques would expect that this bend would cause unwanted reflections and impedance discontinuities. This paper analyses the folded acoustic transmission line using finite-difference, time-domain techniques and shows that the discontinuity can be compensated with by the use of a manufacturable method. PMID:15857045
Numerical techniques in linear duct acoustics. [finite difference and finite element analyses
NASA Technical Reports Server (NTRS)
Baumeister, K. J.
1980-01-01
Both finite difference and finite element analyses of small amplitude (linear) sound propagation in straight and variable area ducts with flow, as might be found in a typical turboject engine duct, muffler, or industrial ventilation system, are reviewed. Both steady state and transient theories are discussed. Emphasis is placed on the advantages and limitations associated with the various numerical techniques. Examples of practical problems are given for which the numerical techniques have been applied.
NASA Technical Reports Server (NTRS)
Masiulaniec, K. C.; Keith, T. G., Jr.; Dewitt, K. J.
1984-01-01
A numerical procedure is presented for analyzing a wide variety of heat conduction problems in multilayered bodies having complex geometry. The method is based on a finite difference solution of the heat conduction equation using a body fitted coordinate system transformation. Solution techniques are described for steady and transient problems with and without internal energy generation. Results are found to compare favorably with several well known solutions.
Tamil, L S; Mitra, S S; Dutta, R; Pereira, J M
1991-03-20
A simple numerical method based on finite differences to solve the scalar wave equation encountered in optical fibers is presented. The method uses the Ricatti transformation to the scalar wave equation and is capable of analyzing fibers of arbitrary refractive index profiles. Achieving errors as low as 0.005% for propagation constants is possible for lower-order modes. However, the error seems to increase for frequencies closer to cutoff.
High-Order Energy Stable WENO Schemes
NASA Technical Reports Server (NTRS)
Yamaleev, Nail K.; Carpenter, Mark H.
2008-01-01
A new third-order Energy Stable Weighted Essentially NonOscillatory (ESWENO) finite difference scheme for scalar and vector linear hyperbolic equations with piecewise continuous initial conditions is developed. The new scheme is proven to be stable in the energy norm for both continuous and discontinuous solutions. In contrast to the existing high-resolution shock-capturing schemes, no assumption that the reconstruction should be total variation bounded (TVB) is explicitly required to prove stability of the new scheme. A rigorous truncation error analysis is presented showing that the accuracy of the 3rd-order ESWENO scheme is drastically improved if the tuning parameters of the weight functions satisfy certain criteria. Numerical results show that the new ESWENO scheme is stable and significantly outperforms the conventional third-order WENO finite difference scheme of Jiang and Shu in terms of accuracy, while providing essentially nonoscillatory solutions near strong discontinuities.
NASA Astrophysics Data System (ADS)
Dou, Hu; Ma, Hongmei; Sun, Yu-Bao
2016-09-01
The finite-difference time-domain method is used to simulate the optical characteristics of an in-plane switching blue phase liquid crystal display. Compared with the matrix optic methods and the refractive method, the finite-difference time-domain method, which is used to directly solve Maxwell’s equations, can consider the lateral variation of the refractive index and obtain an accurate convergence effect. The simulation results show that e-rays and o-rays bend in different directions when the in-plane switching blue phase liquid crystal display is driven by the operating voltage. The finite-difference time-domain method should be used when the distribution of the liquid crystal in the liquid crystal display has a large lateral change. Project supported by the National Natural Science Foundation of China (Grant Nos. 11304074, 61475042, and 11274088), the Natural Science Foundation of Hebei Province, China (Grant Nos. A2015202320 and GCC2014048), and the Key Subject Construction Project of Hebei Province University, China.
NASA Technical Reports Server (NTRS)
Scott, James R.; Atassi, Hafiz M.
1991-01-01
A numerical method is developed for solving periodic, three-dimensional, vortical flows around lifting airfoils in subsonic flow. The first-order method, that is presented, fully accounts for the distortion effects of the nonuniform mean flow on the convected upstream vortical disturbances. The unsteady velocity is split into a vortical component which is a known function of the upstream flow conditions and the Lagrangian coordinates of the mean flow, and an irrotational field whose potential satisfies a nonconstant-coefficient, inhomogeneous, convective wave equation. Using an elliptic coordinate transformation, the unsteady boundary value problem is solved in the frequency domain on grids which are determined as a function of the Mach number and reduced frequency. Extensive comparisons are made with known solutions to unsteady vortical flow problems, and it is seen that the agreement is generally very good for reduced frequencies ranging from 0 up to 4.
Riley, D.J.
1993-04-01
A technique to integrate a dense, locally non-uniform mesh into finite-difference time-domain (FDTD) codes is presented. The method is designed for the full-wave analysis of multi-material layers that are physically thin, but perhaps electrically thick. Such layers are often used for the purpose of suppressing electromagnetic reflections from conducting surfaces. Throughout the non-uniform local mesh, average values for the conductivity and permittivity are used, where as variations in permeability are accommodated by splitting H-field line integrals and enforcing continuity of the normal B field. A unique interpolation scheme provides accuracy and late-time stability for mesh discontinuities as large as 1000 to 1. Application is made to resistive sheets, the absorbing Salisbury screen, crosstalk on printed circuit boards, and apertures that are narrow both in width and depth with regard to a uniform cell. Where appropriate, comparisons are made with the MoM code CARLOS and transmission-line theory. The hybrid mesh formulation has been highly optimized for both vector and parallel-processing on Cray YMP architectures.
Silva, F. da; Hacquin, S.
2005-03-01
We present a novel numerical signal injection technique allowing unidirectional injection of a wave in a wave-guiding structure, applicable to 2D finite-difference time-domain electromagnetic codes, both Maxwell and wave-equation. It is particularly suited to continuous wave radar-like simulations. The scheme gives an unidirectional injection of a signal while being transparent to waves propagating in the opposite direction (directional coupling). The reflected or backscattered waves (returned) are separated from the probing waves allowing direct access to the information on amplitude and phase of the returned wave. It also facilitates the signal processing used to extract the phase derivative (or group delay) when simulating radar systems. Although general, the technique is particularly suited to swept frequency sources (frequency modulated) in the context of reflectometry, a fusion plasma diagnostic. The UTS applications presented here are restricted to fusion plasma reflectometry simulations for different physical situations. This method can, nevertheless, also be used in other dispersive media such as dielectrics, being useful, for example, in the simulation of plasma filled waveguides or directional couplers.
NASA Technical Reports Server (NTRS)
Lansing, Faiza S.; Rascoe, Daniel L.
1993-01-01
This paper presents a modified Finite-Difference Time-Domain (FDTD) technique using a generalized conformed orthogonal grid. The use of the Conformed Orthogonal Grid, Finite Difference Time Domain (GFDTD) enables the designer to match all the circuit dimensions, hence eliminating a major source o error in the analysis.
NASA Technical Reports Server (NTRS)
Hamilton, H. Harris, II; Millman, Daniel R.; Greendyke, Robert B.
1992-01-01
A computer code was developed that uses an implicit finite-difference technique to solve nonsimilar, axisymmetric boundary layer equations for both laminar and turbulent flow. The code can treat ideal gases, air in chemical equilibrium, and carbon tetrafluoride (CF4), which is a useful gas for hypersonic blunt-body simulations. This is the only known boundary layer code that can treat CF4. Comparisons with experimental data have demonstrated that accurate solutions are obtained. The method should prove useful as an analysis tool for comparing calculations with wind tunnel experiments and for making calculations about flight vehicles where equilibrium air chemistry assumptions are valid.
Zherdetsky, Aleksej; Prakonina, Alena; Malony, Allen D.
2014-01-01
The Electrical Impedance Tomography (EIT) and electroencephalography (EEG) forward problems in anisotropic inhomogeneous media like the human head belongs to the class of the three-dimensional boundary value problems for elliptic equations with mixed derivatives. We introduce and explore the performance of several new promising numerical techniques, which seem to be more suitable for solving these problems. The proposed numerical schemes combine the fictitious domain approach together with the finite-difference method and the optimally preconditioned Conjugate Gradient- (CG-) type iterative method for treatment of the discrete model. The numerical scheme includes the standard operations of summation and multiplication of sparse matrices and vector, as well as FFT, making it easy to implement and eligible for the effective parallel implementation. Some typical use cases for the EIT/EEG problems are considered demonstrating high efficiency of the proposed numerical technique. PMID:24527060
Turovets, Sergei; Volkov, Vasily; Zherdetsky, Aleksej; Prakonina, Alena; Malony, Allen D
2014-01-01
The Electrical Impedance Tomography (EIT) and electroencephalography (EEG) forward problems in anisotropic inhomogeneous media like the human head belongs to the class of the three-dimensional boundary value problems for elliptic equations with mixed derivatives. We introduce and explore the performance of several new promising numerical techniques, which seem to be more suitable for solving these problems. The proposed numerical schemes combine the fictitious domain approach together with the finite-difference method and the optimally preconditioned Conjugate Gradient- (CG-) type iterative method for treatment of the discrete model. The numerical scheme includes the standard operations of summation and multiplication of sparse matrices and vector, as well as FFT, making it easy to implement and eligible for the effective parallel implementation. Some typical use cases for the EIT/EEG problems are considered demonstrating high efficiency of the proposed numerical technique. PMID:24527060
NASA Astrophysics Data System (ADS)
Ren, Zhiming; Liu, Yang; Zhang, Qunshan
2014-05-01
Full waveform inversion (FWI) has the potential to provide preferable subsurface model parameters. The main barrier of its applications to real seismic data is heavy computational amount. Numerical modelling methods are involved in both forward modelling and backpropagation of wavefield residuals, which spend most of computational time in FWI. We develop a time-space domain finite-difference (FD) method and adaptive variable-length spatial operator scheme in numerical simulation of viscoacoustic equation and extend them into the viscoacoustic FWI. Compared with conventional FD methods, different operator lengths are adopted for different velocities and quality factors, which can reduce the amount of computation without reducing accuracy. Inversion algorithms also play a significant role in FWI. In conventional single-scale methods, it is likely to converge to local minimums especially when the initial model is far from the real model. To tackle the problem, we introduce the second generation wavelet transform to implement the multiscale FWI. Compared to other multiscale methods, our method has advantages of ease of implementation and better time-frequency local analysis ability. The L2 norm is widely used in FWI and gives invalid model estimates when the data is contaminated with strong non-uniform noises. We apply the L1-norm and the Huber-norm criteria in the time-domain FWI to improve its antinoise ability. Our strategies have been successfully applied in synthetic experiments to both onshore and offshore reflection seismic data. The results of the viscoacoustic Marmousi example indicate that our new FWI scheme consumes smaller computer resources. In addition, the viscoacoustic Overthrust example shows its better convergence and more reasonable velocity and quality factor structures. All these results demonstrate that our method can improve inversion accuracy and computational efficiency of FWI.
NASA Astrophysics Data System (ADS)
Sallares, V.; Kormann, J.; Cobo, P.; Biescas, B.; Carbonell, R.
2007-05-01
Holbrook et al. (2003) demonstrated recently the possibility of visualizing fine structures in the water column, like thermohaline intrusion or internal waves, through seismic exploration experiments. Seismic exploration is becoming a popular technique for providing high-lateral resolution images of the explored area, in contrast with the classical oceanography probes, like XBT or XCDT. In this work we present a wave propagation model based upon a high order finite-differences time-domain (FDTD) scheme which includes special absorbing conditions in the boundaries. FDTD algorithms are known for presenting problems with reflections on the computational edges. Classical boundary conditions, like those of Engquist, provide reflection coefficients or the order of 10-2. However, reflection coefficients of fine structures in the water we are trying to model are about 10-4. Thus, the key point of the algorithm we present is in the implementation of Perfectly Matched Layer (PML) boundary conditions. These consist in zones with high absorption (therefore, very low reflection coefficient). The PML implemented in this scheme consists in a second order algorithm in the time domain, to take advantage of its stability and convergence properties. In this work we specify the propagation algorithm, and compare it results with the with Engquist and PML absorbing boundaries conditions. The PML condition affords reflection coefficients in the numerical edges lower than 10-4. Holbrook, W.S., Paramo, P., Pearse, S. and Schmitt, R.W., 2003. Thermohaline fine structure in an oceanographic front from seismic reflection profiling. Science, 301, 821-824.
Harbaugh, Arlen W.
1992-01-01
The U.S. Geological Survey's Modular Ground-Water Flow Model assumes that model nodes are in the center of cells and that transmissivity is constant within a cell. Based on these assumptions, the model calculates coefficients, called conductance, that are multiplied by head difference to determine flow between cells. Although these are common assumptions in finite-difference models, other assumptions are possible. A new option to the model program reads conductance as input data rather than calculating it. This optional lows the user to calculate conductance outside of the model. The user thus has the flexibility to define conductance using any desired assumptions. For a water-table condition, horizontal conductance must change as water level varies. To handle this situation, the new option reads conductance divided by thickness (CDT) as input data. The model calculates saturated thickness and multiplies it by CDT to obtain conductance. Thus, the user is still free from the assumptions of centered nodes and constant transmissivity in cells. The model option is written in FORTRAN77 and is fully compatible with the existing model. This report documents the new model option; it includes a description of the concepts, detailed input instructions, and a listing of the code.
Hunt, R.J.; Anderson, M.P.; Kelson, V.A.
1998-01-01
This paper demonstrates that analytic element models have potential as powerful screening tools that can facilitate or improve calibration of more complicated finite-difference and finite-element models. We demonstrate how a two-dimensional analytic element model was used to identify errors in a complex three-dimensional finite-difference model caused by incorrect specification of boundary conditions. An improved finite-difference model was developed using boundary conditions developed from a far-field analytic element model. Calibration of a revised finite-difference model was achieved using fewer zones of hydraulic conductivity and lake bed conductance than the original finite-difference model. Calibration statistics were also improved in that simulated base-flows were much closer to measured values. The improved calibration is due mainly to improved specification of the boundary conditions made possible by first solving the far-field problem with an analytic element model.This paper demonstrates that analytic element models have potential as powerful screening tools that can facilitate or improve calibration of more complicated finite-difference and finite-element models. We demonstrate how a two-dimensional analytic element model was used to identify errors in a complex three-dimensional finite-difference model caused by incorrect specification of boundary conditions. An improved finite-difference model was developed using boundary conditions developed from a far-field analytic element model. Calibration of a revised finite-difference model was achieved using fewer zones of hydraulic conductivity and lake bed conductance than the original finite-difference model. Calibration statistics were also improved in that simulated base-flows were much closer to measured values. The improved calibration is due mainly to improved specification of the boundary conditions made possible by first solving the far-field problem with an analytic element model.
A modular three-dimensional finite-difference ground-water flow model
McDonald, Michael G.; Harbaugh, Arlen W.
1988-01-01
This report presents a finite-difference model and its associated modular computer program. The model simulates flow in three dimensions. The report includes detailed explanations of physical and mathematical concepts on which the model is based and an explanation of how those concepts are incorporated in the modular structure of the computer program. The modular structure consists of a Main Program and a series of highly independent subroutines called 'modules.' The modules are grouped into 'packages.' Each package deals with a specific feature of the hydrologic system which is to be simulated, such as flow from rivers or flow into drains, or with a specific method of solving linear equations which describe the flow system, such as the Strongly Implicit Procedure or Slice-Successive Overrelaxation. The division of the program into modules permits the user to examine specific hydrologic features of the model independently. This also facilita development of additional capabilities because new packages can be added to the program without modifying the existing packages. The input and output systems of the computer program are also designed to permit maximum flexibility. Ground-water flow within the aquifer is simulated using a block-centered finite-difference approach. Layers can be simulated as confined, unconfined, or a combination of confined and unconfined. Flow associated with external stresses, such as wells, areal recharge, evapotranspiration, drains, and streams, can also be simulated. The finite-difference equations can be solved using either the Strongly Implicit Procedure or Slice-Successive Overrelaxation. The program is written in FORTRAN 77 and will run without modification on most computers that have a FORTRAN 77 compiler. For each program ,module, this report includes a narrative description, a flow chart, a list of variables, and a module listing.
A Modular Three-Dimensional Finite-Difference Ground-Water Flow Model
McDonald, Michael G.; Harbaugh, Arlen W.; Guo, Weixing; Lu, Guoping
1988-01-01
This report presents a finite-difference model and its associated modular computer program. The model simulates flow in three dimensions. The report includes detailed explanations of physical and mathematical concepts on which the model is based and an explanation of how those concepts are incorporated in the modular structure of the computer program. The modular structure consists of a Main Program and a series of highly independent subroutines called 'modules.' The modules are grouped into 'packages.' Each package deals with a specific feature of the hydrologic system which is to be simulated, such as flow from rivers or flow into drains, or with a specific method of solving linear equations which describe the flow system, such as the Strongly Implicit Procedure or Slice-Successive Overrelaxation. The division of the program into modules permits the user to examine specific hydrologic features of the model independently. This also facilita development of additional capabilities because new packages can be added to the program without modifying the existing packages. The input and output systems of the computer program are also designed to permit maximum flexibility. Ground-water flow within the aquifer is simulated using a block-centered finite-difference approach. Layers can be simulated as confined, unconfined, or a combination of confined and unconfined. Flow associated with external stresses, such as wells, areal recharge, evapotranspiration, drains, and streams, can also be simulated. The finite-difference equations can be solved using either the Strongly Implicit Procedure or Slice-Successive Overrelaxation. The program is written in FORTRAN 77 and will run without modification on most computers that have a FORTRAN 77 compiler. For each program ,module, this report includes a narrative description, a flow chart, a list of variables, and a module listing.
A modular three-dimensional finite-difference ground-water flow model
McDonald, M.G.; Harbaugh, A.W.
1984-01-01
This report presents a finite-difference model and its associated modular computer program. The model simulates flow in three dimensions. The report includes detailed explanations of physical and mathematical concepts on which the model is based and an explanation of how those concepts were incorporated in the modular structure of the computer program. The modular structure consists of a Main Program and a series of highly independent subroutines called 'modules.' The modules are grouped into 'packages.' Each package deals with a specific feature of the hydrologic system which is to be simulated, such as flow from rivers or flow into drains, or with a specific method of solving linear equations which describe the flow system, such as the Strongly Implicit Procedure or Slice-Successive Overrelaxation. The division of the program into modules permits the user to examine specific hydrologic features of the model independently. This also facilitates development of additional capabilities because new modules or packages can be added to the program without modifying the existing modules or packages. The input and output systems of the computer program are also designed to permit maximum flexibility. Ground-water flow within the aquifer is simulated using a block-centered finite-difference approach. Layers can be simulated as confined, unconfined, or a combination of confined and unconfined. Flow from external stresses, such as flow to wells, areal recharge, evapotranspiration, flow to drains, and flow through riverbeds, can also be simulated. The finite-difference equations can be solved using either the Strongly Implicit Procedure or Slice-Successive Overrelaxation. The program is written in FORTRAN '66 and will run without modification on most computers which have a FORTRAN '66 compiler. It will also run, without modification, with most extended FORTRAN '77 compilers and with minor modifications on standard FORTRAN '77 compilers. Documentation presented in this report
A time-space domain stereo finite difference method for 3D scalar wave propagation
NASA Astrophysics Data System (ADS)
Chen, Yushu; Yang, Guangwen; Ma, Xiao; He, Conghui; Song, Guojie
2016-11-01
The time-space domain finite difference methods reduce numerical dispersion effectively by minimizing the error in the joint time-space domain. However, their interpolating coefficients are related with the Courant numbers, leading to significantly extra time costs for loading the coefficients consecutively according to velocity in heterogeneous models. In the present study, we develop a time-space domain stereo finite difference (TSSFD) method for 3D scalar wave equation. The method propagates both the displacements and their gradients simultaneously to keep more information of the wavefields, and minimizes the maximum phase velocity error directly using constant interpolation coefficients for different Courant numbers. We obtain the optimal constant coefficients by combining the truncated Taylor series approximation and the time-space domain optimization, and adjust the coefficients to improve the stability condition. Subsequent investigation shows that the TSSFD can suppress numerical dispersion effectively with high computational efficiency. The maximum phase velocity error of the TSSFD is just 3.09% even with only 2 sampling points per minimum wavelength when the Courant number is 0.4. Numerical experiments show that to generate wavefields with no visible numerical dispersion, the computational efficiency of the TSSFD is 576.9%, 193.5%, 699.0%, and 191.6% of those of the 4th-order and 8th-order Lax-Wendroff correction (LWC) method, the 4th-order staggered grid method (SG), and the 8th-order optimal finite difference method (OFD), respectively. Meanwhile, the TSSFD is compatible to the unsplit convolutional perfectly matched layer (CPML) boundary condition for absorbing artificial boundaries. The efficiency and capability to handle complex velocity models make it an attractive tool in imaging methods such as acoustic reverse time migration (RTM).
Generating meshes for finite-difference analysis using a solid modeler
NASA Astrophysics Data System (ADS)
Laguna, G. W.; White, W. T.; Cabral, B. K.
1987-09-01
One tool used by the Engineering Research Division of LLNL to help analyze the behavior of electronic systems in hostile environments is 3D finite-difference time-domain (FDTD) computation. FDTD codes solve Maxwell's equations,the differential equations of electromagnetism, on a uniform lattice of points. It is this uniform lattice, or mesh, that distinguishes finite-difference codes from other codes. The simple mesh makes FDTD codes computationally more efficient than other codes, which enables them to run larger problems and to run faster (up to thirty times faster than finite-element codes, for example). Therefore, within the Engineering Department at LLNL, Electronics Engineering (EE) has initiated a project to develop a mesh generator that will provide meshes suitable for FDTD analysis. This report describes the results of the first year of EE's FDTD Mesh Generation Project. During this year a preliminary version of an automated mesh generator was built and used to create a mesh of an object of interest to the High-Power Microwave Program, namely an electrically detonatable land mine. The code was verified by meshing basic solids such as spheres and cylinders. Because of the design of the code, there is no software limitation to the size of meshes that can be accommodated. The algorithm with a mesh space of approximately 500,000 cells has been demonstrated. The mesh generator can detect certain objects with walls that are thinner than the width of a cell. The code has internal graphics for viewing objects as they appear prior to being converted to a finite-difference representation. Additionally, via data files, the code is coupled to two external graphics packages for visually checking the meshes, namely TAURUS on the Cray and a new code, IMAGE, on the Silicon Graphics IRIS workstation.
A practical numerical scheme for the ternary Cahn-Hilliard system with a logarithmic free energy
NASA Astrophysics Data System (ADS)
Jeong, Darae; Kim, Junseok
2016-01-01
We consider a practically stable finite difference method for the ternary Cahn-Hilliard system with a logarithmic free energy modeling the phase separation of a three-component mixture. The numerical scheme is based on a linear unconditionally gradient stable scheme by Eyre and is solved by an efficient and accurate multigrid method. The logarithmic function has a singularity at zero. To remove the singularity, we regularize the function near zero by using a quadratic polynomial approximation. We perform a convergence test, a linear stability analysis, and a robustness test of the ternary Cahn-Hilliard equation. We observe that our numerical solutions are convergent, consistent with the exact solutions of linear stability analysis, and stable with practically large enough time steps. Using the proposed numerical scheme, we also study the temporal evolution of morphology patterns during phase separation in one-, two-, and three-dimensional spaces.
Supercomputing of supersonic flows using upwind relaxation and MacCormack schemes
NASA Technical Reports Server (NTRS)
Baysal, O.
1987-01-01
The performance of two numerical solution schemes, (1) an implicit upwind relaxation with a finite-volume discretization (Thomas and Walters, 1985) and (2) an explicit-implicit MacCormack (1981) scheme with a finite-difference discretization, is compared in two-dimensional simulations of supersonic flow past a flat plate with leading edge, a rearward-facing step, a 10-deg compression corner, a NACA 0012 airfoil at high angle of attack, and a cavity. The algebraic turbulence model, the solution methods, and the boundary conditions and SIMD coding are explained, and the results are presented in tables and graphs and characterized with reference to published experimental data. Scheme (1) is found to converge more rapidly and to give more accurate results than (2) in a wide range of problem types.
Higher-Order Compact Schemes for Numerical Simulation of Incompressible Flows
NASA Technical Reports Server (NTRS)
Wilson, Robert V.; Demuren, Ayodeji O.; Carpenter, Mark
1998-01-01
A higher order accurate numerical procedure has been developed for solving incompressible Navier-Stokes equations for 2D or 3D fluid flow problems. It is based on low-storage Runge-Kutta schemes for temporal discretization and fourth and sixth order compact finite-difference schemes for spatial discretization. The particular difficulty of satisfying the divergence-free velocity field required in incompressible fluid flow is resolved by solving a Poisson equation for pressure. It is demonstrated that for consistent global accuracy, it is necessary to employ the same order of accuracy in the discretization of the Poisson equation. Special care is also required to achieve the formal temporal accuracy of the Runge-Kutta schemes. The accuracy of the present procedure is demonstrated by application to several pertinent benchmark problems.
Rotordynamic coefficients for labyrinth seals calculated by means of a finite difference technique
NASA Technical Reports Server (NTRS)
Nordmann, R.; Weiser, P.
1989-01-01
The compressible, turbulent, time dependent and three dimensional flow in a labyrinth seal can be described by the Navier-Stokes equations in conjunction with a turbulence model. Additionally, equations for mass and energy conservation and an equation of state are required. To solve these equations, a perturbation analysis is performed yielding zeroth order equations for centric shaft position and first order equations describing the flow field for small motions around the seal center. For numerical solution a finite difference method is applied to the zeroth and first order equations resulting in leakage and dynamic seal coefficients respectively.
A finite-difference program for stresses in anisotropic, layered plates in bending
NASA Technical Reports Server (NTRS)
Salamon, N. J.
1975-01-01
The interlaminar stresses induced in a layered laminate that is bent into a cylindrical surface are studied. The laminate is modeled as a continuum, and the resulting elasticity equations are solved using the finite difference method. The report sets forth the mathematical framework, presents some preliminary results, and provides a listing and explanation of the computer program. Significant among the results are apparent symmetry relationships that will reduce the numerical size of certain problems and an interlaminar stress behavior having a sharp rise at the free edges.
Polarization-current-based, finite-difference time-domain, near-to-far-field transformation.
Zeng, Yong; Moloney, Jerome V
2009-05-15
A near-to-far-field transformation algorithm for three-dimensional finite-difference time-domain is presented in this Letter. This approach is based directly on the polarization current of the scatterer, not the scattered near fields. It therefore eliminates the numerical errors originating from the spatial offset of the E and H fields, inherent in the standard near-to-far-field transformation. The proposed method is validated via direct comparisons with the analytical Lorentz-Mie solutions of plane waves scattered by large dielectric and metallic spheres with strong forward-scattering lobes. PMID:19448834
NASA Technical Reports Server (NTRS)
Ghil, M.; Balgovind, R.
1979-01-01
The inhomogeneous Cauchy-Riemann equations in a rectangle are discretized by a finite difference approximation. Several different boundary conditions are treated explicitly, leading to algorithms which have overall second-order accuracy. All boundary conditions with either u or v prescribed along a side of the rectangle can be treated by similar methods. The algorithms presented here have nearly minimal time and storage requirements and seem suitable for development into a general-purpose direct Cauchy-Riemann solver for arbitrary boundary conditions.
The electromagnetic modeling of thin apertures using the finite-difference time-domain technique
NASA Technical Reports Server (NTRS)
Demarest, Kenneth R.
1987-01-01
A technique which computes transient electromagnetic responses of narrow apertures in complex conducting scatterers was implemented as an extension of previously developed Finite-Difference Time-Domain (FDTD) computer codes. Although these apertures are narrow with respect to the wavelengths contained within the power spectrum of excitation, this technique does not require significantly more computer resources to attain the increased resolution at the apertures. In the report, an analytical technique which utilizes Babinet's principle to model the apertures is developed, and an FDTD computer code which utilizes this technique is described.
WONDY V: a one-dimensional finite-difference wave-propagation code
Kipp, M.E.; Lawrence, R.J.
1982-06-01
WONDY V solves the finite difference analogs to the Lagrangian equations of motion in one spatial dimension (planar, cylindrical, or spherical). Simulations of explosive detonation, energy deposition, plate impact, and dynamic fracture are possible, using a variety of existing material models. In addition, WONDY has proven to be a powerful tool in the evaluation of new constitutive models. A preprocessor is available to allocate storage arrays commensurate with problem size, and automatic rezoning may be employed to improve resolution. This document provides a description of the equations solved, available material models, operating instructions, and sample problems.
One-dimensional transient finite difference model of an operational salinity gradient solar pond
NASA Technical Reports Server (NTRS)
Hicks, Michael C.; Golding, Peter
1992-01-01
This paper describes the modeling approach used to simulate the transient behavior of a salinity gradient solar pond. A system of finite difference equations are used to generate the time dependent temperature and salinity profiles within the pond. The stability of the pond, as determined by the capacity of the resulting salinity profile to suppress thermal convection within the primary gradient region of the pond, is continually monitored and when necessary adjustments are made to the thickness of the gradient zone. Results of the model are then compared to measurements taken during two representative seasonal periods at the University of Texas at El Paso's (UTEP's) research solar pond.
HEMP 3D -- a finite difference program for calculating elastic-plastic flow
Wilkins, M.L.
1993-05-26
The HEMP 3D program can be used to solve problems in solid mechanics involving dynamic plasticity and time dependent material behavior and problems in gas dynamics. The equations of motion, the conservation equations, and the constitutive relations are solved by finite difference methods following the format of the HEMP computer simulation program formulated in two space dimensions and time. Presented here is an update of the 1975 report on the HEMP 3D numerical technique. The present report includes the sliding surface routines programmed by Robert Gulliford.
Dispersion and stability analysis for a finite difference beam propagation method.
de-Oliva-Rubio, J; Molina-Fernández, I; Godoy-Rubio, R
2008-06-01
Applying continuous and discrete transformation techniques, new analytical expressions to calculate dispersion and stability of a Runge- Kutta Finite Difference Beam Propagation Method (RK-FDBPM) are obtained. These expressions give immediate insight about the discretization errors introduced by the numerical method in the plane-wave spectrum domain. From these expressions a novel strategy to adequately set the mesh steps sizes of the RK-FDBPM is presented. Assessment of the method is performed by studying the propagation in several linear and nonlinear photonic devices for different spatial discretizations.
NASA Technical Reports Server (NTRS)
Doohovskoy, A.
1977-01-01
A change in MACSYMA syntax is proposed to accommodate the operator manipulators necessary to implement direct and indirect methods for the solution of differential equations, calculus of finite differences, and the fractional calculus, as well as their modern counterparts. To illustrate the benefits and convenience of this syntax extension, an example is given to show how MACSYMA's pattern-matching capability can be used to implement a particular set of operator identities which can then be used to obtain exact solutions to nonlinear differential equations.
An adaptive-mesh finite-difference solution method for the Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Luchini, Paolo
1987-02-01
An adjustable variable-spacing grid is presented which permits the addition or deletion of single points during iterative solutions of the Navier-Stokes equations by finite difference methods. The grid is designed for application to two-dimensional steady-flow problems which can be described by partial differential equations whose second derivatives are constrained to the Laplacian operator. An explicit Navier-Stokes equations solution technique defined for use with the grid incorporates a hybrid form of the convective terms. Three methods are developed for automatic modifications of the mesh during calculations.
NASA Astrophysics Data System (ADS)
Adhikari, Achyut; Dev, Kapil; Asundi, Anand
2016-11-01
Wire grid polarizers (WGP), are sub-wavelength gratings with applications in display projection system due to their compact size, wide field of view and long-term stability. Measurement and testing of these structures are important to optimize their use. This is done by first measuring the Mueller matrix of the WGP using a Mueller matrix polarimeter. Next the finite difference time domain (FDTD) method is used to simulate a similar Mueller matrix thus providing the period and step height of the WGP. This approach may lead to more generic determination of sub-wavelength structures including diffractive optical structures.
Modelling migration in multilayer systems by a finite difference method: the spherical symmetry case
NASA Astrophysics Data System (ADS)
Hojbotǎ, C. I.; Toşa, V.; Mercea, P. V.
2013-08-01
We present a numerical model based on finite differences to solve the problem of chemical impurity migration within a multilayer spherical system. Migration here means diffusion of chemical species in conditions of concentration partitioning at layer interfaces due to different solubilities of the migrant in different layers. We detail here the numerical model and discuss the results of its implementation. To validate the method we compare it with cases where an analytic solution exists. We also present an application of our model to a practical problem in which we compute the migration of caprolactam from the packaging multilayer foil into the food.
Garcia-Herranz, Nuria; Cabellos, Oscar; Aragones, Jose M.; Ahnert, Carol
2003-05-15
In order to take into account in a more effective and accurate way the intranodal heterogeneities in coarse-mesh finite-difference (CMFD) methods, a new equivalent parameter generation methodology has been developed and tested. This methodology accounts for the dependence of the nodal homogeneized two-group cross sections and nodal coupling factors, with interface flux discontinuity (IFD) factors that account for heterogeneities on the flux-spectrum and burnup intranodal distributions as well as on neighbor effects.The methodology has been implemented in an analytic CMFD method, rigorously obtained for homogeneous nodes with transverse leakage and generalized now for heterogeneous nodes by including IFD heterogeneity factors. When intranodal mesh node heterogeneity vanishes, the heterogeneous solution tends to the analytic homogeneous nodal solution. On the other hand, when intranodal heterogeneity increases, a high accuracy is maintained since the linear and nonlinear feedbacks on equivalent parameters have been shown to be as a very effective way of accounting for heterogeneity effects in two-group multidimensional coarse-mesh diffusion calculations.
NASA Technical Reports Server (NTRS)
Sun, W.; Loeb, N. G.; Tanev, S.; Videen, G.
2004-01-01
The two-dimensional (2-D) finite-difference time domain (FDTD) method is applied to calculate light scattering and absorption by an arbitrarily shaped infinite column embedded in an absorbing dielectric medium. A uniaxial perfectly matched layer (UPML) absorbing boundary condition (ABC) is used to truncate the computational domain. The single-scattering properties of the infinite column embedded in the absorbing medium, including scattering phase functions, extinction and absorption efficiencies, are derived using an area integration of the internal field. An exact solution for light scattering and absorption by a circular cylinder in an absorbing medium is used to examine the accuracy of the 2-D UPML FDTD code. With use of a cell size of 1/120 incident wavelength in the FDTD calculations, the errors in the extinction and absorption efficiencies and asymmetry factors from the 2-D UPML FDTD are generally smaller than approx .1%. The errors in the scattering phase functions are typically smaller than approx .4%. Using the 2-D UPML FDTD technique, light scattering and absorption by long noncircular columns embedded in absorbing media can be accurately solved.
NASA Technical Reports Server (NTRS)
Martel, Francoise; Wunsch, Carl
1993-01-01
A finite-difference model of the North Atlantic is constructed for the purpose of making an estimate of the circulation through an inverse calculation. The data base is eclectic, and includes hydrography, oxygen, nutrients, current meter and float records, atmospheric momentum, heat and water vapor transfers, as well as estimates of certain integral fluxes. Owing to the available hydrographic database, the model resolution is restricted to 1 deg at best, and is much coarser in many aspects. This limited resolution is a major obstacle to accurate estimates of climatological fluxes. In its final form, there are about 9000 constraints in 29,000 formal unknowns plus 9000 noise unknowns. The system is solved as a tapered least-squares system by a sparse conjugate gradient algorithm. With the exception of a few float velocities, all constraints are found to be consistent within error estimates. The model produces estimates of large-scale fluxes and flux divergences for all conventional properties including heat and nutrients as well as carbon dioxide and alkalinity. Meridional fluxes of carbon are found to be indistinguishable from zero, whereas the North Atlantic tends to export nutrients to the south, but carry heat to the north. Traditional oceanographic depictions of the circulation through combination of nonsynoptic data into steady models may have reached their useful limit in the present calculation, as the conflicts between the data and physical requirements become quantitatively apparent.
Bohling, G.C.; Butler, J.J.
2001-01-01
We have developed a program for inverse analysis of two-dimensional linear or radial groundwater flow problems. The program, 1r2dinv, uses standard finite difference techniques to solve the groundwater flow equation for a horizontal or vertical plane with heterogeneous properties. In radial mode, the program simulates flow to a well in a vertical plane, transforming the radial flow equation into an equivalent problem in Cartesian coordinates. The physical parameters in the model are horizontal or x-direction hydraulic conductivity, anisotropy ratio (vertical to horizontal conductivity in a vertical model, y-direction to x-direction in a horizontal model), and specific storage. The program allows the user to specify arbitrary and independent zonations of these three parameters and also to specify which zonal parameter values are known and which are unknown. The Levenberg-Marquardt algorithm is used to estimate parameters from observed head values. Particularly powerful features of the program are the ability to perform simultaneous analysis of heads from different tests and the inclusion of the wellbore in the radial mode. These capabilities allow the program to be used for analysis of suites of well tests, such as multilevel slug tests or pumping tests in a tomographic format. The combination of information from tests stressing different vertical levels in an aquifer provides the means for accurately estimating vertical variations in conductivity, a factor profoundly influencing contaminant transport in the subsurface. ?? 2001 Elsevier Science Ltd. All rights reserved.
NASA Astrophysics Data System (ADS)
Ying, Jinyong; Xie, Dexuan
2015-10-01
The Poisson-Boltzmann equation (PBE) is one widely-used implicit solvent continuum model for calculating electrostatics of ionic solvated biomolecule. In this paper, a new finite element and finite difference hybrid method is presented to solve PBE efficiently based on a special seven-overlapped box partition with one central box containing the solute region and surrounded by six neighboring boxes. In particular, an efficient finite element solver is applied to the central box while a fast preconditioned conjugate gradient method using a multigrid V-cycle preconditioning is constructed for solving a system of finite difference equations defined on a uniform mesh of each neighboring box. Moreover, the PBE domain, the box partition, and an interface fitted tetrahedral mesh of the central box can be generated adaptively for a given PQR file of a biomolecule. This new hybrid PBE solver is programmed in C, Fortran, and Python as a software tool for predicting electrostatics of a biomolecule in a symmetric 1:1 ionic solvent. Numerical results on two test models with analytical solutions and 12 proteins validate this new software tool, and demonstrate its high performance in terms of CPU time and memory usage.
A moving mesh finite difference method for equilibrium radiation diffusion equations
NASA Astrophysics Data System (ADS)
Yang, Xiaobo; Huang, Weizhang; Qiu, Jianxian
2015-10-01
An efficient moving mesh finite difference method is developed for the numerical solution of equilibrium radiation diffusion equations in two dimensions. The method is based on the moving mesh partial differential equation approach and moves the mesh continuously in time using a system of meshing partial differential equations. The mesh adaptation is controlled through a Hessian-based monitor function and the so-called equidistribution and alignment principles. Several challenging issues in the numerical solution are addressed. Particularly, the radiation diffusion coefficient depends on the energy density highly nonlinearly. This nonlinearity is treated using a predictor-corrector and lagged diffusion strategy. Moreover, the nonnegativity of the energy density is maintained using a cutoff method which has been known in literature to retain the accuracy and convergence order of finite difference approximation for parabolic equations. Numerical examples with multi-material, multiple spot concentration situations are presented. Numerical results show that the method works well for radiation diffusion equations and can produce numerical solutions of good accuracy. It is also shown that a two-level mesh movement strategy can significantly improve the efficiency of the computation.
Feldberg, S.W.
1991-01-01
Commencing in the early 60s the application of explicit finite difference (EFD) methods to the analysis of electrochemical problems paralleled the development and availability of fast, main-frame, digital computers. The appeal of the EFD method has been its simplicity of principle and of application. EFD algorithms, however, are notoriously inefficient for solving certain types of stiff problems (e.g., problems involving a wide dynamic range of time constants). In this presentation the author discusses the principles and some applications of a fast quasi-explicit finite difference (FQEFD) method in which the computational speed is enhanced, by many orders of magnitude in some cases, without compromising the user friendliness which has popularized the EFD method. The method is designed to treat electrochemical responses to a discontinuous (e.g, chronoamperometric) perturbation and utilizes the DuFort-Frankel algorithm (1) with exponentially expanding space (2) and exponentially expanding time grids. (A previously published version of the FQEFD method (3,4) was designed to treat electrochemical responses to a continuous (e.g., cyclic voltammetric) perturbation and utilizes the DuFort-Frankel (3) algorithm in conjunction with an exponentially expanding space grid and a uniform time grid. The development of the basic FQEFD equations was presented there). The protocol for introducing the expanding time grid is straightforward and is discussed. 7 refs., 1 fig. 1 tab.
NASA Astrophysics Data System (ADS)
Zehner, Björn; Hellwig, Olaf; Linke, Maik; Görz, Ines; Buske, Stefan
2016-01-01
3D geological underground models are often presented by vector data, such as triangulated networks representing boundaries of geological bodies and geological structures. Since models are to be used for numerical simulations based on the finite difference method, they have to be converted into a representation discretizing the full volume of the model into hexahedral cells. Often the simulations require a high grid resolution and are done using parallel computing. The storage of such a high-resolution raster model would require a large amount of storage space and it is difficult to create such a model using the standard geomodelling packages. Since the raster representation is only required for the calculation, but not for the geometry description, we present an algorithm and concept for rasterizing geological models on the fly for the use in finite difference codes that are parallelized by domain decomposition. As a proof of concept we implemented a rasterizer library and integrated it into seismic simulation software that is run as parallel code on a UNIX cluster using the Message Passing Interface. We can thus run the simulation with realistic and complicated surface-based geological models that are created using 3D geomodelling software, instead of using a simplified representation of the geological subsurface using mathematical functions or geometric primitives. We tested this set-up using an example model that we provide along with the implemented library.
Development of an advanced finite-difference atmospheric general circulation model
Randall, D.A.
1992-03-01
We have proposed to provide and further develop an advanced finite-difference climate model for use in CHAMMP. The model includes advanced parameterizations of cumulus convection, boundary-layer processes, cloud formation, and land-surface vegetation, as well as parameterizations of radiative transfer and gravity wave drag. Postprocessing codes and a user's guide will also be provided. This research is being conducted in collaboration with Professors C.R. Mechoso and A. Arakawa at the University of California at Los Angeles (UCLA). The following research tasks are being carried out in support of CHAMMP: (1) Provide to CHAMMP a base-line finite-difference model and postprocessing codes for further development by the CHAMMP Science Team; (2) Provide to CHAMMP improved model physics to be developed in the course of our research project; (3) Provide to CHAMMP improved computational methods for use in the model; and, (4) Investigate the performance of current and to-be-developed physical parameterizations and computational methods at very high resolution.
On the Definition of Surface Potentials for Finite-Difference Operators
NASA Technical Reports Server (NTRS)
Tsynkov, S. V.; Bushnell, Dennis M. (Technical Monitor)
2001-01-01
For a class of linear constant-coefficient finite-difference operators of the second order, we introduce the concepts similar to those of conventional single- and double-layer potentials for differential operators. The discrete potentials are defined completely independently of any notion related to the approximation of the continuous potentials on the grid. We rather use all approach based on differentiating, and then inverting the differentiation of a function with surface discontinuity of a particular kind, which is the most general way of introducing surface potentials in the theory of distributions. The resulting finite-difference "surface" potentials appear to be solutions of the corresponding continuous potentials. Primarily, this pertains to the possibility of representing a given solution to the homogeneous equation on the domain as a variety of surface potentials, with the density defined on the domain's boundary. At the same time the discrete surface potentials can be interpreted as one specific realization of the generalized potentials of Calderon's type, and consequently, their approximation properties can be studied independently in the framework of the difference potentials method by Ryaben'kii. The motivation for introducing and analyzing the discrete surface potentials was provided by the problems of active shielding and control of sound, in which the aforementioned source terms that drive the potentials are interpreted as the acoustic control sources that cancel out the unwanted noise on a predetermined region of interest.
A moving mesh finite difference method for equilibrium radiation diffusion equations
Yang, Xiaobo; Huang, Weizhang; Qiu, Jianxian
2015-10-01
An efficient moving mesh finite difference method is developed for the numerical solution of equilibrium radiation diffusion equations in two dimensions. The method is based on the moving mesh partial differential equation approach and moves the mesh continuously in time using a system of meshing partial differential equations. The mesh adaptation is controlled through a Hessian-based monitor function and the so-called equidistribution and alignment principles. Several challenging issues in the numerical solution are addressed. Particularly, the radiation diffusion coefficient depends on the energy density highly nonlinearly. This nonlinearity is treated using a predictor–corrector and lagged diffusion strategy. Moreover, the nonnegativity of the energy density is maintained using a cutoff method which has been known in literature to retain the accuracy and convergence order of finite difference approximation for parabolic equations. Numerical examples with multi-material, multiple spot concentration situations are presented. Numerical results show that the method works well for radiation diffusion equations and can produce numerical solutions of good accuracy. It is also shown that a two-level mesh movement strategy can significantly improve the efficiency of the computation.
Enhancing finite differences with radial basis functions: Experiments on the Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Flyer, Natasha; Barnett, Gregory A.; Wicker, Louis J.
2016-07-01
Polynomials are used together with polyharmonic spline (PHS) radial basis functions (RBFs) to create local RBF-finite-difference (RBF-FD) weights on different node layouts for spatial discretizations that can be viewed as enhancements of the classical finite differences (FD). The presented method replicates the convergence properties of FD but for arbitrary node layouts. It is tested on the 2D compressible Navier-Stokes equations at low Mach number, relevant to atmospheric flows. Test cases are taken from the numerical weather prediction community and solved on bounded domains. Thus, attention is given on how to handle boundaries with the RBF-FD method, as well as a novel implementation for hyperviscosity. Comparisons are done on Cartesian, hexagonal, and quasi-uniform node layouts. Consideration and guidelines are given on PHS order, polynomial degree and stencil size. The main advantages of the present method are: 1) capturing the basic physics of the problem surprisingly well, even at very coarse resolutions, 2) high-order accuracy without the need of tuning a shape parameter, and 3) the inclusion of polynomials eliminates stagnation (saturation) errors. A MATLAB code is given to calculate the differentiation weights for this novel approach.
Simple Numerical Schemes for the Korteweg-deVries Equation
C. J. McKinstrie; M. V. Kozlov
2000-12-01
Two numerical schemes, which simulate the propagation of dispersive non-linear waves, are described. The first is a split-step Fourier scheme for the Korteweg-de Vries (KdV) equation. The second is a finite-difference scheme for the modified KdV equation. The stability and accuracy of both schemes are discussed. These simple schemes can be used to study a wide variety of physical processes that involve dispersive nonlinear waves.
A finite difference Hartree-Fock program for atoms and diatomic molecules
NASA Astrophysics Data System (ADS)
Kobus, Jacek
2013-03-01
The newest version of the two-dimensional finite difference Hartree-Fock program for atoms and diatomic molecules is presented. This is an updated and extended version of the program published in this journal in 1996. It can be used to obtain reference, Hartree-Fock limit values of total energies and multipole moments for a wide range of diatomic molecules and their ions in order to calibrate existing and develop new basis sets, calculate (hyper)polarizabilities (αzz, βzzz, γzzzz, Az,zz, Bzz,zz) of atoms, homonuclear and heteronuclear diatomic molecules and their ions via the finite field method, perform DFT-type calculations using LDA or B88 exchange functionals and LYP or VWN correlations ones or the self-consistent multiplicative constant method, perform one-particle calculations with (smooth) Coulomb and Krammers-Henneberger potentials and take account of finite nucleus models. The program is easy to install and compile (tarball+configure+make) and can be used to perform calculations within double- or quadruple-precision arithmetic. Catalogue identifier: ADEB_v2_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADEB_v2_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public License version 2 No. of lines in distributed program, including test data, etc.: 171196 No. of bytes in distributed program, including test data, etc.: 9481802 Distribution format: tar.gz Programming language: Fortran 77, C. Computer: any 32- or 64-bit platform. Operating system: Unix/Linux. RAM: Case dependent, from few MB to many GB Classification: 16.1. Catalogue identifier of previous version: ADEB_v1_0 Journal reference of previous version: Comput. Phys. Comm. 98(1996)346 Does the new version supersede the previous version?: Yes Nature of problem: The program finds virtually exact solutions of the Hartree-Fock and density functional theory type equations for atoms, diatomic molecules and their ions
NASA Technical Reports Server (NTRS)
Shu, Chi-Wang
1992-01-01
The nonlinear stability of compact schemes for shock calculations is investigated. In recent years compact schemes were used in various numerical simulations including direct numerical simulation of turbulence. However to apply them to problems containing shocks, one has to resolve the problem of spurious numerical oscillation and nonlinear instability. A framework to apply nonlinear limiting to a local mean is introduced. The resulting scheme can be proven total variation (1D) or maximum norm (multi D) stable and produces nice numerical results in the test cases. The result is summarized in the preprint entitled 'Nonlinearly Stable Compact Schemes for Shock Calculations', which was submitted to SIAM Journal on Numerical Analysis. Research was continued on issues related to two and three dimensional essentially non-oscillatory (ENO) schemes. The main research topics include: parallel implementation of ENO schemes on Connection Machines; boundary conditions; shock interaction with hydrogen bubbles, a preparation for the full combustion simulation; and direct numerical simulation of compressible sheared turbulence.
NASA Astrophysics Data System (ADS)
Popov, Anton; Kaus, Boris
2015-04-01
This software project aims at bringing the 3D lithospheric deformation modeling to a qualitatively different level. Our code LaMEM (Lithosphere and Mantle Evolution Model) is based on the following building blocks: * Massively-parallel data-distributed implementation model based on PETSc library * Light, stable and accurate staggered-grid finite difference spatial discretization * Marker-in-Cell pedictor-corector time discretization with Runge-Kutta 4-th order * Elastic stress rotation algorithm based on the time integration of the vorticity pseudo-vector * Staircase-type internal free surface boundary condition without artificial viscosity contrast * Geodynamically relevant visco-elasto-plastic rheology * Global velocity-pressure-temperature Newton-Raphson nonlinear solver * Local nonlinear solver based on FZERO algorithm * Coupled velocity-pressure geometric multigrid preconditioner with Galerkin coarsening Staggered grid finite difference, being inherently Eulerian and rather complicated discretization method, provides no natural treatment of free surface boundary condition. The solution based on the quasi-viscous sticky-air phase introduces significant viscosity contrasts and spoils the convergence of the iterative solvers. In LaMEM we are currently implementing an approximate stair-case type of the free surface boundary condition which excludes the empty cells and restores the solver convergence. Because of the mutual dependence of the stress and strain-rate tensor components, and their different spatial locations in the grid, there is no straightforward way of implementing the nonlinear rheology. In LaMEM we have developed and implemented an efficient interpolation scheme for the second invariant of the strain-rate tensor, that solves this problem. Scalable efficient linear solvers are the key components of the successful nonlinear problem solution. In LaMEM we have a range of PETSc-based preconditioning techniques that either employ a block factorization of
NASA Technical Reports Server (NTRS)
Nordstrom, Jan; Carpenter, Mark H.
1998-01-01
Boundary and interface conditions for high order finite difference methods applied to the constant coefficient Euler and Navier-Stokes equations are derived. The boundary conditions lead to strict and strong stability. The interface conditions are stable and conservative even if the finite difference operators and mesh sizes vary from domain to domain. Numerical experiments show that the new conditions also lead to good results for the corresponding nonlinear problems.
CUDA Fortran acceleration for the finite-difference time-domain method
NASA Astrophysics Data System (ADS)
Hadi, Mohammed F.; Esmaeili, Seyed A.
2013-05-01
A detailed description of programming the three-dimensional finite-difference time-domain (FDTD) method to run on graphical processing units (GPUs) using CUDA Fortran is presented. Two FDTD-to-CUDA thread-block mapping designs are investigated and their performances compared. Comparative assessment of trade-offs between GPU's shared memory and L1 cache is also discussed. This presentation is for the benefit of FDTD programmers who work exclusively with Fortran and are reluctant to port their codes to C in order to utilize GPU computing. The derived CUDA Fortran code is compared with an optimized CPU version that runs on a workstation-class CPU to present a realistic GPU to CPU run time comparison and thus help in making better informed investment decisions on FDTD code redesigns and equipment upgrades. All analyses are mirrored with CUDA C simulations to put in perspective the present state of CUDA Fortran development.
Full-wave finite-difference time-domain simulation of electromagnetic cloaking structures.
Zhao, Yan; Argyropoulos, Christos; Hao, Yang
2008-04-28
This paper proposes a radial dependent dispersive finite-difference time-domain method for the modeling of electromagnetic cloaking structures. The permittivity and permeability of the cloak are mapped to the Drude dispersion model and taken into account in dispersive FDTD simulations. Numerical simulations demonstrate that under ideal conditions, objects placed inside the cloak are 'invisible' to external electromagnetic fields. However for the simplified cloak based on linear transformations, the back scattering has a similar level to the case of a PEC cylinder without any cloak, rendering the object still being 'visible'. It is also demonstrated numerically that the simplified cloak based on high-order transformations can indeed improve the cloaking performance.
Seismic Analysis of a Rockfill Dam by FLAC Finite Difference Code
Miglio, Livia; Pagliaroli, Alessandro; Lanzo, Giuseppe; Miliziano, Salvatore
2008-07-08
The paper presents the results of numerical analyses carried out with FLAC finite difference code aiming at investigating the seismic response of rockfill dams. In particular the hysteretic damping model, recently incorporated within the code, coupled with a perfectly plastic yield criterion, was employed. As first step, 1D and 2D calibration analyses were performed and comparisons with the results supplied by well known linear equivalent and fully non linear codes were carried out. Then the seismic response of E1 Infiernillo rockfill dam was investigated during two weak and strong seismic events. Benefits and shortcomings of using the hysteretic damping model are discussed in the light of the results obtained from calibration studies and field-scale analyses.
Inclusion of lumped elements in finite difference time domain electromagnetic calculations
Thomas, V.A.; Jones, M.E.; Mason, R.J.
1994-12-31
A general approach for including lumped circuit elements in a finite difference, time domain (FD-TD) solution of Maxwell`s equations is presented. The methodology allows the direct access to SPICE to model the lumped circuits, while the full 3-Dimensional solution to Maxwell`s equations provides the electromagnetic field evolution. This type of approach could be used to mode a pulsed power machine by using a SPICE model for the driver and using an electromagnetic PIC code for the plasma/electromagnetics calculation. The evolution of the driver can be made self consistent with the behavior of the plasma load. Other applications are also possible, including modeling of nonlinear microwave circuits (as long as the non-linearities may be expressed in terms of a lumped element) and self-consistent calculation of very high speed computer interconnections and digital circuits.
NASA Astrophysics Data System (ADS)
Yan, Chao; Wu, Yulin; Mei, Zuyan
1991-06-01
A finite difference method for computing 3D incompressible laminar and turbulent boundary layers is presented. The curvilinear semiorthogonal coordinate system is used to express the 3D boundary layer equations. Reynolds stresses are assumed zero for the laminar boundary layer. For the turbulent boundary layer, the Reynolds stresses are expressed through an algebraic turbulence model taking into account nonisotropic eddy viscosity based on the Cebeci-Smith turbulence model and Rotta's turbulent stress formulas. The numerical solution method is described and carried out for a 3D boundary layer of a wing. The results are in good agreement with measured data. The method and its program have high accuracy, require less computer storage, are computationally fast and flexible in use.
NASA Technical Reports Server (NTRS)
Kaul, Upender K. (Inventor)
2009-01-01
Modeling and simulation of free and forced structural vibrations is essential to an overall structural health monitoring capability. In the various embodiments, a first principles finite-difference approach is adopted in modeling a structural subsystem such as a mechanical gear by solving elastodynamic equations in generalized curvilinear coordinates. Such a capability to generate a dynamic structural response is widely applicable in a variety of structural health monitoring systems. This capability (1) will lead to an understanding of the dynamic behavior of a structural system and hence its improved design, (2) will generate a sufficiently large space of normal and damage solutions that can be used by machine learning algorithms to detect anomalous system behavior and achieve a system design optimization and (3) will lead to an optimal sensor placement strategy, based on the identification of local stress maxima all over the domain.
Finite Difference Time Domain Analysis of Underwater Acoustic Lens System for Ambient Noise Imaging
NASA Astrophysics Data System (ADS)
Mori, Kazuyoshi; Miyazaki, Ayano; Ogasawara, Hanako; Yokoyama, Tomoki; Nakamura, Toshiaki
2006-05-01
Much attention has been paid to the new idea of detecting objects using ocean ambient noise. This concept is called ambient noise imaging (ANI). In this study, sound fields focused by an acoustic lens system constructed with a single biconcave lens were analyzed using the finite difference time domain (FDTD) method for realizing an ANI system. The size of the lens aperture that would have sufficient resolution—for example, the beam width is 1° at 60 kHz—was roughly determined by comparing the image points and -3 dB areas of sound pressure fields generated by lenses with various apertures. Then, in another FDTD analysis, we successfully used a lens with a determined aperture to detect rigid target objects in an acoustic noise field generated by a large number of point sources.
Parallel finite-difference time-domain modeling of an opal photonic crystal
NASA Astrophysics Data System (ADS)
Vaccari, Alessandro; Cristoforetti, Luca; Lesina, Antonino Calà; Ramunno, Lora; Chiappini, Andrea; Prudenzano, Francesco; Bozzoli, Alessandro; Calliari, Lucia
2014-07-01
This work describes a computational approach for the optical characterization of an opal photonic crystal (PC). We intend, in particular, to validate our approach by comparing the transmittance of a crystal model, as obtained by numerical simulation, with the transmittance of the same crystal, as measured over 400- to 700-nm wavelength range. We consider an opal PC with a face-centered cubic lattice structure of spherical particles made of polystyrene (a nonabsorptive material with constant relative dielectric permittivity). Light-crystal interaction is simulated by numerically solving Maxwell's equations via the finite-difference time-domain method and by using the Kirchhoff formula to calculate the far field. A method to study the propagating Bloch modes inside the crystal bulk is also sketched.
NASA Technical Reports Server (NTRS)
Kaul, Upender K.
2005-01-01
A three-dimensional numerical solver based on finite-difference solution of three-dimensional elastodynamic equations in generalized curvilinear coordinates has been developed and used to generate data such as radial and tangential stresses over various gear component geometries under rotation. The geometries considered are an annulus, a thin annular disk, and a thin solid disk. The solution is based on first principles and does not involve lumped parameter or distributed parameter systems approach. The elastodynamic equations in the velocity-stress formulation that are considered here have been used in the solution of problems of geophysics where non-rotating Cartesian grids are considered. For arbitrary geometries, these equations along with the appropriate boundary conditions have been cast in generalized curvilinear coordinates in the present study.
Finite Difference Time Domain Analysis for a Sound Field Including a Plate in Water
NASA Astrophysics Data System (ADS)
Saito, Hideaki; Naoi, Jun; Kikuchi, Toshiaki
2004-05-01
In marine research, measures against self-noise of an observatory ship are important. Generally, the self-noise is measured after the completion of ships. It is difficult to predict this noise level beforehand. Then, an attempt is made to determine the noise emitted from various elements of a structure. The finite difference time domain method is applied to obtain sound fields, including that of a plate in water. The time behavior of the sound wave emitted from a sound source placed near the upper part of a plate is investigated. As a result, the reflected and re-radiated waves from the plate including the head wave resulting from the longitudinal and traverse waves in the plate are able to be visualized. In the case of the plate with a branch plate, the suppression of the wave which propagates at the inside of the plate with the length of the branch plate is shown.
NASA Technical Reports Server (NTRS)
Lansing, F. L.
1976-01-01
A numerical procedure was established using the finite-difference technique in the determination of the time-varying temperature distribution of a tubular solar collector under changing solar radiancy and ambient temperature. Three types of spatial discretization processes were considered and compared for their accuracy of computations and for selection of the shortest computer time and cost. The stability criteria of this technique was analyzed in detail to give the critical time increment to ensure stable computations. The results of the numerical analysis were in good agreement with the analytical solution previously reported. The numerical method proved to be a powerful tool in the investigation of the collector sensitivity to two different flow patterns and several flow control mechanisms.
NASA Astrophysics Data System (ADS)
Guda, A. A.; Guda, S. A.; Soldatov, M. A.; Lomachenko, K. A.; Bugaev, A. L.; Lamberti, C.; Gawelda, W.; Bressler, C.; Smolentsev, G.; Soldatov, A. V.; Joly, Y.
2016-05-01
Finite difference method (FDM) implemented in the FDMNES software [Phys. Rev. B, 2001, 63, 125120] was revised. Thorough analysis shows, that the calculated diagonal in the FDM matrix consists of about 96% zero elements. Thus a sparse solver would be more suitable for the problem instead of traditional Gaussian elimination for the diagonal neighbourhood. We have tried several iterative sparse solvers and the direct one MUMPS solver with METIS ordering turned out to be the best. Compared to the Gaussian solver present method is up to 40 times faster and allows XANES simulations for complex systems already on personal computers. We show applicability of the software for metal-organic [Fe(bpy)3]2+ complex both for low spin and high spin states populated after laser excitation.
Finite element-finite difference thermal/structural analysis of large space truss structures
NASA Technical Reports Server (NTRS)
Warren, Andrew H.; Arelt, Joseph E.; Eskew, William F.; Rogers, Karen M.
1992-01-01
A technique of automated and efficient thermal-structural processing of truss structures that interfaces the finite element and finite difference method was developed. The thermal-structural analysis tasks include development of the thermal and structural math models, thermal analysis, development of an interface and data transfer between the models, and finally an evaluation of the thermal stresses and displacements in the structure. Consequently, the objective of the developed technique was to minimize the model development time, in order to assure an automatic transfer of data between the thermal and structural models as well as to minimize the computer resources needed for the analysis itself. The method and techniques described are illustrated on the thermal/structural analysis of the Space Station Freedom main truss.
NASA Astrophysics Data System (ADS)
Putri, Selmi; Arif, Idam; Khotimah, Siti Nurul
2015-04-01
In this study, peritoneal dialysis transport system was numerically simulated using finite difference method. The increase in the intraperitoneal pressure due to coughing has a high value outside the working area of the void volume fraction of the hydrostatic pressure θ(P). Therefore to illustrate the effects of the pressure increment, the pressure of working area is chosen between 1 and 3 mmHg. The effects of increased pressure in peritoneal tissue cause more fluid to flow into the blood vessels and lymph. Furthermore, the increased pressure in peritoneal tissue makes the volumetric flux jv and solute flux js across the tissue also increase. The more fluid flow into the blood vessels and lymph causes the fluid to flow into tissue qv and the glucose flow qs to have more negative value and also decreases the glucose concentration CG in the tissue.
NASA Technical Reports Server (NTRS)
Vinokur, M.
1983-01-01
The class of one-dimensional stretching functions used in finite-difference calculations is studied. For solutions containing a highly localized region of rapid variation, simple criteria for a stretching function are derived using a truncation error analysis. These criteria are used to investigate two types of stretching functions. One an interior stretching function, for which the location and slope of an interior clustering region are specified. The simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine. The other type of function is a two-sided stretching function, for which the arbitrary slopes at the two ends of the one-dimensional interval are specified. The simplest such general function is found to be one based on the inverse tangent. Previously announced in STAR as N80-25055
NASA Technical Reports Server (NTRS)
Vinokur, M.
1979-01-01
The class of one-dimensional stretching functions used in finite-difference calculations is studied. For solutions containing a highly localized region of rapid variation, simple criteria for a stretching function are derived using a truncation error analysis. These criteria are used to investigate two types of stretching functions. One is an interior stretching function, for which the location and slope of an interior clustering region are specified. The simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine. The other type of function is a two-sided stretching function, for which the arbitrary slopes at the two ends of the one-dimensional interval are specified. The simplest such general function is found to be one based on the inverse tangent.
The analysis of reactively loaded microstrip antennas by finite difference time domain modelling
NASA Technical Reports Server (NTRS)
Hilton, G. S.; Beach, M. A.; Railton, C. J.
1990-01-01
In recent years, much interest has been shown in the use of printed circuit antennas in mobile satellite and communications terminals at microwave frequencies. Although such antennas have many advantages in weight and profile size over more conventional reflector/horn configurations, they do, however, suffer from an inherently narrow bandwidth. A way of optimizing the bandwidth of such antennas by an electronic tuning technique using a loaded probe mounted within the antenna structure is examined, and the resulting far-field radiation patterns are shown. Simulation results from a 2D finite difference time domain (FDTD) model for a rectangular microstrip antenna loaded with shorting pins are given and compared to results obtained with an actual antenna. It is hoped that this work will result in a design package for the analysis of microstrip patch antenna elements.
Development of the Finite Difference Time Domain Method on a Lebedev Grid for Anisotropic Materials
NASA Astrophysics Data System (ADS)
Nauta, Marcel D.
The finite-difference time-domain (FDTD) method is derived on a Lebedev grid, instead of the standard Yee grid, to better represent the constitutive relations in anisotropic materials. The Lebedev grid extends the Yee grid by approximating Maxwell's equations with tensor constitutive relations using only central differences. A dispersion relation with stability criteria is derived and it is proven that the Lebedev grid has a consistent calculus. An integral derivation of the update equations illustrates how to appropriately excite the grid. This approach is also used to derive the update equations at planar material interfaces and domain edge PEC. Lebedev grid results are compared with analytical and Yee grid solutions using an equal memory comparison. Numerical results show that the Lebedev grid suffers greater dispersion error but better represents material interfaces. Focus is given to generalizing the concepts that make the Yee grid robust for isotropic materials. Keywords: FDTD, anisotropic materials, Lebedev grid, collocated grids.
Full-wave finite-difference time-domain simulation of electromagnetic cloaking structures.
Zhao, Yan; Argyropoulos, Christos; Hao, Yang
2008-04-28
This paper proposes a radial dependent dispersive finite-difference time-domain method for the modeling of electromagnetic cloaking structures. The permittivity and permeability of the cloak are mapped to the Drude dispersion model and taken into account in dispersive FDTD simulations. Numerical simulations demonstrate that under ideal conditions, objects placed inside the cloak are 'invisible' to external electromagnetic fields. However for the simplified cloak based on linear transformations, the back scattering has a similar level to the case of a PEC cylinder without any cloak, rendering the object still being 'visible'. It is also demonstrated numerically that the simplified cloak based on high-order transformations can indeed improve the cloaking performance. PMID:18545374
Improved finite-difference computation of the van der Waals force: One-dimensional case
Pinto, Fabrizio
2009-10-15
We present an improved demonstration of the calculation of Casimir forces in one-dimensional systems based on the recently proposed numerical imaginary frequency Green's function computation approach. The dispersion force on two thick lossy dielectric slabs separated by an empty gap and placed within a perfectly conducting cavity is obtained from the Green's function of the modified Helmholtz equation by means of an ordinary finite-difference method. In order to demonstrate the possibility to develop algorithms to explore complex geometries in two and three dimensions to higher order in the mesh spacing, we generalize existing classical electromagnetism algebraic methods to generate the difference equations for dielectric boundaries not coinciding with any grid points. Diagnostic tests are presented to monitor the accuracy of our implementation of the method and follow-up applications in higher dimensions are introduced.
Numerical analysis of polarization gratings using the finite-difference time-domain method
Oh, Chulwoo; Escuti, Michael J.
2007-10-15
We report the first full numerical analysis of polarization gratings (PGs), and study their most general properties and limits by using the finite-difference time-domain (FDTD) method. In this way, we avoid limiting assumptions on material properties or grating dimensions (e.g., no paraxial approximations) and provide a more complete understanding of PG diffraction behavior. We identify the fundamental delineation between diffraction regimes (thin versus thick) for anisotropic gratings and determine the conditions for {approx_equal}100% diffraction efficiency in the framework of the coupled-wave {rho} and Q parameters. Diffraction characteristics including the efficiency, spectral response, and polarization sensitivity are investigated for the two primary types of PGs with linear and circular birefringence. The angular response and finite-grating behavior (i.e., pixelation) are also examined. Comparisons with previous analytic approximations, where applicable, show good agreement.
Finite-difference time-domain simulation of thermal noise in open cavities
Andreasen, Jonathan; Cao Hui; Taflove, Allen; Kumar, Prem |; Cao Changqi
2008-02-15
A numerical model based on the finite-difference time-domain (FDTD) method is developed to simulate thermal noise in open cavities owing to output coupling. The absorbing boundary of the FDTD grid is treated as a blackbody, whose thermal radiation penetrates the cavity in the grid. The calculated amount of thermal noise in a one-dimensional dielectric cavity recovers the standard result of the quantum Langevin equation in the Markovian regime. Our FDTD simulation also demonstrates that in the non-Markovian regime the buildup of the intracavity noise field depends on the ratio of the cavity photon lifetime to the coherence time of thermal radiation. The advantage of our numerical method is that the thermal noise is introduced in the time domain without prior knowledge of cavity modes.
Transient analysis of printed lines using finite-difference time-domain method
Ahmed, Shahid
2012-03-29
Comprehensive studies of ultra-wideband pulses and electromagnetic coupling on printed coupled lines have been performed using full-wave 3D finite-difference time-domain analysis. Effects of unequal phase velocities of coupled modes, coupling between line traces, and the frequency dispersion on the waveform fidelity and crosstalk have been investigated in detail. To discriminate the contributions of different mechanisms into pulse evolution, single and coupled microstrip lines without (ϵ_{r} = 1) and with (ϵ_{r} > 1) dielectric substrates have been examined. To consistently compare the performance of the coupled lines with substrates of different permittivities and transients of different characteristic times, a generic metric similar to the electrical wavelength has been introduced. The features of pulse propagation on coupled lines with layered and pedestal substrates and on the irregular traces have been explored. Finally, physical interpretations of the simulation results are discussed in the paper.
Finite difference method to find period-one gait cycles of simple passive walkers
NASA Astrophysics Data System (ADS)
Dardel, Morteza; Safartoobi, Masoumeh; Pashaei, Mohammad Hadi; Ghasemi, Mohammad Hassan; Navaei, Mostafa Kazemi
2015-01-01
Passive dynamic walking refers to a class of bipedal robots that can walk down an incline with no actuation or control input. These bipeds are sensitive to initial conditions due to their style of walking. According to small basin of attraction of passive limit cycles, it is important to start with an initial condition in the basin of attraction of stable walking (limit cycle). This paper presents a study of the simplest passive walker with point and curved feet. A new approach is proposed to find proper initial conditions for a pair of stable and unstable period-one gait limit cycles. This methodology is based on finite difference method which can solve the nonlinear differential equations of motion on a discrete time. Also, to investigate the physical configurations of the walkers and the environmental influence such as the slope angle, the parameter analysis is applied. Numerical simulations reveal the performance of the presented method in finding two stable and unstable gait patterns.
A study of the efficiency of various Navier-Stokes solvers. [finite difference methods
NASA Technical Reports Server (NTRS)
Atias, M.; Wolfshtein, M.; Israeli, M.
1975-01-01
A comparative study of the efficiency of some finite difference methods for the solution of the Navier-Stokes equations was conducted. The study was restricted to the two-dimensional steady, uniform property vorticity-stream function equations. The comparisons were drawn by recording the CPU time required to obtain a solution as well as the accuracy of this solution using five numerical methods: central differences, first order upwind differences, second order upwind differences, exponential differences, and an ADI solution of the central difference equations. Solutions were obtained for two test cases: a recirculating eddy inside a square cavity with a moving top, and an impinging jet flow. The results show that whenever the central difference method is stable it generates results with a given accuracy for less CPU time than any other method.
Finite Difference Time Domain Electromagnetic Scattering from Frequency-Dependent Lossy Materials
NASA Technical Reports Server (NTRS)
Luebbers, Raymond J.; Beggs, John H.
1991-01-01
During this effort the tasks specified in the Statement of Work have been successfully completed. The extension of Finite Difference Time Domain (FDTD) to more complicated materials has been made. A three-dimensional FDTD code capable of modeling interactions with both dispersive dielectric and magnetic materials has been written, validated, and documented. This code is efficient and is capable of modeling interesting targets using a modest computer work station platform. However, in addition to the tasks in the Statement of Work, a significant number of other FDTD extensions and calculations have been made. RCS results for two different plate geometries have been reported. The FDTD method has been extended to computing far zone time domain results in two dimensions. Finally, the capability to model nonlinear materials has been incorporated into FDTD and validated. The FDTD computer codes developed have been supplied, along with documentation, and preprints describing the other FDTD advances have been included with this report as attachments.
An Analysis on 3d Marine Csem Responses Based on a Finite Difference Method
NASA Astrophysics Data System (ADS)
Han, N.; Nam, M.; Kim, H.
2010-12-01
Three-dimensional (3D) marine controlled-source electromagnetic (CSEM) data are analyzed using a modeling algorithm based on a finite difference method. The algorithm employs the secondary-field formulation of a vector Helmholtz equation for electric fields to avoid singularity problems. Primary fields are calculated analytically using a numerical filter for the Hankel transform for a three-layered 1D background model, that consists of air, sea and sub-seafloor; the model includes the air layer to consider air waves. Several numerical filters for the Hankel transform are compared in terms of their accuracy and computation time. Using the analytically-calculated primary fields, we compute secondary fields using a finite difference method with a staggered grid. The grid defines electric fields along cell edges while magnetic fields at cell faces. We verified the developed modeling algorithm using not only 1D analytic solutions but also responses for a 3D model, that are computed by other algorithms. Using disk models, this study analyzes marine CSEM data for horizontal and vertical electric and magnetic dipole sources to determine the most effective source-receiver configuration for the exploration of 3D thin and resistive hydrocarbon targets. Numerical results show that marine CSEM has exciting potential for oilfield characterization. Further, air waves should be properly considered in modeling and interpretation of marine CSEM data because they have great effects on marine CSEM data. For an analysis on bathymetry effects, a stepwise-bathymetry model was constructed. Bathymetry causes significant effects on marine CSEM data because transmitter and receivers are located very far each other. We propose a bathymetry correction method for a proper interpretation of marine CSEM data distorted by bathymetry.
Lloyd, Jeffrey T.; Clayton, John D.; Austin, Ryan A.; McDowell, David L.
2015-07-10
Background: The shock response of metallic single crystals can be captured using a micro-mechanical description of the thermoelastic-viscoplastic material response; however, using a such a description within the context of traditional numerical methods may introduce a physical artifacts. Advantages and disadvantages of complex material descriptions, in particular the viscoplastic response, must be framed within approximations introduced by numerical methods. Methods: Three methods of modeling the shock response of metallic single crystals are summarized: finite difference simulations, steady wave simulations, and algebraic solutions of the Rankine-Hugoniot jump conditions. For the former two numerical techniques, a dislocation density based framework describes themore » rate- and temperature-dependent shear strength on each slip system. For the latter analytical technique, a simple (two-parameter) rate- and temperature-independent linear hardening description is necessarily invoked to enable simultaneous solution of the governing equations. For all models, the same nonlinear thermoelastic energy potential incorporating elastic constants of up to order 3 is applied. Results: Solutions are compared for plate impact of highly symmetric orientations (all three methods) and low symmetry orientations (numerical methods only) of aluminum single crystals shocked to 5 GPa (weak shock regime) and 25 GPa (overdriven regime). Conclusions: For weak shocks, results of the two numerical methods are very similar, regardless of crystallographic orientation. For strong shocks, artificial viscosity affects the finite difference solution, and effects of transverse waves for the lower symmetry orientations not captured by the steady wave method become important. The analytical solution, which can only be applied to highly symmetric orientations, provides reasonable accuracy with regards to prediction of most variables in the final shocked state but, by construction, does not provide
Field Test of a Hybrid Finite-Difference and Analytic Element Regional Model.
Abrams, D B; Haitjema, H M; Feinstein, D T; Hunt, R J
2016-01-01
Regional finite-difference models often have cell sizes that are too large to sufficiently model well-stream interactions. Here, a steady-state hybrid model is applied whereby the upper layer or layers of a coarse MODFLOW model are replaced by the analytic element model GFLOW, which represents surface waters and wells as line and point sinks. The two models are coupled by transferring cell-by-cell leakage obtained from the original MODFLOW model to the bottom of the GFLOW model. A real-world test of the hybrid model approach is applied on a subdomain of an existing model of the Lake Michigan Basin. The original (coarse) MODFLOW model consists of six layers, the top four of which are aggregated into GFLOW as a single layer, while the bottom two layers remain part of MODFLOW in the hybrid model. The hybrid model and a refined "benchmark" MODFLOW model simulate similar baseflows. The hybrid and benchmark models also simulate similar baseflow reductions due to nearby pumping when the well is located within the layers represented by GFLOW. However, the benchmark model requires refinement of the model grid in the local area of interest, while the hybrid approach uses a gridless top layer and is thus unaffected by grid discretization errors. The hybrid approach is well suited to facilitate cost-effective retrofitting of existing coarse grid MODFLOW models commonly used for regional studies because it leverages the strengths of both finite-difference and analytic element methods for predictions in mildly heterogeneous systems that can be simulated with steady-state conditions.
GPU-accelerated 3D neutron diffusion code based on finite difference method
Xu, Q.; Yu, G.; Wang, K.
2012-07-01
Finite difference method, as a traditional numerical solution to neutron diffusion equation, although considered simpler and more precise than the coarse mesh nodal methods, has a bottle neck to be widely applied caused by the huge memory and unendurable computation time it requires. In recent years, the concept of General-Purpose computation on GPUs has provided us with a powerful computational engine for scientific research. In this study, a GPU-Accelerated multi-group 3D neutron diffusion code based on finite difference method was developed. First, a clean-sheet neutron diffusion code (3DFD-CPU) was written in C++ on the CPU architecture, and later ported to GPUs under NVIDIA's CUDA platform (3DFD-GPU). The IAEA 3D PWR benchmark problem was calculated in the numerical test, where three different codes, including the original CPU-based sequential code, the HYPRE (High Performance Pre-conditioners)-based diffusion code and CITATION, were used as counterpoints to test the efficiency and accuracy of the GPU-based program. The results demonstrate both high efficiency and adequate accuracy of the GPU implementation for neutron diffusion equation. A speedup factor of about 46 times was obtained, using NVIDIA's Geforce GTX470 GPU card against a 2.50 GHz Intel Quad Q9300 CPU processor. Compared with the HYPRE-based code performing in parallel on an 8-core tower server, the speedup of about 2 still could be observed. More encouragingly, without any mathematical acceleration technology, the GPU implementation ran about 5 times faster than CITATION which was speeded up by using the SOR method and Chebyshev extrapolation technique. (authors)
An iterative finite difference method for solving the quantum hydrodynamic equations of motion
Kendrick, Brian K
2010-01-01
The quantum hydrodynamic equations of motion associated with the de Broglie-Bohm description of quantum mechanics describe a time evolving probability density whose 'fluid' elements evolve as a correlated ensemble of particle trajectories. These equations are intuitively appealing due to their similarities with classical fluid dynamics and the appearance of a generalized Newton's equation of motion in which the total force contains both a classical and quantum contribution. However, the direct numerical solution of the quantum hydrodynamic equations (QHE) is fraught with challenges: the probability 'fluid' is highly-compressible, it has zero viscosity, the quantum potential ('pressure') is non-linear, and if that weren't enough the quantum potential can also become singular during the course of the calculations. Collectively these properties are responsible for the notorious numerical instabilities associated with a direct numerical solution of the QHE. The most successful and stable numerical approach that has been used to date is based on the Moving Least Squares (MLS) algorithm. The improved stability of this approach is due to the repeated local least squares fitting which effectively filters or reduces the numerical noise which tends to accumulate with time. However, this method is also subject to instabilities if it is pushed too hard. In addition, the stability of the MLS approach often comes at the expense of reduced resolution or fidelity of the calculation (i.e., the MLS filtering eliminates the higher-frequency components of the solution which may be of interest). Recently, a promising new solution method has been developed which is based on an iterative solution of the QHE using finite differences. This method (referred to as the Iterative Finite Difference Method or IFDM) is straightforward to implement, computationally efficient, stable, and its accuracy and convergence properties are well understood. A brief overview of the IFDM will be presented
Overview of finite difference Hartree-Fock method algorithm, implementation and application
NASA Astrophysics Data System (ADS)
Kobus, J.
2012-12-01
Two-dimensional, finite difference Hartree-Fock method has been in constant usage and development over the last two decades. The method has proved stable and efficient enough to be applied to dozens of diatomic molecules, even to systems as large as the thorium fluoride. Its latest version is presented and the dependence of its accuracy on the grid size and efficiency on the overrelaxation parameters are discussed. The method has been mainly used to develop and calibrate sequences of universal even-tempered and polarization-consistent basis sets and assess basis set truncation and superposition errors. Its modified version has proved useful in testing various exchange-correlation potentials within the density functional theory. The method has turned out to be a valuable source of reference values of total energies, multipole moments, static polarizabilities and hyperpolarizabilities (αzz, βzzz, γzzzz, Az,zz and Bzz,zz) for atoms, diatomic molecules and their ions. Recently, it has been modified to allow to calculate the electrical properties of homonuclear molecules and the results for the Li2, N2, F2 and O2 systems are presented. Electrical properties of the AlF, CS, KCl diatomics and of highly ionized krypton atom (Kr+32) are reported as well. Accuracy of both the matrix Hartree-Fock employing universal even-tempered basis sets and the finite difference Hartree-Fock methods is discussed and the basis set superposition errors of the dipole polarizability and the first hyperpolarizability of the FH molecule is reexamined. Basis set superposition errors are also discussed in case of the dipole polarizability and the second hyperpolarizability of the F2 system.
Lloyd, Jeffrey T.; Clayton, John D.; Austin, Ryan A.; McDowell, David L.
2015-07-10
Background: The shock response of metallic single crystals can be captured using a micro-mechanical description of the thermoelastic-viscoplastic material response; however, using a such a description within the context of traditional numerical methods may introduce a physical artifacts. Advantages and disadvantages of complex material descriptions, in particular the viscoplastic response, must be framed within approximations introduced by numerical methods. Methods: Three methods of modeling the shock response of metallic single crystals are summarized: finite difference simulations, steady wave simulations, and algebraic solutions of the Rankine-Hugoniot jump conditions. For the former two numerical techniques, a dislocation density based framework describes the rate- and temperature-dependent shear strength on each slip system. For the latter analytical technique, a simple (two-parameter) rate- and temperature-independent linear hardening description is necessarily invoked to enable simultaneous solution of the governing equations. For all models, the same nonlinear thermoelastic energy potential incorporating elastic constants of up to order 3 is applied. Results: Solutions are compared for plate impact of highly symmetric orientations (all three methods) and low symmetry orientations (numerical methods only) of aluminum single crystals shocked to 5 GPa (weak shock regime) and 25 GPa (overdriven regime). Conclusions: For weak shocks, results of the two numerical methods are very similar, regardless of crystallographic orientation. For strong shocks, artificial viscosity affects the finite difference solution, and effects of transverse waves for the lower symmetry orientations not captured by the steady wave method become important. The analytical solution, which can only be applied to highly symmetric orientations, provides reasonable accuracy with regards to prediction of most variables in the final shocked state but, by construction, does not provide insight
NASA Technical Reports Server (NTRS)
Seidel, D. A.; Bennett, R. M.; Ricketts, R. H.
1983-01-01
A time-marching finite difference code, XTRAN3S, that solves the three-dimensional transonic small perturbation equation for flow over isolated wings has recently been developed. During initial applications of the program, problems were encountered in the prediction of unsteady forces. The use of a revised grid and force calculation scheme improved those predictions. Comparisons are made between predicted and experimental pressure data for a rectangular supercritical wing. Comparisons of steady and unsteady data at freestream Mach number = 0.700 show good agreement between calculated and experimental values. A comparison of steady data at freestream Mach number = 0.825 shows poor agreement between calculations and experiment. Program difficulties have been encountered with swept and tapered configurations.
Mayer, A
2006-10-01
We present a finite-difference scheme for computing the Green's function of a one-dimensional crystal. The method enables one to derive the band structure and the density of states of this type of structures, whatever the particular values of the potential energy. The technique also enables one to compute the influence of defects on the density of states and on the scattering of the eigenstates of the crystal. The technique is applied to the Krönig-Penney potential. In particular, we study the bound states of a square potential introduced in the crystal and their influence on the conductance of the system. We also determine the surface states induced by a termination of the Krönig-Penney potential. Our results turn out to be in excellent agreement with analytical expressions, which proves their validity and the versatility of the technique.
Comparative study of numerical schemes of TVD3, UNO3-ACM and optimized compact scheme
NASA Technical Reports Server (NTRS)
Lee, Duck-Joo; Hwang, Chang-Jeon; Ko, Duck-Kon; Kim, Jae-Wook
1995-01-01
Three different schemes are employed to solve the benchmark problem. The first one is a conventional TVD-MUSCL (Monotone Upwind Schemes for Conservation Laws) scheme. The second scheme is a UNO3-ACM (Uniformly Non-Oscillatory Artificial Compression Method) scheme. The third scheme is an optimized compact finite difference scheme modified by us: the 4th order Runge Kutta time stepping, the 4th order pentadiagonal compact spatial discretization with the maximum resolution characteristics. The problems of category 1 are solved by using the second (UNO3-ACM) and third (Optimized Compact) schemes. The problems of category 2 are solved by using the first (TVD3) and second (UNO3-ACM) schemes. The problem of category 5 is solved by using the first (TVD3) scheme. It can be concluded from the present calculations that the Optimized Compact scheme and the UN03-ACM show good resolutions for category 1 and category 2 respectively.
NASA Astrophysics Data System (ADS)
Lin, Xue-lei; Lu, Xin; Ng, Micheal K.; Sun, Hai-Wei
2016-10-01
A fast accurate approximation method with multigrid solver is proposed to solve a two-dimensional fractional sub-diffusion equation. Using the finite difference discretization of fractional time derivative, a block lower triangular Toeplitz matrix is obtained where each main diagonal block contains a two-dimensional matrix for the Laplacian operator. Our idea is to make use of the block ɛ-circulant approximation via fast Fourier transforms, so that the resulting task is to solve a block diagonal system, where each diagonal block matrix is the sum of a complex scalar times the identity matrix and a Laplacian matrix. We show that the accuracy of the approximation scheme is of O (ɛ). Because of the special diagonal block structure, we employ the multigrid method to solve the resulting linear systems. The convergence of the multigrid method is studied. Numerical examples are presented to illustrate the accuracy of the proposed approximation scheme and the efficiency of the proposed solver.
3-D geoelectrical modelling using finite-difference: a new boundary conditions improvement
NASA Astrophysics Data System (ADS)
Maineult, A.; Schott, J.-J.; Ardiot, A.
2003-04-01
Geoelectrical prospecting is a well-known and frequently used method for quantitative and non-destructive subsurface exploration until depths of a few hundreds metres. Thus archeological objects can be efficiently detected as their resistivities often contrast with those of the surrounding media. Nevertheless using the geoelectrical prospecting method has long been restricted due to inhability to model correctly arbitrarily-shaped structures. The one-dimensional modelling and inversion have long been classical, but are of no interest for the majority of field data, since the natural distribution of resistivity is rarely homogeneous or tabular. Since the 1970's some authors developed discrete methods in order to solve the two and three-dimensional problem, using mathematical tools such as finite-element or finite-difference. The finite-difference approach is quite simple, easily understandable and programmable. Since the work of Dey and Morrison (1979), this approach has become quite popular. Nevertheless, one of its major drawbacks is the difficulty to establish satisfying boundary conditions. Recently Lowry et al. (1989) and Zhao and Yedlin (1996) suggested some refinements on the improvement of the boundary problem. We propose a new betterment, based on the splitting of the potential into two terms, the potential due to a reference tabular medium and a secondary potential caused by a disturbance of this medium. The surface response of a tabular medium has long been known (see for example Koefoed 1979). Here we developed the analytical solution for the electrical tabular potential everywhere in the medium, in order to establish more satisfying boundary conditions. The response of the perturbation, that is to say the object of interest, is then solved using volume-difference and preconditioned conjugate gradient. Finally the grid is refined one or more times in the perturbed domain in order to ameliorate the precision. This method of modelling is easy to implement
Mimetic finite difference method for the stokes problem on polygonal meshes
Lipnikov, K; Beirao Da Veiga, L; Gyrya, V; Manzini, G
2009-01-01
Various approaches to extend the finite element methods to non-traditional elements (pyramids, polyhedra, etc.) have been developed over the last decade. Building of basis functions for such elements is a challenging task and may require extensive geometry analysis. The mimetic finite difference (MFD) method has many similarities with low-order finite element methods. Both methods try to preserve fundamental properties of physical and mathematical models. The essential difference is that the MFD method uses only the surface representation of discrete unknowns to build stiffness and mass matrices. Since no extension inside the mesh element is required, practical implementation of the MFD method is simple for polygonal meshes that may include degenerate and non-convex elements. In this article, we develop a MFD method for the Stokes problem on arbitrary polygonal meshes. The method is constructed for tensor coefficients, which will allow to apply it to the linear elasticity problem. The numerical experiments show the second-order convergence for the velocity variable and the first-order for the pressure.
Bringuier, Jonathan N; Mittra, Raj
2012-01-01
A rigorous full-wave solution, via the Finite-Difference-Time-Domain (FDTD) method, is performed in an attempt to obtain realistic communication channel models for on-body wireless transmission in Body-Area-Networks (BANs), which are local data networks using the human body as a propagation medium. The problem of modeling the coupling between body mounted antennas is often not amenable to attack by hybrid techniques owing to the complex nature of the human body. For instance, the time-domain Green's function approach becomes more involved when the antennas are not conformal. Furthermore, the human body is irregular in shape and has dispersion properties that are unique. One consequence of this is that we must resort to modeling the antenna network mounted on the body in its entirety, and the number of degrees of freedom (DoFs) can be on the order of billions. Even so, this type of problem can still be modeled by employing a parallel version of the FDTD algorithm running on a cluster. Lastly, we note that the results of rigorous simulation of BANs can serve as benchmarks for comparison with the abundance of measurement data. PMID:23012575
Inkinen, Satu I; Liukkonen, Jukka; Malo, Markus K H; Virén, Tuomas; Jurvelin, Jukka S; Töyräs, Juha
2016-07-01
Measurement of ultrasound backscattering is a promising diagnostic technique for arthroscopic evaluation of articular cartilage. However, contribution of collagen and chondrocytes on ultrasound backscattering and speed of sound in cartilage is not fully understood and is experimentally difficult to study. Agarose hydrogels have been used in tissue engineering applications of cartilage. Therefore, the aim of this study was to simulate the propagation of high frequency ultrasound (40 MHz) in agarose scaffolds with varying concentrations of chondrocytes (1 to 32 × 10(6) cells/ml) and collagen (1.56-200 mg/ml) using transversely isotropic two-dimensional finite difference time domain method (FDTD). Backscatter and speed of sound were evaluated from the simulated pulse-echo and through transmission measurements, respectively. Ultrasound backscatter increased with increasing collagen and chondrocyte concentrations. Furthermore, speed of sound increased with increasing collagen concentration. However, this was not observed with increasing chondrocyte concentrations. The present study suggests that the FDTD method may have some applicability in simulations of ultrasound scattering and propagation in constructs containing collagen and chondrocytes. Findings of this study indicate the significant role of collagen and chondrocytes as ultrasound scatterers and can aid in development of modeling approaches for understanding how cartilage architecture affects to the propagation of high frequency ultrasound. PMID:27475127
3D Finite-Difference Modeling of Scattered Teleseismic Wavefields in a Subduction Zone
NASA Astrophysics Data System (ADS)
Morozov, I. B.; Zheng, H.
2005-12-01
For a teleseismic array targeting subducting crust in a zone of active subduction, scattering from the zone underlying the trench result in subhorizontally-propagating waves that could be difficult to distinguish from converted P- and S- wave backscattered from the surface. Because back-scattered modes often provide the most spectacular images of subducting slabs, it is important to understand their differences from the arrivals scattered from the trench zone. To investigate the detailed teleseismic wavefield in a subduction zone environment, we performed a full-waveform, 3-D visco-elastic finite-difference modeling of teleseismic wave propagation using a Beowulf cluster. The synthetics show strong scattering from the trench zone, dominated by the mantle and crustal P-waves propagating at 6.2-8.1.km/s and slower. These scattered waves occupy the same time and moveout intervals as the backscattered modes, and also have similar amplitudes. Although their amplitude decay characters are different, with the uncertainties in the velocity and density structure of the subduction zone, unambiguous distinguishing of these modes appears difficult. However, under minimal assumptions (in particular, without invoking slab dehydration), recent observations of receiver function amplitudes decreasing away from the trench favor the interpretation of trench-zone scattering.
Three-dimensional finite difference viscoelastic wave modelling including surface topography
NASA Astrophysics Data System (ADS)
Hestholm, Stig
1999-12-01
I have undertaken 3-D finite difference (FD) modelling of seismic scattering fromfree-surface topography. Exact free-surface boundary conditions for arbitrary 3-D topographies have been derived for the particle velocities. The boundary conditions are combined with a velocity-stress formulation of the full viscoelastic wave equations. A curved grid represents the physical medium and its upper boundary represents the free-surface topography. The wave equations are numerically discretized by an eighth-order FD method on a staggered grid in space, and a leap-frog technique and the Crank-Nicholson method in time. I simulate scattering from teleseismic P waves by using plane incident wave fronts and real topography from a 60 x 60 km area that includes the NORESS array of seismic receiver stations in southeastern Norway. Synthetic snapshots and seismograms of the wavefield show clear conversion from P to Rg (short-period fundamental-mode Rayleigh) waves in areas of rough topography, which is consistent with numerous observations. By parallelization on fast supercomputers, it is possible to model higher frequencies and/or larger areas than before.
Fakhari, Abbas; Lee, Taehun
2014-03-01
An adaptive-mesh-refinement (AMR) algorithm for the finite-difference lattice Boltzmann method (FDLBM) is presented in this study. The idea behind the proposed AMR is to remove the need for a tree-type data structure. Instead, pointer attributes are used to determine the neighbors of a certain block via appropriate adjustment of its children identifications. As a result, the memory and time required for tree traversal are completely eliminated, leaving us with an efficient algorithm that is easier to implement and use on parallel machines. To allow different mesh sizes at separate parts of the computational domain, the Eulerian formulation of the streaming process is invoked. As a result, there is no need for rescaling the distribution functions or using a temporal interpolation at the fine-coarse grid boundaries. The accuracy and efficiency of the proposed FDLBM AMR are extensively assessed by investigating a variety of vorticity-dominated flow fields, including Taylor-Green vortex flow, lid-driven cavity flow, thin shear layer flow, and the flow past a square cylinder.
An analysis of finite-difference and finite-volume formulations of conservation laws
NASA Technical Reports Server (NTRS)
Vinokur, Marcel
1986-01-01
Finite-difference and finite-volume formulations are analyzed in order to clear up the confusion concerning their application to the numerical solution of conservation laws. A new coordinate-free formulation of systems of conservation laws is developed, which clearly distinguishes the role of physical vectors from that of algebraic vectors which characterize the system. The analysis considers general types of equations--potential, Euler, and Navier-Stokes. Three-dimensional unsteady flows with time-varying grids are described using a single, consistent nomeclature for both formulations. Grid motion due to a non-inertial reference frame as well as flow adaptation is covered. In comparing the two formulations, it is found useful to distinguish between differences in numerical methods and differences in grid definition. The former plays a role for non-Cartesian grids, and results in only cosmetic differences in the manner in which geometric terms are handled. The differences in grid definition for the two formulations is found to be more important, since it affects the manner in which boundary conditions, zonal procedures, and grid singularities are handled at computational boundaries. The proper interpretation of strong and weak conservation-law forms for quasi-one-dimensional and axisymmetric flows is brought out.
NASA Astrophysics Data System (ADS)
Brissaud, Q.; Garcia, R.; Martin, R.; Komatitsch, D.
2015-12-01
The acoustic and gravity waves propagating in the planetary atmospheres have been studied intensively as markers of specific phenomena (tectonic events, explosions) or as contributors to the atmosphere dynamics. To get a better understanding of the physic behind these dynamic processes, both acoustic and gravity waves propagation should be modeled in an attenuating and windy 3D atmosphere from the ground to the upper thermosphere. Thus, In order to provide an efficient numerical tool at the regional or the global scale a high order finite difference time domain (FDTD) approach is proposed that relies on the linearized compressible Navier-Stokes equations (Landau 1959) with non constant physical parameters (density, viscosities and speed of sound) and background velocities (wind). One significant benefit from this code is its versatility. Indeed, it handles both acoustic and gravity waves in the same simulation that enables one to observe correlations between the two. Simulations will also be performed on 2D/3D realistic cases such as tsunamis in a full MSISE-00 atmosphere and gravity-wave generation through atmospheric explosions. Computations are validated by comparison to well-known analytical solutions based on dispersion relations in specific benchmark cases (atmospheric explosion and bottom displacement forcing).
NASA Astrophysics Data System (ADS)
Choi, S. J.; Kim, J.; Shin, S.
2014-12-01
In this presentation, a new non-hydrostatic (NH) dynamical core using the spectral element method (SEM) in the horizontal discretization and the finite difference method (FDM) in the vertical discretization will be presented. By using horizontal SEM, which decomposes the physical domain into smaller pieces with a small communication stencil, we can achieve a high level of scalability. Also by using vertical FDM, we provide an easy way for coupling the dynamics and existing physics packages. The Euler equations used here are in a flux form based on the hybrid sigma hydrostatic pressure vertical coordinate, which are similar to those used in the Weather Research and Forecasting (WRF) model. Within these Euler equations, we use a time-split third-order Runge-Kutta (RK3) for the time discretization. In order to establish robustness, firstly the NH dynamical core is verified in a simplified two dimensional (2D) slice framework by conducting widely used standard benchmark tests, and then we verify the global three dimensional (3D) dynamical core on the cubed-sphere grid with several test cases introduced by Dynamical Core Model Intercomparison Project (DCMIP).
Transfer-matrix approach for finite-difference time-domain simulation of periodic structures.
Deinega, Alexei; Belousov, Sergei; Valuev, Ilya
2013-11-01
Optical properties of periodic structures can be calculated using the transfer-matrix approach, which establishes a relation between amplitudes of the wave incident on a structure with transmitted or reflected waves. The transfer matrix can be used to obtain transmittance and reflectance spectra of finite periodic structures as well as eigenmodes of infinite structures. Traditionally, calculation of the transfer matrix is performed in the frequency domain and involves linear algebra. In this work, we present a technique for calculation of the transfer matrix using the finite-difference time-domain (FDTD) method and show the way of its implementation in FDTD code. To illustrate the performance of our technique we calculate the transmittance spectra for opal photonic crystal slabs consisting of multiple layers of spherical scatterers. Our technique can be used for photonic band structure calculations. It can also be combined with existing FDTD methods for the analysis of periodic structures at an oblique incidence, as well as for modeling point sources in a periodic environment. PMID:24329377
Global synthetic seismograms using a 2-D finite-difference method
NASA Astrophysics Data System (ADS)
Li, Dunzhu; Helmberger, Don; Clayton, Robert W.; Sun, Daoyuan
2014-05-01
Two-dimensional (2-D) finite-difference (FD) synthetics, which fill the gap between fast 1-D analytic synthetics and time-consuming full 3-D synthetics in our ability to model seismograms, have been used in many studies. We address several issues involving 2-D FD methods in generating global synthetic seismograms. These include: (1) interfacing point source excitation for earthquakes with 2-D FD methods; (2) out-of-plane spreading corrections and (3) reducing the spherical Earth to the flattened models. The first issue is tackled using two methods, a `transparent source box' approach and a moment tensor excitation approach, where each has its own advantages. Moreover, our `source box' excitation does not have the late-time drift problem that occurred in previous studies. The out-of-plane geometric spreading correction is accounted for by estimating the ray parameter and applying a post-simulation filter to 2-D synthetics. Finally, parameters of the Earth-flattening transformation are discussed and validated. The effectiveness of this method is demonstrated by comparing our synthetics with frequency-wavenumber summation, normal-mode and 3-D spectral-element synthetics.
NASA Astrophysics Data System (ADS)
Zhang, Zhenguo; Zhang, Wei; Li, Hong; Chen, Xiaofei
2013-03-01
Simulating seismic waves with uniform grid in heterogeneous high-velocity contrast media requires small-grid spacing determined by the global minimal velocity, which leads to huge number of grid points and small time step. To reduce the computational cost, discontinuous grids that use a finer grid at the shallow low-velocity region and a coarser grid at high-velocity regions are needed. In this paper, we present a discontinuous grid implementation for the collocated-grid finite-difference (FD) methods to increase the efficiency of seismic wave modelling. The grid spacing ratio n could be an arbitrary integer n ≥ 2. To downsample the wavefield from the finer grid to the coarser grid, our implementation can simply take the values on the finer grid without employing a downsampling filter for grid spacing ratio n = 2 to achieve stable results for long-time simulation. For grid spacing ratio n ≥ 3, the Gaussian filter should be used as the downsampling filter to get a stable simulation. To interpolate the wavefield from the coarse grid to the finer grid, the trilinear interpolation is used. Combining the efficiency of discontinuous grid with the flexibility of collocated-grid FD method on curvilinear grids, our method can simulate large-scale high-frequency strong ground motion of real earthquake with consideration of surface topography.
NASA Astrophysics Data System (ADS)
Fukuchi, Tsugio
2014-06-01
The finite difference method (FDM) based on Cartesian coordinate systems can be applied to numerical analyses over any complex domain. A complex domain is usually taken to mean that the geometry of an immersed body in a fluid is complex; here, it means simply an analytical domain of arbitrary configuration. In such an approach, we do not need to treat the outer and inner boundaries differently in numerical calculations; both are treated in the same way. Using a method that adopts algebraic polynomial interpolations in the calculation around near-wall elements, all the calculations over irregular domains reduce to those over regular domains. Discretization of the space differential in the FDM is usually derived using the Taylor series expansion; however, if we use the polynomial interpolation systematically, exceptional advantages are gained in deriving high-order differences. In using the polynomial interpolations, we can numerically solve the Poisson equation freely over any complex domain. Only a particular type of partial differential equation, Poisson's equations, is treated; however, the arguments put forward have wider generality in numerical calculations using the FDM.
NASA Technical Reports Server (NTRS)
Ryan, Deirdre A.; Langdon, H. Scott; Beggs, John H.; Steich, David J.; Luebbers, Raymond J.; Kunz, Karl S.
1992-01-01
The approach chosen to model steady state scattering from jet engines with moving turbine blades is based upon the Finite Difference Time Domain (FDTD) method. The FDTD method is a numerical electromagnetic program based upon the direct solution in the time domain of Maxwell's time dependent curl equations throughout a volume. One of the strengths of this method is the ability to model objects with complicated shape and/or material composition. General time domain functions may be used as source excitations. For example, a plane wave excitation may be specified as a pulse containing many frequencies and at any incidence angle to the scatterer. A best fit to the scatterer is accomplished using cubical cells in the standard cartesian implementation of the FDTD method. The material composition of the scatterer is determined by specifying its electrical properties at each cell on the scatterer. Thus, the FDTD method is a suitable choice for problems with complex geometries evaluated at multiple frequencies. It is assumed that the reader is familiar with the FDTD method.
Light Scattering by Gaussian Particles: A Solution with Finite-Difference Time Domain Technique
NASA Technical Reports Server (NTRS)
Sun, W.; Nousiainen, T.; Fu, Q.; Loeb, N. G.; Videen, G.; Muinonen, K.
2003-01-01
The understanding of single-scattering properties of complex ice crystals has significance in atmospheric radiative transfer and remote-sensing applications. In this work, light scattering by irregularly shaped Gaussian ice crystals is studied with the finite-difference time-domain (FDTD) technique. For given sample particle shapes and size parameters in the resonance region, the scattering phase matrices and asymmetry factors are calculated. It is found that the deformation of the particle surface can significantly smooth the scattering phase functions and slightly reduce the asymmetry factors. The polarization properties of irregular ice crystals are also significantly different from those of spherical cloud particles. These FDTD results could provide a reference for approximate light-scattering models developed for irregular particle shapes and can have potential applications in developing a much simpler practical light scattering model for ice clouds angular-distribution models and for remote sensing of ice clouds and aerosols using polarized light. (copyright) 2003 Elsevier Science Ltd. All rights reserved.
An analysis of finite-difference and finite-volume formulations of conservation laws
NASA Technical Reports Server (NTRS)
Vinokur, Marcel
1989-01-01
Finite-difference and finite-volume formulations are analyzed in order to clear up the confusion concerning their application to the numerical solution of conservation laws. A new coordinate-free formulation of systems of conservation laws is developed, which clearly distinguishes the role of physical vectors from that of algebraic vectors which characterize the system. The analysis considers general types of equations: potential, Euler, and Navier-Stokes. Three-dimensional unsteady flows with time-varying grids are described using a single, consistent nomenclature for both formulations. Grid motion due to a non-inertial reference frame as well as flow adaptation is covered. In comparing the two formulations, it is found useful to distinguish between differences in numerical methods and differences in grid definition. The former plays a role for non-Cartesian grids, and results in only cosmetic differences in the manner in which geometric terms are handled. The differences in grid definition for the two formulations is found to be more important, since it affects the manner in which boundary conditions, zonal procedures, and grid singularities are handled at computational boundaries. The proper interpretation of strong and weak conservation-law forms for quasi-one-dimensional and axisymmetric flows is brought out.
Simulation of optical devices using parallel finite-difference time-domain method
NASA Astrophysics Data System (ADS)
Li, Kang; Kong, Fanmin; Mei, Liangmo; Liu, Xin
2005-11-01
This paper presents a new parallel finite-difference time-domain (FDTD) numerical method in a low-cost network environment to stimulate optical waveguide characteristics. The PC motherboard based cluster is used, as it is relatively low-cost, reliable and has high computing performance. Four clusters are networked by fast Ethernet technology. Due to the simplicity nature of FDTD algorithm, a native Ethernet packet communication mechanism is used to reduce the overhead of the communication between the adjacent clusters. To validate the method, a microcavity ring resonator based on semiconductor waveguides is chosen as an instance of FDTD parallel computation. Speed-up rate under different division density is calculated. From the result we can conclude that when the decomposing size reaches a certain point, a good parallel computing speed up will be maintained. This simulation shows that through the overlapping of computation and communication method and controlling the decomposing size, the overhead of the communication of the shared data will be conquered. The result indicates that the implementation can achieve significant speed up for the FDTD algorithm. This will enable us to tackle the larger real electromagnetic problem by the low-cost PC clusters.
Bringuier, Jonathan N; Mittra, Raj
2012-01-01
A rigorous full-wave solution, via the Finite-Difference-Time-Domain (FDTD) method, is performed in an attempt to obtain realistic communication channel models for on-body wireless transmission in Body-Area-Networks (BANs), which are local data networks using the human body as a propagation medium. The problem of modeling the coupling between body mounted antennas is often not amenable to attack by hybrid techniques owing to the complex nature of the human body. For instance, the time-domain Green's function approach becomes more involved when the antennas are not conformal. Furthermore, the human body is irregular in shape and has dispersion properties that are unique. One consequence of this is that we must resort to modeling the antenna network mounted on the body in its entirety, and the number of degrees of freedom (DoFs) can be on the order of billions. Even so, this type of problem can still be modeled by employing a parallel version of the FDTD algorithm running on a cluster. Lastly, we note that the results of rigorous simulation of BANs can serve as benchmarks for comparison with the abundance of measurement data.
NASA Astrophysics Data System (ADS)
Brissaud, Quentin; Martin, Roland; Garcia, Raphaël F.; Komatitsch, Dimitri
2016-07-01
Acoustic and gravity waves propagating in planetary atmospheres have been studied intensively as markers of specific phenomena such as tectonic events or explosions or as contributors to atmosphere dynamics. To get a better understanding of the physics behind these dynamic processes, both acoustic and gravity waves propagation should be modelled in a 3-D attenuating and windy atmosphere extending from the ground to the upper thermosphere. Thus, in order to provide an efficient numerical tool at the regional or global scale, we introduce a finite difference in the time domain (FDTD) approach that relies on the linearized compressible Navier-Stokes equations with a background flow (wind). One significant benefit of such a method is its versatility because it handles both acoustic and gravity waves in the same simulation, which enables one to observe interactions between them. Simulations can be performed for 2-D or 3-D realistic cases such as tsunamis in a full MSISE-00 atmosphere or gravity-wave generation by atmospheric explosions. We validate the computations by comparing them to analytical solutions based on dispersion relations in specific benchmark cases: an atmospheric explosion, and a ground displacement forcing.
Acceleration of 3D Finite Difference AWP-ODC for seismic simulation on GPU Fermi Architecture
NASA Astrophysics Data System (ADS)
Zhou, J.; Cui, Y.; Choi, D.
2011-12-01
AWP-ODC, a highly scalable parallel finite-difference application, enables petascale 3D earthquake calculations. This application generates realistic dynamic earthquake source description and detailed physics-based anelastic ground motions at frequencies pertinent to safe building design. In 2010, the code achieved M8, a full dynamical simulation of a magnitude-8 earthquake on the southern San Andreas fault up to 2-Hz, the largest-ever earthquake simulation. Building on the success of the previous work, we have implemented CUDA on AWP-ODC to accelerate wave propagation on GPU platform. Our CUDA development aims on aggressive parallel efficiency, optimized global and shared memory access to make the best use of GPU memory hierarchy. The benchmark on NVIDIA Tesla C2050 graphics cards demonstrated many tens of speedup in single precision compared to serial implementation at a testing problem size, while an MPI-CUDA implementation is in the progress to extend our solver to multi-GPU clusters. Our CUDA implementation has been carefully verified for accuracy.
Jeong, Hyok; Lam, Yiu Wai
2012-01-01
The finite difference time domain (FDTD) method is a numerical technique that is straight forward to implement for the simulation of acoustic propagation. For room acoustics applications, the implementation of efficient source excitation and frequency dependent boundary conditions on arbitrary geometry can be seen as two of the most significant problems. This paper deals with the source implementation problem. Among existing source implementation methods, the hard source implementation is the simplest and computationally most efficient. Unfortunately, it generates a large low-frequency modulation in the measured time response. This paper presents a detailed investigation into these side effects. Surprisingly, some of these side effects are found to exist even if a transparent source implementation is used. By combing a time limited approach with a class of more natural source pulse function, this paper develops a source implementation method in FDTD that is as simple and computationally as efficient as a hard source implementation and yet capable of producing results that are virtually the same as a true transparent source. It is believed that the source implementation method developed in this paper will provide an improvement to the practical usability of the FDTD method for room acoustic simulation. PMID:22280589
Unsteady streamflow simulation using a linear implicit finite-difference model
Land, Larry F.
1978-01-01
A computer program for simulating one-dimensional subcritical, gradually varied, unsteady flow in a stream has been developed and documented. Given upstream and downstream boundary conditions and channel geometry data, roughness coefficients, stage, and discharge can be calculated anywhere within the reach as a function of time. The program uses a linear implicit finite-difference technique that discritizes the partial differential equations. Then it arranges the coefficients of the continuity and momentum equations into a pentadiagonal matrix for solution. Because it is a reasonable compromise between computational accuracy, speed and ease of use,the technique is one of the most commonly used. The upstream boundary condition is a depth hydrograph. However, options also allow the boundary condition to be discharge or water-surface elevation. The downstream boundary condition is a depth which may be constant, self-setting, or unsteady. The reach may be divided into uneven increments and the cross sections may be nonprismatic and may vary from one to the other. Tributary and lateral inflow may enter the reach. The digital model will simulate such common problems as (1) flood waves, (2) releases from dams, and (3) channels where storage is a consideration. It may also supply the needed flow information for mass-transport simulation. (Woodard-USGS)
Ground motion simulations in Marmara (Turkey) region from 3D finite difference method
NASA Astrophysics Data System (ADS)
Aochi, Hideo; Ulrich, Thomas; Douglas, John
2016-04-01
In the framework of the European project MARSite (2012-2016), one of the main contributions from our research team was to provide ground-motion simulations for the Marmara region from various earthquake source scenarios. We adopted a 3D finite difference code, taking into account the 3D structure around the Sea of Marmara (including the bathymetry) and the sea layer. We simulated two moderate earthquakes (about Mw4.5) and found that the 3D structure improves significantly the waveforms compared to the 1D layer model. Simulations were carried out for different earthquakes (moderate point sources and large finite sources) in order to provide shake maps (Aochi and Ulrich, BSSA, 2015), to study the variability of ground-motion parameters (Douglas & Aochi, BSSA, 2016) as well as to provide synthetic seismograms for the blind inversion tests (Diao et al., GJI, 2016). The results are also planned to be integrated in broadband ground-motion simulations, tsunamis generation and simulations of triggered landslides (in progress by different partners). The simulations are freely shared among the partners via the internet and the visualization of the results is diffused on the project's homepage. All these simulations should be seen as a reference for this region, as they are based on the latest knowledge that obtained during the MARSite project, although their refinement and validation of the model parameters and the simulations are a continuing research task relying on continuing observations. The numerical code used, the models and the simulations are available on demand.
Transfer-matrix approach for finite-difference time-domain simulation of periodic structures.
Deinega, Alexei; Belousov, Sergei; Valuev, Ilya
2013-11-01
Optical properties of periodic structures can be calculated using the transfer-matrix approach, which establishes a relation between amplitudes of the wave incident on a structure with transmitted or reflected waves. The transfer matrix can be used to obtain transmittance and reflectance spectra of finite periodic structures as well as eigenmodes of infinite structures. Traditionally, calculation of the transfer matrix is performed in the frequency domain and involves linear algebra. In this work, we present a technique for calculation of the transfer matrix using the finite-difference time-domain (FDTD) method and show the way of its implementation in FDTD code. To illustrate the performance of our technique we calculate the transmittance spectra for opal photonic crystal slabs consisting of multiple layers of spherical scatterers. Our technique can be used for photonic band structure calculations. It can also be combined with existing FDTD methods for the analysis of periodic structures at an oblique incidence, as well as for modeling point sources in a periodic environment.
Trescott, Peter C.; Pinder, George Francis; Larson, S.P.
1976-01-01
The model will simulate ground-water flow in an artesian aquifer, a water-table aquifer, or a combined artesian and water-table aquifer. The aquifer may be heterogeneous and anisotropic and have irregular boundaries. The source term in the flow equation may include well discharge, constant recharge, leakage from confining beds in which the effects of storage are considered, and evapotranspiration as a linear function of depth to water. The theoretical development includes presentation of the appropriate flow equations and derivation of the finite-difference approximations (written for a variable grid). The documentation emphasizes the numerical techniques that can be used for solving the simultaneous equations and describes the results of numerical experiments using these techniques. Of the three numerical techniques available in the model, the strongly implicit procedure, in general, requires less computer time and has fewer numerical difficulties than do the iterative alternating direction implicit procedure and line successive overrelaxation (which includes a two-dimensional correction procedure to accelerate convergence). The documentation includes a flow chart, program listing, an example simulation, and sections on designing an aquifer model and requirements for data input. It illustrates how model results can be presented on the line printer and pen plotters with a program that utilizes the graphical display software available from the Geological Survey Computer Center Division. In addition the model includes options for reading input data from a disk and writing intermediate results on a disk.
Bringuier, Jonathan N.; Mittra, Raj
2012-01-01
A rigorous full-wave solution, via the Finite-Difference-Time-Domain (FDTD) method, is performed in an attempt to obtain realistic communication channel models for on-body wireless transmission in Body-Area-Networks (BANs), which are local data networks using the human body as a propagation medium. The problem of modeling the coupling between body mounted antennas is often not amenable to attack by hybrid techniques owing to the complex nature of the human body. For instance, the time-domain Green's function approach becomes more involved when the antennas are not conformal. Furthermore, the human body is irregular in shape and has dispersion properties that are unique. One consequence of this is that we must resort to modeling the antenna network mounted on the body in its entirety, and the number of degrees of freedom (DoFs) can be on the order of billions. Even so, this type of problem can still be modeled by employing a parallel version of the FDTD algorithm running on a cluster. Lastly, we note that the results of rigorous simulation of BANs can serve as benchmarks for comparison with the abundance of measurement data. PMID:23012575
Composite centered schemes for multidimensional conservation laws
Liska, R.; Wendroff, B.
1998-05-08
The oscillations of a centered second order finite difference scheme and the excessive diffusion of a first order centered scheme can be overcome by global composition of the two, that is by performing cycles consisting of several time steps of the second order method followed by one step of the diffusive method. The authors show the effectiveness of this approach on some test problems in two and three dimensions.
Fu, Q.; Sun, W.B.; Yang, P.
1998-09-01
An accurate parameterization is presented for the infrared radiative properties of cirrus clouds. For the single-scattering calculations, a composite scheme is developed for randomly oriented hexagonal ice crystals by comparing results from Mie theory, anomalous diffraction theory (ADT), the geometric optics method (GOM), and the finite-difference time domain technique. This scheme employs a linear combination of single-scattering properties from the Mie theory, ADT, and GOM, which is accurate for a wide range of size parameters. Following the approach of Q. Fu, the extinction coefficient, absorption coefficient, and asymmetry factor are parameterized as functions of the cloud ice water content and generalized effective size (D{sub ge}). The present parameterization of the single-scattering properties of cirrus clouds is validated by examining the bulk radiative properties for a wide range of atmospheric conditions. Compared with reference results, the typical relative error in emissivity due to the parameterization is {approximately}2.2%. The accuracy of this parameterization guarantees its reliability in applications to climate models. The present parameterization complements the scheme for the solar radiative properties of cirrus clouds developed by Q. Fu for use in numerical models.
NASA Astrophysics Data System (ADS)
Fu, Qiang; Yang, Ping; Sun, W. B.
1998-09-01
An accurate parameterization is presented for the infrared radiative properties of cirrus clouds. For the single-scattering calculations, a composite scheme is developed for randomly oriented hexagonal ice crystals by comparing results from Mie theory, anomalous diffraction theory (ADT), the geometric optics method (GOM), and the finite-difference time domain technique. This scheme employs a linear combination of single-scattering properties from the Mie theory, ADT, and GOM, which is accurate for a wide range of size parameters. Following the approach of Q. Fu, the extinction coefficient, absorption coefficient, and asymmetry factor are parameterized as functions of the cloud ice water content and generalized effective size (Dge). The present parameterization of the single-scattering properties of cirrus clouds is validated by examining the bulk radiative properties for a wide range of atmospheric conditions. Compared with reference results, the typical relative error in emissivity due to the parameterization is 2.2%. The accuracy of this parameterization guarantees its reliability in applications to climate models. The present parameterization complements the scheme for the solar radiative properties of cirrus clouds developed by Q. Fu for use in numerical models.
Stable explicit schemes for equations of Schroedinger type
NASA Technical Reports Server (NTRS)
Mickens, Ronald E.
1989-01-01
A method for constructing explicit finite-difference schemes which can be used to solve Schroedinger-type partial-differential equations is presented. A forward Euler scheme that is conditionally stable is given by the procedure. The results presented are based on the analysis of the simplest Schroedinger type equation.
NASA Astrophysics Data System (ADS)
Lin, M. C.; Loverich, J.; Stoltz, P. H.; Nieter, C.
2013-10-01
This work introduces a conformal finite difference time domain (CFDTD) particle-in-cell (PIC) method with an improved field emission algorithm to accurately and efficiently study field emission devices. The CFDTD method is based on the Dey-Mittra algorithm or cut-cell algorithm, as implemented in the Vorpal code. For the field emission algorithm, we employ the elliptic function v(y) found by Forbes and a new fitting function t(y)2 for the Fowler-Nordheim (FN) equation. With these improved correction factors, field emission of electrons from a cathode surface is much closer to the prediction of the exact FN formula derived by Murphy and Good. This work was supported in part by both the U.S. Department of Defense under Grant No. FA9451-07-C-0025 and the U.S. Department of Energy under Grant No. DE-SC0004436.
An implicit finite-difference code for a two-equation turbulence model for three-dimensional flows
NASA Technical Reports Server (NTRS)
Kaul, U. K.
1985-01-01
An implicit finite difference code was developed which solves the transport equations for the turbulence kinetic energy and its dissipation rate in generalized coordinates in three dimensions. The finite difference equations are solved using the Beam-Warming algorithm. The kinetic energy-dissipation code, KEM, provides the closure; i.e., the turbulent viscosity for calculation of either compressible or incompressible flows. Turbulent internal flow over a backward-facing step has been calculated using the present code in conjunction with the Incompressible Navier-Stokes Code, INS3D. The results are in good agreement with experiments and two dimensional computations of other researchers.
NASA Astrophysics Data System (ADS)
Huang, Chun-Wen Paul
Small printed antennas are becoming one of the most popular designs in personal wireless communication systems, because these antennas feature light weight, small size, high frequency operation, and high transmission efficiency. Recently, a new type of printed meander line antenna was introduced. In this dissertation, the characteristics of a class of novel meander line antennas are presented and analyzed in detail using the finite difference time domain technique. The presented designs of the meander line antennas feature small dimensions (less than 77 x 11 x 3.17 mm3), approximately 50 Ω input impedance, dual or multiple frequency bands, and operate within the 0.9-3.0 GHz range on a comparably small ground plane (59 x 25.4 mm 2). Empirical design equations and the analysis for the input impedance and operating frequencies of different designs of meander line antennas are also provided. Validations of the numerical codes used in this investigation are made by comparing the computed FDTD results for known geometries with numerical results by other methods and measurements. Prototypes of optimized printed taper meander line antennas are built, and measurements of their return loss are compared with the computational results obtained using the FDTD technique. Very good agreement is found between FDTD numerical analysis and comparison results by other numerical methods and measurements. Optimal designs are achieved by applying dual printed sleeve tuners to equal segment ratio and taper meander line antennas, which cover the frequency range 1.25 GHz to 3.0 GHz with VSWR less than 1.4 and bandwidth varied within 120-340 MHz These optimal designs may effectively support the upper PCS bands and the future 3 rd generation cellular applications.
NASA Astrophysics Data System (ADS)
Mejdoubi, Abdelilah; Brosseau, Christian
2006-03-01
Currently, there is a great interest in tailoring the polarization properties of composite materials with the goal of controlling the dielectric behavior. This paper reports finite-difference time-domain (FDTD) modeling of the dielectric behavior of two-dimensional (2D) lossless two-phase heterostructures. More specifically, we present extensive results of 2D FDTD computations on the quasistatic effective permittivity of a single inclusion, with arbitrarily complex geometry (regular polygons and fractals), embedded in a plane. The uniaxial perfectly matched layer-absorbing boundary condition is found adequate for truncating the boundary of the 2D space because it leads to only very small backreflections. The effectiveness of the method is demonstrated by the variety of geometries modeled, i.e., regular polygons and fractals, and permittivity contrast ratios which allows us to distinguish between effects of surface fraction and effects of morphology. Our calculations show that geometrical effects can give rise to significant modifications of the surface fraction dependence of the permittivity. The results are compared with Maxwell-Garnett (MG) and symmetric Bruggeman (SBG) formulas. As expected the effective permittivity in the situations considered here deviates from the MG and SBG results at high surface fractions and/or high permittivity ratios between the inclusion and the host medium. In addition, the results show that a two-phase composite containing a fractal-boundary inclusion, e.g., Koch's snowflake, can have a permittivity which is several tens of percent lower between the first and the fourth iteration of the structure at a fixed perimeter-to-surface ratio. This feature is consistent with the fact that as the surface fraction becomes higher, the inclusion rough boundaries dominate the overall geometry. We believe that simplified modeling such as the modeling done here can serve as a useful purpose in understanding the interplay between the structure and
NASA Astrophysics Data System (ADS)
Lipnikov, K.; Manzini, G.; Svyatskiy, D.
2011-04-01
The maximum principle is one of the most important properties of solutions of partial differential equations. Its numerical analog, the discrete maximum principle (DMP), is one of the most difficult properties to achieve in numerical methods, especially when the computational mesh is distorted to adapt and conform to the physical domain or the problem coefficients are highly heterogeneous and anisotropic. Violation of the DMP may lead to numerical instabilities such as oscillations and to unphysical solutions such as heat flow from a cold material to a hot one. In this work, we investigate sufficient conditions to ensure the monotonicity of the mimetic finite difference (MFD) method on two- and three-dimensional meshes. These conditions result in a set of general inequalities for the elements of the mass matrix of every mesh element. Efficient solutions are devised for meshes consisting of simplexes, parallelograms and parallelepipeds, and orthogonal locally refined elements as those used in the AMR methodology. On simplicial meshes, it turns out that the MFD method coincides with the mixed-hybrid finite element methods based on the low-order Raviart-Thomas vector space. Thus, in this case we recover the well-established conventional angle conditions of such approximations. Instead, in the other cases a suitable design of the MFD method allows us to formulate a monotone discretization for which the existence of a DMP can be theoretically proved. Moreover, on meshes of parallelograms we establish a connection with a similar monotonicity condition proposed for the Multi-Point Flux Approximation (MPFA) methods. Numerical experiments confirm the effectiveness of the considered monotonicity conditions.
Three-dimensional finite-difference modeling of non-linear ground notion
Jones, E.M.; Olsen, K.B.
1997-08-01
We present a hybrid finite-difference technique capable of modeling non-linear soil amplification from the 3-D finite-fault radiation pattern for earthquakes in arbitrary earth models. The method is applied to model non-linear effects in the soils of the San Fernando Valley (SFV) from the 17 January 1994 M 6.7 Northridge earthquake. 0-7 Hz particle velocities are computed for an area of 17 km by 19 km immediately above the causative fault and 5 km below the surface where peak strike-parallel, strike-perpendicular, vertical, and total velocities reach values of 71 cm/s, 145 cm/s, 152 cm/s, and 180 cm/s, respectively. Selected Green`s functions and a soil model for the SFV are used to compute the approximate stress level during the earthquake, and comparison to the values for near-surface alluvium at the U.S. Nevada Test Site suggests that the non-linear regime may have been entered. We use selected values from the simulated particle velocity distribution at 5 km depth to compute the non-linear response in a soil column below a site within the Van Norman Complex in SFV, where the strongest ground motion was recorded. Since site-specific non- linear material parameters from the SFV are currently unavailable, values are taken from analyses of observed Test Site ground motions. Preliminary results show significant reduction of spectral velocities at the surface normalized to the peak source velocity due to non-linear effects when the peak velocity increases from 32 cm/s (approximately linear case) to 64 cm/s (30-92%), 93 cm/s (7-83%), and 124 cm/s (2-70%). The largest reduction occurs for frequencies above 1 Hz.
A NON-OSCILLATORY SCHEME FOR OPEN CHANNEL FLOWS. (R825200)
In modeling shocks in open channel flows, the traditional finite difference schemes become inefficient and warrant special numerical treatment for smooth computations. This paper provides a general introduction to the non-oscillatory high-resolution methodology, coupled with the ...
NASA Astrophysics Data System (ADS)
Jang, Jihyeon; Hong, Song-You
2016-04-01
The spectral method is generally assumed to provide better numerical accuracy than the finite difference method. However, the majority of regional models use finite discretization methods due to the difficulty of specifying time-dependent lateral boundary conditions in spectral models. This study evaluates the behavior of nonhydrostatic dynamics with a spectral discretization. To this end, Juang's nonhydrostatic dynamical core for the National Centers for Environmental Prediction (NCEP) regional spectral model has been implemented into the Regional Model Program (RMP) of the Global/Regional Integrated Model system (GRIMs). The behavior of the nonhydrostatic RMP is validated, and compared with that of the hydrostatic core in 2-D idealized experiments: the mountain wave, rising thermal bubble, and density current experiments. The nonhydrostatic effect in the RMP is further validated in comparison with the results from the Weather Research and Forecasting (WRF) model, which uses a finite difference method. The analyses of the experimental results from the RMP generally follow the characteristics found in previous studies without any discernible difference. For example, in both the RMP and the WRF model, the eastward-tilted propagation of mountain waves is very similar in the nonhydrostatic core experiments. Both nonhydrostatic models also efficiently reproduce the motion and deformation of the warm and cold bubbles, but the RMP results contain more small-scale noise. In a 1-km real-case simulation testbed, the lee waves that originate over the eastern flank of the Korean peninsula travel further eastward in the WRF model than in the RMP. It is found that differences of small-scale wave characteristics between the RMP and WRF model are mainly from the numerical techniques used, such as the accuracy of the advection scheme and the magnitude of the numerical diffusion, rather than from discrepancies in the spatial discretization.
NASA Astrophysics Data System (ADS)
Koltenbah, Benjamin E. C.; Parazzoli, Claudio G.; Greegor, Robert B.; Dowell, David H.
2002-07-01
Recent interest in advanced laser light sources has stimulated development of accelerator systems of intermediate beam energy, 100-200 MeV, and high charge, 1-10 nC, for high power FEL applications and high energy, 1-2 GeV, high charge, SASE-FEL applications. The current generation of beam transport codes which were developed for high-energy, low-charge beams with low self-fields are inadequate to address this energy and charge regime, and better computational tools are required to accurately calculate self-fields. To that end, we have developed a new version of PARMELA, named PARMELA_B and written in Fortran 95, which includes a coherent synchrotron radiation (CSR) routine and an improved, generalized space charge (SC) routine. An electron bunch is simulated by a collection of macro-particles, which traverses a series of beam line elements. At each time step through the calculation, the momentum of each particle is updated due to the presence of external and self-fields. The self-fields are due to CSR and SC. For the CSR calculations, the macro-particles are further combined into macro-particle-bins that follow the central trajectory of the bend. The energy change through the time step is calculated from expressions derived from the Liénard-Wiechart formulae, and from this energy change the particle's momentum is updated. For the SC calculations, we maintain the same rest-frame-electrostatic approach of the original PARMELA; however, we employ a finite difference Poisson equation solver instead of the symmetrical ring algorithm of the original code. In this way, we relax the symmetry assumptions in the original code. This method is based upon standard numerical procedures and conserves momentum to first order. The SC computational grid is adaptive and conforms to the size of the pulse as it evolves through the calculation. We provide descriptions of these two algorithms, validation comparisons with other CSR and SC methods, and a limited comparison with
Sasaki, Akira; Kojo, Masashi; Hirose, Kikuji; Goto, Hidekazu
2011-11-01
The path-integral renormalization group and direct energy minimization method of practical first-principles electronic structure calculations for multi-body systems within the framework of the real-space finite-difference scheme are introduced. These two methods can handle higher dimensional systems with consideration of the correlation effect. Furthermore, they can be easily extended to the multicomponent quantum systems which contain more than two kinds of quantum particles. The key to the present methods is employing linear combinations of nonorthogonal Slater determinants (SDs) as multi-body wavefunctions. As one of the noticeable results, the same accuracy as the variational Monte Carlo method is achieved with a few SDs. This enables us to study the entire ground state consisting of electrons and nuclei without the need to use the Born-Oppenheimer approximation. Recent activities on methodological developments aiming towards practical calculations such as the implementation of auxiliary field for Coulombic interaction, the treatment of the kinetic operator in imaginary-time evolutions, the time-saving double-grid technique for bare-Coulomb atomic potentials and the optimization scheme for minimizing the total-energy functional are also introduced. As test examples, the total energy of the hydrogen molecule, the atomic configuration of the methylene and the electronic structures of two-dimensional quantum dots are calculated, and the accuracy, availability and possibility of the present methods are demonstrated.
NASA Technical Reports Server (NTRS)
Hosny, W. M.; Tabakoff, W.
1975-01-01
A two-dimensional finite difference numerical technique is presented to determine the temperature distribution in a solid blade of a radial guide vane. A computer program is written in Fortran IV for IBM 370/165 computer. The computer results obtained from these programs have a similar behavior and trend as those obtained by experimental results.
Mattila, Keijo Kalervo; Hegele Júnior, Luiz Adolfo; Philippi, Paulo Cesar
2014-01-01
We propose isotropic finite differences for high-accuracy approximation of high-rank derivatives. These finite differences are based on direct application of lattice-Boltzmann stencils. The presented finite-difference expressions are valid in any dimension, particularly in two and three dimensions, and any lattice-Boltzmann stencil isotropic enough can be utilized. A theoretical basis for the proposed utilization of lattice-Boltzmann stencils in the approximation of high-rank derivatives is established. In particular, the isotropy and accuracy properties of the proposed approximations are derived directly from this basis. Furthermore, in this formal development, we extend the theory of Hermite polynomial tensors in the case of discrete spaces and present expressions for the discrete inner products between monomials and Hermite polynomial tensors. In addition, we prove an equivalency between two approaches for constructing lattice-Boltzmann stencils. For the numerical verification of the presented finite differences, we introduce 5th-, 6th-, and 8th-order two-dimensional lattice-Boltzmann stencils. PMID:24688360
Finite-Difference Seismic Modeling of Discrete Fractures in a San Juan Basin Gas Reservoir
NASA Astrophysics Data System (ADS)
Daley, T. M.; Nihei, K. T.; Myer, L. R.; Majer, E. L.; Queen, J. H.; Fortuna, M. A.; Murphy, J. O.; Coates, R. T.
2001-12-01
As part of a Dept. of Energy sponsored program in fractured gas production, we are conducting numerical modeling of seismic wave propagation in fractured media. The current modeling algorithm is a 2-D, anisotropic, elastic, finite-difference implementation. Fractures are discrete (one grid point wide), vertical, and are described by two parameters, the normal and tangential fracture stiffness, which are converted to anisotropic, elastic constants and placed in an isotropic background. A five-layer, 2250 m2 model with 3 m grid point spacing is used study the effects of fracturing on two scales: long, compliant fractures (i.e. joints) at wide spacing (650 m) and short, stiff fractures at narrower spacing (21 m). The fracture spacing is approximately equal to bed thickness. The fracture stiffness value for the stiff, short fractures was derived from a conceptual model of regularly spaced, infinitely thin openings which are 30 % of the fracture length. The joints were arbitrarily assigned a stiffness 10 times lower (more compliant). The normal and tangential stiffness were assumed equal (for a model of gas-filled fractures). The layer properties (P- and S-velocity and density) and the model's scale are based on well information from the San Juan basin, focusing on the Mesa Verde unit and its Cliffhouse sandstone member. Surface seismic (including CMP gathers) and VSP geometries, as modeled, were based on field data acquired in the basin. The model results (including wavefield time snapshots, and two-component seismograms) show discrete P- and S-wave scattered events from the compliant joints which have large amplitude P-to-S converted phases. These converted waves can be observed in surface seismic acquisition geometry when they are reflected by the horizontal velocity interfaces. In VSP geometry the downgoing fracture-scattered phases can be directly observed. The closely spaced, stiffer fractures generate multiple scattering which is observed as lower amplitude