Many particle approximation of the Aw-Rascle-Zhang second order model for vehicular traffic.

PubMed

Francesco, Marco Di; Fagioli, Simone; Rosini, Massimiliano D

2017-02-01

We consider the follow-the-leader approximation of the Aw-Rascle-Zhang (ARZ) model for traffic flow in a multi population formulation. We prove rigorous convergence to weak solutions of the ARZ system in the many particle limit in presence of vacuum. The result is based on uniform BV estimates on the discrete particle velocity. We complement our result with numerical simulations of the particle method compared with some exact solutions to the Riemann problem of the ARZ system.

A fast numerical scheme for causal relativistic hydrodynamics with dissipation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Takamoto, Makoto, E-mail: takamoto@tap.scphys.kyoto-u.ac.jp; Inutsuka, Shu-ichiro

2011-08-01

Highlights: {yields} We have developed a new multi-dimensional numerical scheme for causal relativistic hydrodynamics with dissipation. {yields} Our new scheme can calculate the evolution of dissipative relativistic hydrodynamics faster and more effectively than existing schemes. {yields} Since we use the Riemann solver for solving the advection steps, our method can capture shocks very accurately. - Abstract: In this paper, we develop a stable and fast numerical scheme for relativistic dissipative hydrodynamics based on Israel-Stewart theory. Israel-Stewart theory is a stable and causal description of dissipation in relativistic hydrodynamics although it includes relaxation process with the timescale for collision of constituentmore » particles, which introduces stiff equations and makes practical numerical calculation difficult. In our new scheme, we use Strang's splitting method, and use the piecewise exact solutions for solving the extremely short timescale problem. In addition, since we split the calculations into inviscid step and dissipative step, Riemann solver can be used for obtaining numerical flux for the inviscid step. The use of Riemann solver enables us to capture shocks very accurately. Simple numerical examples are shown. The present scheme can be applied to various high energy phenomena of astrophysics and nuclear physics.« less

Shadow poles in coupled-channel problems calculated with the Berggren basis

NASA Astrophysics Data System (ADS)

Id Betan, R. M.; Kruppa, A. T.; Vertse, T.

2018-02-01

Background: In coupled-channels models the poles of the scattering S matrix are located on different Riemann sheets. Physical observables are affected mainly by poles closest to the physical region but sometimes shadow poles have considerable effect too. Purpose: The purpose of this paper is to show that in coupled-channels problems all poles of the S matrix can be located by an expansion in terms of a properly constructed complex-energy basis. Method: The Berggren basis is used for expanding the coupled-channels solutions. Results: The locations of the poles of the S matrix for the Cox potential, constructed for coupled-channels problems, were numerically calculated and compared with the exact ones. In a nuclear physics application the Jπ=3 /2+ resonant poles of 5He were calculated in a phenomenological two-channel model. The properties of both the normal and shadow resonances agree with previous findings. Conclusions: We have shown that, with an appropriately chosen Berggren basis, all poles of the S matrix including the shadow poles can be determined. We have found that the shadow pole of 5He migrates between Riemann sheets if the coupling strength is varied.

Calculation of Water Entry Problem for Free-falling Bodies Using a Developed Cartesian Cut Cell Mesh

NASA Astrophysics Data System (ADS)

Wenhua, Wang; Yanying, Wang

2010-05-01

This paper describes the development of free surface capturing method on Cartesian cut cell mesh to water entry problem for free-falling bodies with body-fluid interaction. The incompressible Euler equations for a variable density fluid system are presented as governing equations and the free surface is treated as a contact discontinuity by using free surface capturing method. In order to be convenient for dealing with the problem with moving body boundary, the Cartesian cut cell technique is adopted for generating the boundary-fitted mesh around body edge by cutting solid regions out of a background Cartesian mesh. Based on this mesh system, governing equations are discretized by finite volume method, and at each cell edge inviscid flux is evaluated by means of Roe's approximate Riemann solver. Furthermore, for unsteady calculation in time domain, a time accurate solution is achieved by a dual time-stepping technique with artificial compressibility method. For the body-fluid interaction, the projection method of momentum equations and exact Riemann solution are applied in the calculation of fluid pressure on the solid boundary. Finally, the method is validated by test case of water entry for free-falling bodies.

The Riemann problem for longitudinal motion in an elastic-plastic bar

DOE Office of Scientific and Technical Information (OSTI.GOV)

Trangenstein, J.A.; Pember, R.B.

In this paper the analytical solution to the Riemann problem for the Antman-Szymczak model of longitudinal motion in an elastic-plastic bar is constructed. The model involves two surfaces corresponding to plastic yield in tension and compression, and exhibits the appropriate limiting behavior for total compressions. The solution of the Riemann problem involves discontinuous changes in characteristic speeds due to transitions from elastic to plastic response. Illustrations are presented, in both state-space and self-similar coordinates, of the variety of possible solutions to the Riemann problem for possible use with numerical algorithms.

Riemann's and Helmholtz-Lie's problems of space from Weyl's relativistic perspective

NASA Astrophysics Data System (ADS)

Bernard, Julien

2018-02-01

I reconstruct Riemann's and Helmholtz-Lie's problems of space, from some perspectives that allow for a fruitful comparison with Weyl. In Part II. of his inaugural lecture, Riemann justifies that the infinitesimal metric is the square root of a quadratic form. Thanks to Finsler geometry, I clarify both the implicit and explicit hypotheses used for this justification. I explain that Riemann-Finsler's kind of method is also appropriate to deal with indefinite metrics. Nevertheless, Weyl shares with Helmholtz a strong commitment to the idea that the notion of group should be at the center of the foundations of geometry. Riemann missed this point, and that is why, according to Weyl, he dealt with the problem of space in a "too formal" way. As a consequence, to solve the problem of space, Weyl abandoned Riemann-Finsler's methods for group-theoretical ones. However, from a philosophical point of view, I show that Weyl and Helmholtz are in strong opposition. The meditation on Riemann's inaugural lecture, and its clear methodological separation between the infinitesimal and the finite parts of the problem of space, must have been crucial for Weyl, while searching for strong epistemological foundations for the group-theoretical methods, avoiding Helmholtz's unjustified transition from the finite to the infinitesimal.

Assessment of a high-resolution central scheme for the solution of the relativistic hydrodynamics equations

NASA Astrophysics Data System (ADS)

Lucas-Serrano, A.; Font, J. A.; Ibáñez, J. M.; Martí, J. M.

2004-12-01

We assess the suitability of a recent high-resolution central scheme developed by \\cite{kurganov} for the solution of the relativistic hydrodynamic equations. The novelty of this approach relies on the absence of Riemann solvers in the solution procedure. The computations we present are performed in one and two spatial dimensions in Minkowski spacetime. Standard numerical experiments such as shock tubes and the relativistic flat-faced step test are performed. As an astrophysical application the article includes two-dimensional simulations of the propagation of relativistic jets using both Cartesian and cylindrical coordinates. The simulations reported clearly show the capabilities of the numerical scheme of yielding satisfactory results, with an accuracy comparable to that obtained by the so-called high-resolution shock-capturing schemes based upon Riemann solvers (Godunov-type schemes), even well inside the ultrarelativistic regime. Such a central scheme can be straightforwardly applied to hyperbolic systems of conservation laws for which the characteristic structure is not explicitly known, or in cases where a numerical computation of the exact solution of the Riemann problem is prohibitively expensive. Finally, we present comparisons with results obtained using various Godunov-type schemes as well as with those obtained using other high-resolution central schemes which have recently been reported in the literature.

Inverse scattering transform and soliton classification of the coupled modified Korteweg-de Vries equation

NASA Astrophysics Data System (ADS)

Wu, Jianping; Geng, Xianguo

2017-12-01

The inverse scattering transform of the coupled modified Korteweg-de Vries equation is studied by the Riemann-Hilbert approach. In the direct scattering process, the spectral analysis of the Lax pair is performed, from which a Riemann-Hilbert problem is established for the equation. In the inverse scattering process, by solving Riemann-Hilbert problems corresponding to the reflectionless cases, three types of multi-soliton solutions are obtained. The multi-soliton classification is based on the zero structures of the Riemann-Hilbert problem. In addition, some figures are given to illustrate the soliton characteristics of the coupled modified Korteweg-de Vries equation.

The Riemann-Hilbert problem for nonsymmetric systems

NASA Astrophysics Data System (ADS)

Greenberg, W.; Zweifel, P. F.; Paveri-Fontana, S.

1991-12-01

A comparison of the Riemann-Hilbert problem and the Wiener-Hopf factorization problem arising in the solution of half-space singular integral equations is presented. Emphasis is on the factorization of functions lacking the reflection symmetry usual in transport theory.

A degeneration of two-phase solutions of the focusing nonlinear Schrödinger equation via Riemann-Hilbert problems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bertola, Marco, E-mail: Marco.Bertola@concordia.ca; Centre de Recherches Mathématiques, Université de Montréal, Montréal, Québec H3C 3J7; SISSA/ISAS, via Bonomea 265, Trieste

2015-06-15

Two-phase solutions of focusing NLS equation are classically constructed out of an appropriate Riemann surface of genus two and expressed in terms of the corresponding theta-function. We show here that in a certain limiting regime, such solutions reduce to some elementary ones called “Solitons on unstable condensate.” This degeneration turns out to be conveniently studied by means of basic tools from the theory of Riemann-Hilbert problems. In particular, no acquaintance with Riemann surfaces and theta-function is required for such analysis.

Visualization Techniques Applied to 155-mm Projectile Analysis

DTIC Science & Technology

2014-11-01

semi-infinite Riemann problems are used in CFD++ to provide upwind flux information to the underlying transport scheme. Approximate Riemann solvers ...characteristics-based inflow/outflow boundary condition, which is based on solving a Riemann problem at the boundary. 2.3 Numerics Rolling/spinning is the...the solution files generated by the computational fluid dynamics (CFD) solver for the time-accurate rolling simulations at each timestep for the Mach

On Exact Solutions of Rarefaction-Rarefaction Interactions in Compressible Isentropic Flow

NASA Astrophysics Data System (ADS)

Jenssen, Helge Kristian

2017-12-01

Consider the interaction of two centered rarefaction waves in one-dimensional, compressible gas flow with pressure function p(ρ )=a^2ρ ^γ with γ >1. The classic hodograph approach of Riemann provides linear 2nd order equations for the time and space variables t, x as functions of the Riemann invariants r, s within the interaction region. It is well known that t( r, s) can be given explicitly in terms of the hypergeometric function. We present a direct calculation (based on works by Darboux and Martin) of this formula, and show how the same approach provides an explicit formula for x( r, s) in terms of Appell functions (two-variable hypergeometric functions). Motivated by the issue of vacuum and total variation estimates for 1-d Euler flows, we then use the explicit t-solution to monitor the density field and its spatial variation in interactions of two centered rarefaction waves. It is found that the variation is always non-monotone, and that there is an overall increase in density variation if and only if γ >3. We show that infinite duration of the interaction is characterized by approach toward vacuum in the interaction region, and that this occurs if and only if the Riemann problem defined by the extreme initial states generates a vacuum. Finally, it is verified that the minimal density in such interactions decays at rate O(1)/ t.

A two-dimensional Riemann solver with self-similar sub-structure - Alternative formulation based on least squares projection

NASA Astrophysics Data System (ADS)

Balsara, Dinshaw S.; Vides, Jeaniffer; Gurski, Katharine; Nkonga, Boniface; Dumbser, Michael; Garain, Sudip; Audit, Edouard

2016-01-01

Just as the quality of a one-dimensional approximate Riemann solver is improved by the inclusion of internal sub-structure, the quality of a multidimensional Riemann solver is also similarly improved. Such multidimensional Riemann problems arise when multiple states come together at the vertex of a mesh. The interaction of the resulting one-dimensional Riemann problems gives rise to a strongly-interacting state. We wish to endow this strongly-interacting state with physically-motivated sub-structure. The self-similar formulation of Balsara [16] proves especially useful for this purpose. While that work is based on a Galerkin projection, in this paper we present an analogous self-similar formulation that is based on a different interpretation. In the present formulation, we interpret the shock jumps at the boundary of the strongly-interacting state quite literally. The enforcement of the shock jump conditions is done with a least squares projection (Vides, Nkonga and Audit [67]). With that interpretation, we again show that the multidimensional Riemann solver can be endowed with sub-structure. However, we find that the most efficient implementation arises when we use a flux vector splitting and a least squares projection. An alternative formulation that is based on the full characteristic matrices is also presented. The multidimensional Riemann solvers that are demonstrated here use one-dimensional HLLC Riemann solvers as building blocks. Several stringent test problems drawn from hydrodynamics and MHD are presented to show that the method works. Results from structured and unstructured meshes demonstrate the versatility of our method. The reader is also invited to watch a video introduction to multidimensional Riemann solvers on http://www.nd.edu/ dbalsara/Numerical-PDE-Course.

Riemann correlator in de Sitter including loop corrections from conformal fields

DOE Office of Scientific and Technical Information (OSTI.GOV)

Fröb, Markus B.; Verdaguer, Enric; Roura, Albert, E-mail: mfroeb@ffn.ub.edu, E-mail: albert.roura@uni-ulm.de, E-mail: enric.verdaguer@ub.edu

2014-07-01

The Riemann correlator with appropriately raised indices characterizes in a gauge-invariant way the quantum metric fluctuations around de Sitter spacetime including loop corrections from matter fields. Specializing to conformal fields and employing a method that selects the de Sitter-invariant vacuum in the Poincaré patch, we obtain the exact result for the Riemann correlator through order H{sup 4}/m{sub p}{sup 4}. The result is expressed in a manifestly de Sitter-invariant form in terms of maximally symmetric bitensors. Its behavior for both short and long distances (sub- and superhorizon scales) is analyzed in detail. Furthermore, by carefully taking the flat-space limit, the explicitmore » result for the Riemann correlator for metric fluctuations around Minkowki spacetime is also obtained. Although the main focus is on free scalar fields (our calculation corresponds then to one-loop order in the matter fields), the result for general conformal field theories is also derived.« less

Lagrangian solution of supersonic real gas flows

NASA Technical Reports Server (NTRS)

Loh, Ching-Yuen; Liou, Meng-Sing

1993-01-01

The present extention of a Lagrangian approach of the Riemann solution procedure, which was originally proposed for perfect gases, to real gases, is nontrivial and requires the development of an exact real-gas Riemann solver for the Lagrangian form of the conservation laws. Calculations including complex wave interactions of various types were conducted to test the accuracy and robustness of the approach. Attention is given to the case of 2D oblique waves' capture, where a slip line is clearly in evidence; the real gas effect is demonstrated in the case of a generic engine nozzle.

Nonlinear Conservation Laws and Finite Volume Methods

NASA Astrophysics Data System (ADS)

Leveque, Randall J.

Introduction Software Notation Classification of Differential Equations Derivation of Conservation Laws The Euler Equations of Gas Dynamics Dissipative Fluxes Source Terms Radiative Transfer and Isothermal Equations Multi-dimensional Conservation Laws The Shock Tube Problem Mathematical Theory of Hyperbolic Systems Scalar Equations Linear Hyperbolic Systems Nonlinear Systems The Riemann Problem for the Euler Equations Numerical Methods in One Dimension Finite Difference Theory Finite Volume Methods Importance of Conservation Form - Incorrect Shock Speeds Numerical Flux Functions Godunov's Method Approximate Riemann Solvers High-Resolution Methods Other Approaches Boundary Conditions Source Terms and Fractional Steps Unsplit Methods Fractional Step Methods General Formulation of Fractional Step Methods Stiff Source Terms Quasi-stationary Flow and Gravity Multi-dimensional Problems Dimensional Splitting Multi-dimensional Finite Volume Methods Grids and Adaptive Refinement Computational Difficulties Low-Density Flows Discrete Shocks and Viscous Profiles Start-Up Errors Wall Heating Slow-Moving Shocks Grid Orientation Effects Grid-Aligned Shocks Magnetohydrodynamics The MHD Equations One-Dimensional MHD Solving the Riemann Problem Nonstrict Hyperbolicity Stiffness The Divergence of B Riemann Problems in Multi-dimensional MHD Staggered Grids The 8-Wave Riemann Solver Relativistic Hydrodynamics Conservation Laws in Spacetime The Continuity Equation The 4-Momentum of a Particle The Stress-Energy Tensor Finite Volume Methods Multi-dimensional Relativistic Flow Gravitation and General Relativity References

A 3D finite element ALE method using an approximate Riemann solution

DOE PAGES

Chiravalle, V. P.; Morgan, N. R.

2016-08-09

Arbitrary Lagrangian–Eulerian finite volume methods that solve a multidimensional Riemann-like problem at the cell center in a staggered grid hydrodynamic (SGH) arrangement have been proposed. This research proposes a new 3D finite element arbitrary Lagrangian–Eulerian SGH method that incorporates a multidimensional Riemann-like problem. Here, two different Riemann jump relations are investigated. A new limiting method that greatly improves the accuracy of the SGH method on isentropic flows is investigated. A remap method that improves upon a well-known mesh relaxation and remapping technique in order to ensure total energy conservation during the remap is also presented. Numerical details and test problemmore » results are presented.« less

A 3D finite element ALE method using an approximate Riemann solution

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chiravalle, V. P.; Morgan, N. R.

Arbitrary Lagrangian–Eulerian finite volume methods that solve a multidimensional Riemann-like problem at the cell center in a staggered grid hydrodynamic (SGH) arrangement have been proposed. This research proposes a new 3D finite element arbitrary Lagrangian–Eulerian SGH method that incorporates a multidimensional Riemann-like problem. Here, two different Riemann jump relations are investigated. A new limiting method that greatly improves the accuracy of the SGH method on isentropic flows is investigated. A remap method that improves upon a well-known mesh relaxation and remapping technique in order to ensure total energy conservation during the remap is also presented. Numerical details and test problemmore » results are presented.« less

Homotopy decomposition method for solving one-dimensional time-fractional diffusion equation

NASA Astrophysics Data System (ADS)

Abuasad, Salah; Hashim, Ishak

2018-04-01

In this paper, we present the homotopy decomposition method with a modified definition of beta fractional derivative for the first time to find exact solution of one-dimensional time-fractional diffusion equation. In this method, the solution takes the form of a convergent series with easily computable terms. The exact solution obtained by the proposed method is compared with the exact solution obtained by using fractional variational homotopy perturbation iteration method via a modified Riemann-Liouville derivative.

A discontinuous Galerkin method for poroelastic wave propagation: The two-dimensional case

NASA Astrophysics Data System (ADS)

Dudley Ward, N. F.; Lähivaara, T.; Eveson, S.

2017-12-01

In this paper, we consider a high-order discontinuous Galerkin (DG) method for modelling wave propagation in coupled poroelastic-elastic media. The upwind numerical flux is derived as an exact solution for the Riemann problem including the poroelastic-elastic interface. Attenuation mechanisms in both Biot's low- and high-frequency regimes are considered. The current implementation supports non-uniform basis orders which can be used to control the numerical accuracy element by element. In the numerical examples, we study the convergence properties of the proposed DG scheme and provide experiments where the numerical accuracy of the scheme under consideration is compared to analytic and other numerical solutions.

A simple extension of Roe's scheme for real gases

DOE Office of Scientific and Technical Information (OSTI.GOV)

Arabi, Sina, E-mail: sina.arabi@polymtl.ca; Trépanier, Jean-Yves; Camarero, Ricardo

The purpose of this paper is to develop a highly accurate numerical algorithm to model real gas flows in local thermodynamic equilibrium (LTE). The Euler equations are solved using a finite volume method based on Roe's flux difference splitting scheme including real gas effects. A novel algorithm is proposed to calculate the Jacobian matrix which satisfies the flux difference splitting exactly in the average state for a general equation of state. This algorithm increases the robustness and accuracy of the method, especially around the contact discontinuities and shock waves where the gas properties jump appreciably. The results are compared withmore » an exact solution of the Riemann problem for the shock tube which considers the real gas effects. In addition, the method is applied to a blunt cone to illustrate the capability of the proposed extension in solving two dimensional flows.« less

CHOLLA: A New Massively Parallel Hydrodynamics Code for Astrophysical Simulation

NASA Astrophysics Data System (ADS)

Schneider, Evan E.; Robertson, Brant E.

2015-04-01

We present Computational Hydrodynamics On ParaLLel Architectures (Cholla ), a new three-dimensional hydrodynamics code that harnesses the power of graphics processing units (GPUs) to accelerate astrophysical simulations. Cholla models the Euler equations on a static mesh using state-of-the-art techniques, including the unsplit Corner Transport Upwind algorithm, a variety of exact and approximate Riemann solvers, and multiple spatial reconstruction techniques including the piecewise parabolic method (PPM). Using GPUs, Cholla evolves the fluid properties of thousands of cells simultaneously and can update over 10 million cells per GPU-second while using an exact Riemann solver and PPM reconstruction. Owing to the massively parallel architecture of GPUs and the design of the Cholla code, astrophysical simulations with physically interesting grid resolutions (≳2563) can easily be computed on a single device. We use the Message Passing Interface library to extend calculations onto multiple devices and demonstrate nearly ideal scaling beyond 64 GPUs. A suite of test problems highlights the physical accuracy of our modeling and provides a useful comparison to other codes. We then use Cholla to simulate the interaction of a shock wave with a gas cloud in the interstellar medium, showing that the evolution of the cloud is highly dependent on its density structure. We reconcile the computed mixing time of a turbulent cloud with a realistic density distribution destroyed by a strong shock with the existing analytic theory for spherical cloud destruction by describing the system in terms of its median gas density.

Riemann tensor of motion vision revisited.

PubMed

Brill, M

2001-07-02

This note shows that the Riemann-space interpretation of motion vision developed by Barth and Watson is neither necessary for their results, nor sufficient to handle an intrinsic coordinate problem. Recasting the Barth-Watson framework as a classical velocity-solver (as in computer vision) solves these problems.

On a new class of completely integrable nonlinear wave equations. I. Infinitely many conservation laws

NASA Astrophysics Data System (ADS)

Nutku, Y.

1985-06-01

We point out a class of nonlinear wave equations which admit infinitely many conserved quantities. These equations are characterized by a pair of exact one-forms. The implication that they are closed gives rise to equations, the characteristics and Riemann invariants of which are readily obtained. The construction of the conservation laws requires the solution of a linear second-order equation which can be reduced to canonical form using the Riemann invariants. The hodograph transformation results in a similar linear equation. We discuss also the symplectic structure and Bäcklund transformations associated with these equations.

The Riemann problem for the relativistic full Euler system with generalized Chaplygin proper energy density-pressure relation

NASA Astrophysics Data System (ADS)

Shao, Zhiqiang

2018-04-01

The relativistic full Euler system with generalized Chaplygin proper energy density-pressure relation is studied. The Riemann problem is solved constructively. The delta shock wave arises in the Riemann solutions, provided that the initial data satisfy some certain conditions, although the system is strictly hyperbolic and the first and third characteristic fields are genuinely nonlinear, while the second one is linearly degenerate. There are five kinds of Riemann solutions, in which four only consist of a shock wave and a centered rarefaction wave or two shock waves or two centered rarefaction waves, and a contact discontinuity between the constant states (precisely speaking, the solutions consist in general of three waves), and the other involves delta shocks on which both the rest mass density and the proper energy density simultaneously contain the Dirac delta function. It is quite different from the previous ones on which only one state variable contains the Dirac delta function. The formation mechanism, generalized Rankine-Hugoniot relation and entropy condition are clarified for this type of delta shock wave. Under the generalized Rankine-Hugoniot relation and entropy condition, we establish the existence and uniqueness of solutions involving delta shocks for the Riemann problem.

Some issues in the simulation of two-phase flows: The relative velocity

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gräbel, J.; Hensel, S.; Ueberholz, P.

In this paper we compare numerical approximations for solving the Riemann problem for a hyperbolic two-phase flow model in two-dimensional space. The model is based on mixture parameters of state where the relative velocity between the two-phase systems is taken into account. This relative velocity appears as a main discontinuous flow variable through the complete wave structure and cannot be recovered correctly by some numerical techniques when simulating the associated Riemann problem. Simulations are validated by comparing the results of the numerical calculation qualitatively with OpenFOAM software. Simulations also indicate that OpenFOAM is unable to resolve the relative velocity associatedmore » with the Riemann problem.« less

Non-uniqueness of admissible weak solutions to the Riemann problem for isentropic Euler equations

NASA Astrophysics Data System (ADS)

Chiodaroli, Elisabetta; Kreml, Ondřej

2018-04-01

We study the Riemann problem for multidimensional compressible isentropic Euler equations. Using the framework developed in Chiodaroli et al (2015 Commun. Pure Appl. Math. 68 1157–90), and based on the techniques of De Lellis and Székelyhidi (2010 Arch. Ration. Mech. Anal. 195 225–60), we extend the results of Chiodaroli and Kreml (2014 Arch. Ration. Mech. Anal. 214 1019–49) and prove that it is possible to characterize a set of Riemann data, giving rise to a self-similar solution consisting of one admissible shock and one rarefaction wave, for which the problem also admits infinitely many admissible weak solutions.

Assessment of numerical methods for the solution of fluid dynamics equations for nonlinear resonance systems

NASA Technical Reports Server (NTRS)

Przekwas, A. J.; Yang, H. Q.

1989-01-01

The capability of accurate nonlinear flow analysis of resonance systems is essential in many problems, including combustion instability. Classical numerical schemes are either too diffusive or too dispersive especially for transient problems. In the last few years, significant progress has been made in the numerical methods for flows with shocks. The objective was to assess advanced shock capturing schemes on transient flows. Several numerical schemes were tested including TVD, MUSCL, ENO, FCT, and Riemann Solver Godunov type schemes. A systematic assessment was performed on scalar transport, Burgers' and gas dynamic problems. Several shock capturing schemes are compared on fast transient resonant pipe flow problems. A system of 1-D nonlinear hyperbolic gas dynamics equations is solved to predict propagation of finite amplitude waves, the wave steepening, formation, propagation, and reflection of shocks for several hundred wave cycles. It is shown that high accuracy schemes can be used for direct, exact nonlinear analysis of combustion instability problems, preserving high harmonic energy content for long periods of time.

The piecewise parabolic method for Riemann problems in nonlinear elasticity.

PubMed

Zhang, Wei; Wang, Tao; Bai, Jing-Song; Li, Ping; Wan, Zhen-Hua; Sun, De-Jun

2017-10-18

We present the application of Harten-Lax-van Leer (HLL)-type solvers on Riemann problems in nonlinear elasticity which undergoes high-load conditions. In particular, the HLLD ("D" denotes Discontinuities) Riemann solver is proved to have better robustness and efficiency for resolving complex nonlinear wave structures compared with the HLL and HLLC ("C" denotes Contact) solvers, especially in the shock-tube problem including more than five waves. Also, Godunov finite volume scheme is extended to higher order of accuracy by means of piecewise parabolic method (PPM), which could be used with HLL-type solvers and employed to construct the fluxes. Moreover, in the case of multi material components, level set algorithm is applied to track the interface between different materials, while the interaction of interfaces is realized through HLLD Riemann solver combined with modified ghost method. As seen from the results of both the solid/solid "stick" problem with the same material at the two sides of contact interface and the solid/solid "slip" problem with different materials at the two sides, this scheme composed of HLLD solver, PPM and level set algorithm can capture the material interface effectively and suppress spurious oscillations therein significantly.

Numerical Conformal Mapping Using Cross-Ratios and Delaunay Triangulation

NASA Technical Reports Server (NTRS)

Driscoll, Tobin A.; Vavasis, Stephen A.

1996-01-01

We propose a new algorithm for computing the Riemann mapping of the unit disk to a polygon, also known as the Schwarz-Christoffel transformation. The new algorithm, CRDT, is based on cross-ratios of the prevertices, and also on cross-ratios of quadrilaterals in a Delaunay triangulation of the polygon. The CRDT algorithm produces an accurate representation of the Riemann mapping even in the presence of arbitrary long, thin regions in the polygon, unlike any previous conformal mapping algorithm. We believe that CRDT can never fail to converge to the correct Riemann mapping, but the correctness and convergence proof depend on conjectures that we have so far not been able to prove. We demonstrate convergence with computational experiments. The Riemann mapping has applications to problems in two-dimensional potential theory and to finite-difference mesh generation. We use CRDT to produce a mapping and solve a boundary value problem on long, thin regions for which no other algorithm can solve these problems.

Whitham modulation theory for the Kadomtsev- Petviashvili equation.

PubMed

Ablowitz, Mark J; Biondini, Gino; Wang, Qiao

2017-08-01

The genus-1 Kadomtsev-Petviashvili (KP)-Whitham system is derived for both variants of the KP equation; namely the KPI and KPII equations. The basic properties of the KP-Whitham system, including symmetries, exact reductions and its possible complete integrability, together with the appropriate generalization of the one-dimensional Riemann problem for the Korteweg-de Vries equation are discussed. Finally, the KP-Whitham system is used to study the linear stability properties of the genus-1 solutions of the KPI and KPII equations; it is shown that all genus-1 solutions of KPI are linearly unstable, while all genus-1 solutions of KPII are linearly stable within the context of Whitham theory.

Whitham modulation theory for the Kadomtsev- Petviashvili equation

NASA Astrophysics Data System (ADS)

Ablowitz, Mark J.; Biondini, Gino; Wang, Qiao

2017-08-01

The genus-1 Kadomtsev-Petviashvili (KP)-Whitham system is derived for both variants of the KP equation; namely the KPI and KPII equations. The basic properties of the KP-Whitham system, including symmetries, exact reductions and its possible complete integrability, together with the appropriate generalization of the one-dimensional Riemann problem for the Korteweg-de Vries equation are discussed. Finally, the KP-Whitham system is used to study the linear stability properties of the genus-1 solutions of the KPI and KPII equations; it is shown that all genus-1 solutions of KPI are linearly unstable, while all genus-1 solutions of KPII are linearly stable within the context of Whitham theory.

Quantum spectral curve of the N=6 supersymmetric Chern-Simons theory.

PubMed

Cavaglià, Andrea; Fioravanti, Davide; Gromov, Nikolay; Tateo, Roberto

2014-07-11

Recently, it was shown that the spectrum of anomalous dimensions and other important observables in planar N=4 supersymmetric Yang-Mills theory are encoded into a simple nonlinear Riemann-Hilbert problem: the Pμ system or quantum spectral curve. In this Letter, we extend this formulation to the N=6 supersymmetric Chern-Simons theory introduced by Aharony, Bergman, Jafferis, and Maldacena. This may be an important step towards the exact determination of the interpolating function h(λ) characterizing the integrability of this model. We also discuss a surprising relation between the quantum spectral curves for the N=4 supersymmetric Yang-Mills theory and the N=6 supersymmetric Chern-Simons theory considered here.

A procedure to construct exact solutions of nonlinear fractional differential equations.

PubMed

Güner, Özkan; Cevikel, Adem C

2014-01-01

We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions.

Analytical approach for the fractional differential equations by using the extended tanh method

NASA Astrophysics Data System (ADS)

Pandir, Yusuf; Yildirim, Ayse

2018-07-01

In this study, we consider analytical solutions of space-time fractional derivative foam drainage equation, the nonlinear Korteweg-de Vries equation with time and space-fractional derivatives and time-fractional reaction-diffusion equation by using the extended tanh method. The fractional derivatives are defined in the modified Riemann-Liouville context. As a result, various exact analytical solutions consisting of trigonometric function solutions, kink-shaped soliton solutions and new exact solitary wave solutions are obtained.

Lie symmetry analysis, exact solutions and conservation laws for the time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation

NASA Astrophysics Data System (ADS)

Baleanu, Dumitru; Inc, Mustafa; Yusuf, Abdullahi; Aliyu, Aliyu Isa

2018-06-01

In this work, we investigate the Lie symmetry analysis, exact solutions and conservation laws (Cls) to the time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera (CDGDK) equation with Riemann-Liouville (RL) derivative. The time fractional CDGDK is reduced to nonlinear ordinary differential equation (ODE) of fractional order. New exact traveling wave solutions for the time fractional CDGDK are obtained by fractional sub-equation method. In the reduced equation, the derivative is in Erdelyi-Kober (EK) sense. Ibragimov's nonlocal conservation method is applied to construct Cls for time fractional CDGDK.

Exact solutions of fractional mBBM equation and coupled system of fractional Boussinesq-Burgers

NASA Astrophysics Data System (ADS)

Javeed, Shumaila; Saif, Summaya; Waheed, Asif; Baleanu, Dumitru

2018-06-01

The new exact solutions of nonlinear fractional partial differential equations (FPDEs) are established by adopting first integral method (FIM). The Riemann-Liouville (R-L) derivative and the local conformable derivative definitions are used to deal with the fractional order derivatives. The proposed method is applied to get exact solutions for space-time fractional modified Benjamin-Bona-Mahony (mBBM) equation and coupled time-fractional Boussinesq-Burgers equation. The suggested technique is easily applicable and effectual which can be implemented successfully to obtain the solutions for different types of nonlinear FPDEs.

Quasi-periodic Solutions of the Kaup-Kupershmidt Hierarchy

NASA Astrophysics Data System (ADS)

Geng, Xianguo; Wu, Lihua; He, Guoliang

2013-08-01

Based on solving the Lenard recursion equations and the zero-curvature equation, we derive the Kaup-Kupershmidt hierarchy associated with a 3×3 matrix spectral problem. Resorting to the characteristic polynomial of the Lax matrix for the Kaup-Kupershmidt hierarchy, we introduce a trigonal curve {K}_{m-1} and present the corresponding Baker-Akhiezer function and meromorphic function on it. The Abel map is introduced to straighten out the Kaup-Kupershmidt flows. With the aid of the properties of the Baker-Akhiezer function and the meromorphic function and their asymptotic expansions, we arrive at their explicit Riemann theta function representations. The Riemann-Jacobi inversion problem is achieved by comparing the asymptotic expansion of the Baker-Akhiezer function and its Riemann theta function representation, from which quasi-periodic solutions of the entire Kaup-Kupershmidt hierarchy are obtained in terms of the Riemann theta functions.

Comparative study of high-resolution shock-capturing schemes for a real gas

NASA Technical Reports Server (NTRS)

Montagne, J.-L.; Yee, H. C.; Vinokur, M.

1987-01-01

Recently developed second-order explicit shock-capturing methods, in conjunction with generalized flux-vector splittings, and a generalized approximate Riemann solver for a real gas are studied. The comparisons are made on different one-dimensional Riemann (shock-tube) problems for equilibrium air with various ranges of Mach numbers, densities and pressures. Six different Riemann problems are considered. These tests provide a check on the validity of the generalized formulas, since theoretical prediction of their properties appears to be difficult because of the non-analytical form of the state equation. The numerical results in the supersonic and low-hypersonic regimes indicate that these produce good shock-capturing capability and that the shock resolution is only slightly affected by the state equation of equilibrium air. The difference in shock resolution between the various methods varies slightly from one Riemann problem to the other, but the overall accuracy is very similar. For the one-dimensional case, the relative efficiency in terms of operation count for the different methods is within 30%. The main difference between the methods lies in their versatility in being extended to multidimensional problems with efficient implicit solution procedures.

Quantum spectral curve for arbitrary state/operator in AdS5/CFT4

NASA Astrophysics Data System (ADS)

Gromov, Nikolay; Kazakov, Vladimir; Leurent, Sébastien; Volin, Dmytro

2015-09-01

We give a derivation of quantum spectral curve (QSC) — a finite set of Riemann-Hilbert equations for exact spectrum of planar N=4 SYM theory proposed in our recent paper Phys. Rev. Lett. 112 (2014). We also generalize this construction to all local single trace operators of the theory, in contrast to the TBA-like approaches worked out only for a limited class of states. We reveal a rich algebraic and analytic structure of the QSC in terms of a so called Q-system — a finite set of Baxter-like Q-functions. This new point of view on the finite size spectral problem is shown to be completely compatible, though in a far from trivial way, with already known exact equations (analytic Y-system/TBA, or FiNLIE). We use the knowledge of this underlying Q-system to demonstrate how the classical finite gap solutions and the asymptotic Bethe ansatz emerge from our formalism in appropriate limits.

A Procedure to Construct Exact Solutions of Nonlinear Fractional Differential Equations

PubMed Central

Güner, Özkan; Cevikel, Adem C.

2014-01-01

We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions. PMID:24737972

Helmholtz, Riemann, and the Sirens: Sound, Color, and the "Problem of Space"

NASA Astrophysics Data System (ADS)

Pesic, Peter

2013-09-01

Emerging from music and the visual arts, questions about hearing and seeing deeply affected Hermann Helmholtz's and Bernhard Riemann's contributions to what became called the "problem of space [ Raumproblem]," which in turn influenced Albert Einstein's approach to general relativity. Helmholtz's physiological investigations measured the time dependence of nerve conduction and mapped the three-dimensional manifold of color sensation. His concurrent studies on hearing illuminated musical evidence through experiments with mechanical sirens that connect audible with visible phenomena, especially how the concept of frequency unifies motion, velocity, and pitch. Riemann's critique of Helmholtz's work on hearing led Helmholtz to respond and study Riemann's then-unpublished lecture on the foundations of geometry. During 1862-1870, Helmholtz applied his findings on the manifolds of hearing and seeing to the Raumproblem by supporting the quadratic distance relation Riemann had assumed as his fundamental hypothesis about geometrical space. Helmholtz also drew a "close analogy … in all essential relations between the musical scale and space." These intersecting studies of hearing and seeing thus led to reconsideration and generalization of the very concept of "space," which Einstein shaped into the general manifold of relativistic space-time.

Formulation of dynamical theory of X-ray diffraction for perfect crystals in the Laue case using the Riemann surface.

PubMed

Saka, Takashi

2016-05-01

The dynamical theory for perfect crystals in the Laue case was reformulated using the Riemann surface, as used in complex analysis. In the two-beam approximation, each branch of the dispersion surface is specified by one sheet of the Riemann surface. The characteristic features of the dispersion surface are analytically revealed using four parameters, which are the real and imaginary parts of two quantities specifying the degree of departure from the exact Bragg condition and the reflection strength. By representing these parameters on complex planes, these characteristics can be graphically depicted on the Riemann surface. In the conventional case, the absorption is small and the real part of the reflection strength is large, so the formulation is the same as the traditional analysis. However, when the real part of the reflection strength is small or zero, the two branches of the dispersion surface cross, and the dispersion relationship becomes similar to that of the Bragg case. This is because the geometrical relationships among the parameters are similar in both cases. The present analytical method is generally applicable, irrespective of the magnitudes of the parameters. Furthermore, the present method analytically revealed many characteristic features of the dispersion surface and will be quite instructive for further numerical calculations of rocking curves.

High Fidelity Modeling of Field Reversed Configuration (FRC) Thrusters

DTIC Science & Technology

2015-10-01

dispersion depends on the Riemann solver • Variables are allowed to be discontinuous at the cell interfaces Advantages - Method is conservative...release; distribution unlimited Discontinuous Galerkin (2) • Riemann problems are solved at each interface to compute fluxes • The source of dissipation

MUSTA fluxes for systems of conservation laws

NASA Astrophysics Data System (ADS)

Toro, E. F.; Titarev, V. A.

2006-08-01

This paper is about numerical fluxes for hyperbolic systems and we first present a numerical flux, called GFORCE, that is a weighted average of the Lax-Friedrichs and Lax-Wendroff fluxes. For the linear advection equation with constant coefficient, the new flux reduces identically to that of the Godunov first-order upwind method. Then we incorporate GFORCE in the framework of the MUSTA approach [E.F. Toro, Multi-Stage Predictor-Corrector Fluxes for Hyperbolic Equations. Technical Report NI03037-NPA, Isaac Newton Institute for Mathematical Sciences, University of Cambridge, UK, 17th June, 2003], resulting in a version that we call GMUSTA. For non-linear systems this gives results that are comparable to those of the Godunov method in conjunction with the exact Riemann solver or complete approximate Riemann solvers, noting however that in our approach, the solution of the Riemann problem in the conventional sense is avoided. Both the GFORCE and GMUSTA fluxes are extended to multi-dimensional non-linear systems in a straightforward unsplit manner, resulting in linearly stable schemes that have the same stability regions as the straightforward multi-dimensional extension of Godunov's method. The methods are applicable to general meshes. The schemes of this paper share with the family of centred methods the common properties of being simple and applicable to a large class of hyperbolic systems, but the schemes of this paper are distinctly more accurate. Finally, we proceed to the practical implementation of our numerical fluxes in the framework of high-order finite volume WENO methods for multi-dimensional non-linear hyperbolic systems. Numerical results are presented for the Euler equations and for the equations of magnetohydrodynamics.

On numerical instabilities of Godunov-type schemes for strong shocks

NASA Astrophysics Data System (ADS)

Xie, Wenjia; Li, Wei; Li, Hua; Tian, Zhengyu; Pan, Sha

2017-12-01

It is well known that low diffusion Riemann solvers with minimal smearing on contact and shear waves are vulnerable to shock instability problems, including the carbuncle phenomenon. In the present study, we concentrate on exploring where the instability grows out and how the dissipation inherent in Riemann solvers affects the unstable behaviors. With the help of numerical experiments and a linearized analysis method, it has been found that the shock instability is strongly related to the unstable modes of intermediate states inside the shock structure. The consistency of mass flux across the normal shock is needed for a Riemann solver to capture strong shocks stably. The famous carbuncle phenomenon is interpreted as the consequence of the inconsistency of mass flux across the normal shock for a low diffusion Riemann solver. Based on the results of numerical experiments and the linearized analysis, a robust Godunov-type scheme with a simple cure for the shock instability is suggested. With only the dissipation corresponding to shear waves introduced in the vicinity of strong shocks, the instability problem is circumvented. Numerical results of several carefully chosen strong shock wave problems are investigated to demonstrate the robustness of the proposed scheme.

Fractional analysis for nonlinear electrical transmission line and nonlinear Schroedinger equations with incomplete sub-equation

NASA Astrophysics Data System (ADS)

Fendzi-Donfack, Emmanuel; Nguenang, Jean Pierre; Nana, Laurent

2018-02-01

We use the fractional complex transform with the modified Riemann-Liouville derivative operator to establish the exact and generalized solutions of two fractional partial differential equations. We determine the solutions of fractional nonlinear electrical transmission lines (NETL) and the perturbed nonlinear Schroedinger (NLS) equation with the Kerr law nonlinearity term. The solutions are obtained for the parameters in the range (0<α≤1) of the derivative operator and we found the traditional solutions for the limiting case of α =1. We show that according to the modified Riemann-Liouville derivative, the solutions found can describe physical systems with memory effect, transient effects in electrical systems and nonlinear transmission lines, and other systems such as optical fiber.

The nonconvex multi-dimensional Riemann problem for Hamilton-Jacobi equations

NASA Technical Reports Server (NTRS)

Bardi, Martino; Osher, Stanley

1991-01-01

Simple inequalities are presented for the viscosity solution of a Hamilton-Jacobi equation in N space dimensions when neither the initial data nor the Hamiltonian need be convex (or concave). The initial data are uniformly Lipschitz and can be written as the sum of a convex function in a group of variables and a concave function in the remaining variables, therefore including the nonconvex Riemann problem. The inequalities become equalities wherever a 'maxmin' equals a 'minmax', and thus a representation formula for this problem is obtained, generalizing the classical Hopi formulas.

The Baker-Akhiezer Function and Factorization of the Chebotarev-Khrapkov Matrix

NASA Astrophysics Data System (ADS)

Antipov, Yuri A.

2014-10-01

A new technique is proposed for the solution of the Riemann-Hilbert problem with the Chebotarev-Khrapkov matrix coefficient {G(t) = α1(t)I + α2(t)Q(t)} , {α1(t), α2(t) in H(L)} , I = diag{1, 1}, Q(t) is a {2×2} zero-trace polynomial matrix. This problem has numerous applications in elasticity and diffraction theory. The main feature of the method is the removal of essential singularities of the solution to the associated homogeneous scalar Riemann-Hilbert problem on the hyperelliptic surface of an algebraic function by means of the Baker-Akhiezer function. The consequent application of this function for the derivation of the general solution to the vector Riemann-Hilbert problem requires the finding of the {ρ} zeros of the Baker-Akhiezer function ({ρ} is the genus of the surface). These zeros are recovered through the solution to the associated Jacobi problem of inversion of abelian integrals or, equivalently, the determination of the zeros of the associated degree-{ρ} polynomial and solution of a certain linear algebraic system of {ρ} equations.

An acoustic-convective splitting-based approach for the Kapila two-phase flow model

DOE Office of Scientific and Technical Information (OSTI.GOV)

Eikelder, M.F.P. ten, E-mail: m.f.p.teneikelder@tudelft.nl; Eindhoven University of Technology, Department of Mathematics and Computer Science, P.O. Box 513, 5600 MB Eindhoven; Daude, F.

In this paper we propose a new acoustic-convective splitting-based numerical scheme for the Kapila five-equation two-phase flow model. The splitting operator decouples the acoustic waves and convective waves. The resulting two submodels are alternately numerically solved to approximate the solution of the entire model. The Lagrangian form of the acoustic submodel is numerically solved using an HLLC-type Riemann solver whereas the convective part is approximated with an upwind scheme. The result is a simple method which allows for a general equation of state. Numerical computations are performed for standard two-phase shock tube problems. A comparison is made with a non-splittingmore » approach. The results are in good agreement with reference results and exact solutions.« less

A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations

NASA Astrophysics Data System (ADS)

Bhrawy, A. H.; Doha, E. H.; Baleanu, D.; Ezz-Eldien, S. S.

2015-07-01

In this paper, an efficient and accurate spectral numerical method is presented for solving second-, fourth-order fractional diffusion-wave equations and fractional wave equations with damping. The proposed method is based on Jacobi tau spectral procedure together with the Jacobi operational matrix for fractional integrals, described in the Riemann-Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations. The validity and effectiveness of the method are demonstrated by solving five numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier.

A critical assessment of flux and source term closures in shallow water models with porosity for urban flood simulations

NASA Astrophysics Data System (ADS)

Guinot, Vincent

2017-11-01

The validity of flux and source term formulae used in shallow water models with porosity for urban flood simulations is assessed by solving the two-dimensional shallow water equations over computational domains representing periodic building layouts. The models under assessment are the Single Porosity (SP), the Integral Porosity (IP) and the Dual Integral Porosity (DIP) models. 9 different geometries are considered. 18 two-dimensional initial value problems and 6 two-dimensional boundary value problems are defined. This results in a set of 96 fine grid simulations. Analysing the simulation results leads to the following conclusions: (i) the DIP flux and source term models outperform those of the SP and IP models when the Riemann problem is aligned with the main street directions, (ii) all models give erroneous flux closures when is the Riemann problem is not aligned with one of the main street directions or when the main street directions are not orthogonal, (iii) the solution of the Riemann problem is self-similar in space-time when the street directions are orthogonal and the Riemann problem is aligned with one of them, (iv) a momentum balance confirms the existence of the transient momentum dissipation model presented in the DIP model, (v) none of the source term models presented so far in the literature allows all flow configurations to be accounted for(vi) future laboratory experiments aiming at the validation of flux and source term closures should focus on the high-resolution, two-dimensional monitoring of both water depth and flow velocity fields.

The Riemann-Hilbert approach to the Helmholtz equation in a quarter-plane: Neumann, Robin and Dirichlet boundary conditions

NASA Astrophysics Data System (ADS)

Its, Alexander; Its, Elizabeth

2018-04-01

We revisit the Helmholtz equation in a quarter-plane in the framework of the Riemann-Hilbert approach to linear boundary value problems suggested in late 1990s by A. Fokas. We show the role of the Sommerfeld radiation condition in Fokas' scheme.

A Riemann-Hilbert approach to the inverse problem for the Stark operator on the line

NASA Astrophysics Data System (ADS)

Its, A.; Sukhanov, V.

2016-05-01

The paper is concerned with the inverse scattering problem for the Stark operator on the line with a potential from the Schwartz class. In our study of the inverse problem, we use the Riemann-Hilbert formalism. This allows us to overcome the principal technical difficulties which arise in the more traditional approaches based on the Gel’fand-Levitan-Marchenko equations, and indeed solve the problem. We also produce a complete description of the relevant scattering data (which have not been obtained in the previous works on the Stark operator) and establish the bijection between the Schwartz class potentials and the scattering data.

Elementary wave interactions in blood flow through artery

NASA Astrophysics Data System (ADS)

Raja Sekhar, T.; Minhajul

2017-10-01

In this paper, we consider the Riemann problem and interaction of elementary waves for the quasilinear hyperbolic system of conservation laws that arises in blood flow through arteries. We study the properties of solution involving shocks and rarefaction waves and establish the existence and uniqueness conditions. We show that the Riemann problem is solvable for arbitrary initial data under certain condition and construct the condition for no-feasible solution. Finally, we present numerical examples with different initial data and discuss all possible interactions of elementary waves.

On Riemann boundary value problems for null solutions of the two dimensional Helmholtz equation

NASA Astrophysics Data System (ADS)

Bory Reyes, Juan; Abreu Blaya, Ricardo; Rodríguez Dagnino, Ramón Martin; Kats, Boris Aleksandrovich

2018-01-01

The Riemann boundary value problem (RBVP to shorten notation) in the complex plane, for different classes of functions and curves, is still widely used in mathematical physics and engineering. For instance, in elasticity theory, hydro and aerodynamics, shell theory, quantum mechanics, theory of orthogonal polynomials, and so on. In this paper, we present an appropriate hyperholomorphic approach to the RBVP associated to the two dimensional Helmholtz equation in R^2 . Our analysis is based on a suitable operator calculus.

Semiclassical limit of the focusing NLS: Whitham equations and the Riemann-Hilbert Problem approach

NASA Astrophysics Data System (ADS)

Tovbis, Alexander; El, Gennady A.

2016-10-01

The main goal of this paper is to put together: a) the Whitham theory applicable to slowly modulated N-phase nonlinear wave solutions to the focusing nonlinear Schrödinger (fNLS) equation, and b) the Riemann-Hilbert Problem approach to particular solutions of the fNLS in the semiclassical (small dispersion) limit that develop slowly modulated N-phase nonlinear wave in the process of evolution. Both approaches have their own merits and limitations. Understanding of the interrelations between them could prove beneficial for a broad range of problems involving the semiclassical fNLS.

Thermodynamic limit of random partitions and dispersionless Toda hierarchy

NASA Astrophysics Data System (ADS)

Takasaki, Kanehisa; Nakatsu, Toshio

2012-01-01

We study the thermodynamic limit of random partition models for the instanton sum of 4D and 5D supersymmetric U(1) gauge theories deformed by some physical observables. The physical observables correspond to external potentials in the statistical model. The partition function is reformulated in terms of the density function of Maya diagrams. The thermodynamic limit is governed by a limit shape of Young diagrams associated with dominant terms in the partition function. The limit shape is characterized by a variational problem, which is further converted to a scalar-valued Riemann-Hilbert problem. This Riemann-Hilbert problem is solved with the aid of a complex curve, which may be thought of as the Seiberg-Witten curve of the deformed U(1) gauge theory. This solution of the Riemann-Hilbert problem is identified with a special solution of the dispersionless Toda hierarchy that satisfies a pair of generalized string equations. The generalized string equations for the 5D gauge theory are shown to be related to hidden symmetries of the statistical model. The prepotential and the Seiberg-Witten differential are also considered.

Entanglement Hamiltonians for Chiral Fermions with Zero Modes.

PubMed

Klich, Israel; Vaman, Diana; Wong, Gabriel

2017-09-22

In this Letter, we study the effect of topological zero modes on entanglement Hamiltonians and the entropy of free chiral fermions in (1+1)D. We show how Riemann-Hilbert solutions combined with finite rank perturbation theory allow us to obtain exact expressions for entanglement Hamiltonians. In the absence of the zero mode, the resulting entanglement Hamiltonians consist of local and bilocal terms. In the periodic sector, the presence of a zero mode leads to an additional nonlocal contribution to the entanglement Hamiltonian. We derive an exact expression for this term and for the resulting change in the entanglement entropy.

Exact solutions for STO and (3+1)-dimensional KdV-ZK equations using (G‧/G2) -expansion method

NASA Astrophysics Data System (ADS)

Bibi, Sadaf; Mohyud-Din, Syed Tauseef; Ullah, Rahmat; Ahmed, Naveed; Khan, Umar

This article deals with finding some exact solutions of nonlinear fractional differential equations (NLFDEs) by applying a relatively new method known as (G‧/G2) -expansion method. Solutions of space-time fractional Sharma-Tasso-Olever (STO) equation of fractional order and (3+1)-dimensional KdV-Zakharov Kuznetsov (KdV-ZK) equation of fractional order are reckoned to demonstrate the validity of this method. The fractional derivative version of modified Riemann-Liouville, linked with Fractional complex transform is employed to transform fractional differential equations into the corresponding ordinary differential equations.

New higher-order Godunov code for modelling performance of two-stage light gas guns

NASA Technical Reports Server (NTRS)

Bogdanoff, D. W.; Miller, R. J.

1995-01-01

A new quasi-one-dimensional Godunov code for modeling two-stage light gas guns is described. The code is third-order accurate in space and second-order accurate in time. A very accurate Riemann solver is used. Friction and heat transfer to the tube wall for gases and dense media are modeled and a simple nonequilibrium turbulence model is used for gas flows. The code also models gunpowder burn in the first-stage breech. Realistic equations of state (EOS) are used for all media. The code was validated against exact solutions of Riemann's shock-tube problem, impact of dense media slabs at velocities up to 20 km/sec, flow through a supersonic convergent-divergent nozzle and burning of gunpowder in a closed bomb. Excellent validation results were obtained. The code was then used to predict the performance of two light gas guns (1.5 in. and 0.28 in.) in service at the Ames Research Center. The code predictions were compared with measured pressure histories in the powder chamber and pump tube and with measured piston and projectile velocities. Very good agreement between computational fluid dynamics (CFD) predictions and measurements was obtained. Actual powder-burn rates in the gun were found to be considerably higher (60-90 percent) than predicted by the manufacturer and the behavior of the piston upon yielding appears to differ greatly from that suggested by low-strain rate tests.

Exact solutions of magnetohydrodynamics for describing different structural disturbances in solar wind

NASA Astrophysics Data System (ADS)

Grib, S. A.; Leora, S. N.

2016-03-01

We use analytical methods of magnetohydrodynamics to describe the behavior of cosmic plasma. This approach makes it possible to describe different structural fields of disturbances in solar wind: shock waves, direction discontinuities, magnetic clouds and magnetic holes, and their interaction with each other and with the Earth's magnetosphere. We note that the wave problems of solar-terrestrial physics can be efficiently solved by the methods designed for solving classical problems of mathematical physics. We find that the generalized Riemann solution particularly simplifies the consideration of secondary waves in the magnetosheath and makes it possible to describe in detail the classical solutions of boundary value problems. We consider the appearance of a fast compression wave in the Earth's magnetosheath, which is reflected from the magnetosphere and can nonlinearly overturn to generate a back shock wave. We propose a new mechanism for the formation of a plateau with protons of increased density and a magnetic field trough in the magnetosheath due to slow secondary shock waves. Most of our findings are confirmed by direct observations conducted on spacecrafts (WIND, ACE, Geotail, Voyager-2, SDO and others).

A flux splitting scheme with high-resolution and robustness for discontinuities

NASA Technical Reports Server (NTRS)

Wada, Yasuhiro; Liou, Meng-Sing

1994-01-01

A flux splitting scheme is proposed for the general nonequilibrium flow equations with an aim at removing numerical dissipation of Van-Leer-type flux-vector splittings on a contact discontinuity. The scheme obtained is also recognized as an improved Advection Upwind Splitting Method (AUSM) where a slight numerical overshoot immediately behind the shock is eliminated. The proposed scheme has favorable properties: high-resolution for contact discontinuities; conservation of enthalpy for steady flows; numerical efficiency; applicability to chemically reacting flows. In fact, for a single contact discontinuity, even if it is moving, this scheme gives the numerical flux of the exact solution of the Riemann problem. Various numerical experiments including that of a thermo-chemical nonequilibrium flow were performed, which indicate no oscillation and robustness of the scheme for shock/expansion waves. A cure for carbuncle phenomenon is discussed as well.

Exp-function method for solving fractional partial differential equations.

PubMed

Zheng, Bin

2013-01-01

We extend the Exp-function method to fractional partial differential equations in the sense of modified Riemann-Liouville derivative based on nonlinear fractional complex transformation. For illustrating the validity of this method, we apply it to the space-time fractional Fokas equation and the nonlinear fractional Sharma-Tasso-Olver (STO) equation. As a result, some new exact solutions for them are successfully established.

Universal moduli spaces of Riemann surfaces

NASA Astrophysics Data System (ADS)

Ji, Lizhen; Jost, Jürgen

2017-04-01

We construct a moduli space for Riemann surfaces that is universal in the sense that it represents compact Riemann surfaces of any finite genus. This moduli space is a connected complex subspace of an infinite dimensional complex space, and is stratified according to genus such that each stratum has a compact closure, and it carries a metric and a measure that induce a Riemannian metric and a finite volume measure on each stratum. Applications to the Plateau-Douglas problem for minimal surfaces of varying genus and to the partition function of Bosonic string theory are outlined. The construction starts with a universal moduli space of Abelian varieties. This space carries a structure of an infinite dimensional locally symmetric space which is of interest in its own right. The key to our construction of the universal moduli space then is the Torelli map that assigns to every Riemann surface its Jacobian and its extension to the Satake-Baily-Borel compactifications.

Well balancing of the SWE schemes for moving-water steady flows

NASA Astrophysics Data System (ADS)

Caleffi, Valerio; Valiani, Alessandro

2017-08-01

In this work, the exact reproduction of a moving-water steady flow via the numerical solution of the one-dimensional shallow water equations is studied. A new scheme based on a modified version of the HLLEM approximate Riemann solver (Dumbser and Balsara (2016) [18]) that exactly preserves the total head and the discharge in the simulation of smooth steady flows and that correctly dissipates mechanical energy in the presence of hydraulic jumps is presented. This model is compared with a selected set of schemes from the literature, including models that exactly preserve quiescent flows and models that exactly preserve moving-water steady flows. The comparison highlights the strengths and weaknesses of the different approaches. In particular, the results show that the increase in accuracy in the steady state reproduction is counterbalanced by a reduced robustness and numerical efficiency of the models. Some solutions to reduce these drawbacks, at the cost of increased algorithm complexity, are presented.

An exact solution for ideal dam-break floods on steep slopes

USGS Publications Warehouse

Ancey, C.; Iverson, R.M.; Rentschler, M.; Denlinger, R.P.

2008-01-01

The shallow-water equations are used to model the flow resulting from the sudden release of a finite volume of frictionless, incompressible fluid down a uniform slope of arbitrary inclination. The hodograph transformation and Riemann's method make it possible to transform the governing equations into a linear system and then deduce an exact analytical solution expressed in terms of readily evaluated integrals. Although the solution treats an idealized case never strictly realized in nature, it is uniquely well-suited for testing the robustness and accuracy of numerical models used to model shallow-water flows on steep slopes. Copyright 2008 by the American Geophysical Union.

Investigation of shock waves in the relativistic Riemann problem: A comparison of viscous fluid dynamics to kinetic theory

NASA Astrophysics Data System (ADS)

Bouras, I.; Molnár, E.; Niemi, H.; Xu, Z.; El, A.; Fochler, O.; Greiner, C.; Rischke, D. H.

2010-08-01

We solve the relativistic Riemann problem in viscous matter using the relativistic Boltzmann equation and the relativistic causal dissipative fluid-dynamical approach of Israel and Stewart. Comparisons between these two approaches clarify and point out the regime of validity of second-order fluid dynamics in relativistic shock phenomena. The transition from ideal to viscous shocks is demonstrated by varying the shear viscosity to entropy density ratio η/s. We also find that a good agreement between these two approaches requires a Knudsen number Kn<1/2.

Investigation of shock waves in the relativistic Riemann problem: A comparison of viscous fluid dynamics to kinetic theory

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bouras, I.; El, A.; Fochler, O.

2010-08-15

We solve the relativistic Riemann problem in viscous matter using the relativistic Boltzmann equation and the relativistic causal dissipative fluid-dynamical approach of Israel and Stewart. Comparisons between these two approaches clarify and point out the regime of validity of second-order fluid dynamics in relativistic shock phenomena. The transition from ideal to viscous shocks is demonstrated by varying the shear viscosity to entropy density ratio {eta}/s. We also find that a good agreement between these two approaches requires a Knudsen number Kn<1/2.

A sharp interface method for compressible liquid–vapor flow with phase transition and surface tension

DOE Office of Scientific and Technical Information (OSTI.GOV)

Fechter, Stefan, E-mail: stefan.fechter@iag.uni-stuttgart.de; Munz, Claus-Dieter, E-mail: munz@iag.uni-stuttgart.de; Rohde, Christian, E-mail: Christian.Rohde@mathematik.uni-stuttgart.de

The numerical approximation of non-isothermal liquid–vapor flow within the compressible regime is a difficult task because complex physical effects at the phase interfaces can govern the global flow behavior. We present a sharp interface approach which treats the interface as a shock-wave like discontinuity. Any mixing of fluid phases is avoided by using the flow solver in the bulk regions only, and a ghost-fluid approach close to the interface. The coupling states for the numerical solution in the bulk regions are determined by the solution of local two-phase Riemann problems across the interface. The Riemann solution accounts for the relevantmore » physics by enforcing appropriate jump conditions at the phase boundary. A wide variety of interface effects can be handled in a thermodynamically consistent way. This includes surface tension or mass/energy transfer by phase transition. Moreover, the local normal speed of the interface, which is needed to calculate the time evolution of the interface, is given by the Riemann solution. The interface tracking itself is based on a level-set method. The focus in this paper is the description of the two-phase Riemann solver and its usage within the sharp interface approach. One-dimensional problems are selected to validate the approach. Finally, the three-dimensional simulation of a wobbling droplet and a shock droplet interaction in two dimensions are shown. In both problems phase transition and surface tension determine the global bulk behavior.« less

Adaptive Osher-type scheme for the Euler equations with highly nonlinear equations of state

NASA Astrophysics Data System (ADS)

Lee, Bok Jik; Toro, Eleuterio F.; Castro, Cristóbal E.; Nikiforakis, Nikolaos

2013-08-01

For the numerical simulation of detonation of condensed phase explosives, a complex equation of state (EOS), such as the Jones-Wilkins-Lee (JWL) EOS or the Cochran-Chan (C-C) EOS, are widely used. However, when a conservative scheme is used for solving the Euler equations with such equations of state, a spurious solution across the contact discontinuity, a well known phenomenon in multi-fluid systems, arises even for single materials. In this work, we develop a generalised Osher-type scheme in an adaptive primitive-conservative framework to overcome the aforementioned difficulties. Resulting numerical solutions are compared with the exact solutions and with the numerical solutions from the Godunov method in conjunction with the exact Riemann solver for the Euler equations with Mie-Grüneisen form of equations of state, such as the JWL and the C-C equations of state. The adaptive scheme is extended to second order and its empirical convergence rates are presented, verifying second order accuracy for smooth solutions. Through a suite of several tests problems in one and two space dimensions we illustrate the failure of conservative schemes and the capability of the methods of this paper to overcome the difficulties.

Focusing of noncircular self-similar shock waves.

PubMed

Betelu, S I; Aronson, D G

2001-08-13

We study the focusing of noncircular shock waves in a perfect gas. We construct an explicit self-similar solution by combining three convergent plane waves with regular shock reflections between them. We then show, with a numerical Riemann solver, that there are initial conditions with smooth shocks whose intermediate asymptotic stage is described by the exact solution. Unlike the focusing of circular shocks, our self-similar shocks have bounded energy density.

An HLLC Riemann solver for resistive relativistic magnetohydrodynamics

NASA Astrophysics Data System (ADS)

Miranda-Aranguren, S.; Aloy, M. A.; Rembiasz, T.

2018-05-01

We present a new approximate Riemann solver for the augmented system of equations of resistive relativistic magnetohydrodynamics that belongs to the family of Harten-Lax-van Leer contact wave (HLLC) solvers. In HLLC solvers, the solution is approximated by two constant states flanked by two shocks separated by a contact wave. The accuracy of the new approximate solver is calibrated through 1D and 2D test problems.

The Invar tensor package: Differential invariants of Riemann

NASA Astrophysics Data System (ADS)

Martín-García, J. M.; Yllanes, D.; Portugal, R.

2008-10-01

The long standing problem of the relations among the scalar invariants of the Riemann tensor is computationally solved for all 6ṡ10 objects with up to 12 derivatives of the metric. This covers cases ranging from products of up to 6 undifferentiated Riemann tensors to cases with up to 10 covariant derivatives of a single Riemann. We extend our computer algebra system Invar to produce within seconds a canonical form for any of those objects in terms of a basis. The process is as follows: (1) an invariant is converted in real time into a canonical form with respect to the permutation symmetries of the Riemann tensor; (2) Invar reads a database of more than 6ṡ10 relations and applies those coming from the cyclic symmetry of the Riemann tensor; (3) then applies the relations coming from the Bianchi identity, (4) the relations coming from commutations of covariant derivatives, (5) the dimensionally-dependent identities for dimension 4, and finally (6) simplifies invariants that can be expressed as product of dual invariants. Invar runs on top of the tensor computer algebra systems xTensor (for Mathematica) and Canon (for Maple). Program summaryProgram title:Invar Tensor Package v2.0 Catalogue identifier:ADZK_v2_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZK_v2_0.html Program obtainable from:CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.:3 243 249 No. of bytes in distributed program, including test data, etc.:939 Distribution format:tar.gz Programming language:Mathematica and Maple Computer:Any computer running Mathematica versions 5.0 to 6.0 or Maple versions 9 and 11 Operating system:Linux, Unix, Windows XP, MacOS RAM:100 Mb Word size:64 or 32 bits Supplementary material:The new database of relations is much larger than that for the previous version and therefore has not been included in the distribution. To obtain the Mathematica and Maple database files click on this link. Classification:1.5, 5 Does the new version supersede the previous version?:Yes. The previous version (1.0) only handled algebraic invariants. The current version (2.0) has been extended to cover differential invariants as well. Nature of problem:Manipulation and simplification of scalar polynomial expressions formed from the Riemann tensor and its covariant derivatives. Solution method:Algorithms of computational group theory to simplify expressions with tensors that obey permutation symmetries. Tables of syzygies of the scalar invariants of the Riemann tensor. Reasons for new version:With this new version, the user can manipulate differential invariants of the Riemann tensor. Differential invariants are required in many physical problems in classical and quantum gravity. Summary of revisions:The database of syzygies has been expanded by a factor of 30. New commands were added in order to deal with the enlarged database and to manipulate the covariant derivative. Restrictions:The present version only handles scalars, and not expressions with free indices. Additional comments:The distribution file for this program is over 53 Mbytes and therefore is not delivered directly when download or Email is requested. Instead a html file giving details of how the program can be obtained is sent. Running time:One second to fully reduce any monomial of the Riemann tensor up to degree 7 or order 10 in terms of independent invariants. The Mathematica notebook included in the distribution takes approximately 5 minutes to run.

Eigenfunctions and Eigenvalues for a Scalar Riemann-Hilbert Problem Associated to Inverse Scattering

NASA Astrophysics Data System (ADS)

Pelinovsky, Dmitry E.; Sulem, Catherine

A complete set of eigenfunctions is introduced within the Riemann-Hilbert formalism for spectral problems associated to some solvable nonlinear evolution equations. In particular, we consider the time-independent and time-dependent Schrödinger problems which are related to the KdV and KPI equations possessing solitons and lumps, respectively. Non-standard scalar products, orthogonality and completeness relations are derived for these problems. The complete set of eigenfunctions is used for perturbation theory and bifurcation analysis of eigenvalues supported by the potentials under perturbations. We classify two different types of bifurcations of new eigenvalues and analyze their characteristic features. One type corresponds to thresholdless generation of solitons in the KdV equation, while the other predicts a threshold for generation of lumps in the KPI equation.

Exact ghost-free bigravitational waves

NASA Astrophysics Data System (ADS)

Ayón-Beato, Eloy; Higuita-Borja, Daniel; Méndez-Zavaleta, Julio A.; Velázquez-Rodríguez, Gerardo

2018-04-01

We study the propagation of exact gravitational waves in the ghost-free bimetric theory. Our focus is on type-N spacetimes compatible with the cosmological constants provided by the bigravity interaction potential, and particularly in the single class known by allowing at least a Killing symmetry: the AdS waves. They have the advantage of being represented by a generalized Kerr-Schild transformation from AdS spacetime. This entails a notorious simplification in bigravity by allowing to straightforwardly compute any power of its interaction square root matrix, opening the door to explore physically meaningful exact configurations. For these exact gravitational waves the complex dynamical structure of bigravity decomposes into elementary exact massless or massive excitations propagating on AdS. We use a complexified formulation of the Euler-Darboux equations to provide for the first time the general solutions to the massive version of the Siklos equation which rules the resulting AdS-wave dynamics, using an integral representation originally due to Poisson. Inspired by this progress, we tackle the subtle problem of how matter couples to bigravity and, more concretely, if this occurs through a composite metric, which is hard to handle in a general setting. Surprisingly, the Kerr-Schild ansatz brings again a huge simplification in how the related energy-momentum tensors are calculated. This allows us to explicitly characterize AdS waves supported by either a massless free scalar field or a wavefront-homogeneous Maxwell field. Considering the most general allowed Maxwell source instead is a highly nontrivial task, which we accomplish by again exploiting the complexified Euler-Darboux description and taking advantage of the classical Riemann method. In fact, this eventually allows us to find the most general configurations for any matter source.

An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension)

NASA Technical Reports Server (NTRS)

Powell, Kenneth G.

1994-01-01

An approximate Riemann solver is developed for the governing equations of ideal magnetohydrodynamics (MHD). The Riemann solver has an eight-wave structure, where seven of the waves are those used in previous work on upwind schemes for MHD, and the eighth wave is related to the divergence of the magnetic field. The structure of the eighth wave is not immediately obvious from the governing equations as they are usually written, but arises from a modification of the equations that is presented in this paper. The addition of the eighth wave allows multidimensional MHD problems to be solved without the use of staggered grids or a projection scheme, one or the other of which was necessary in previous work on upwind schemes for MHD. A test problem made up of a shock tube with rotated initial conditions is solved to show that the two-dimensional code yields answers consistent with the one-dimensional methods developed previously.

Lie symmetry analysis, conservation laws and exact solutions of the time-fractional generalized Hirota-Satsuma coupled KdV system

NASA Astrophysics Data System (ADS)

Saberi, Elaheh; Reza Hejazi, S.

2018-02-01

In the present paper, Lie point symmetries of the time-fractional generalized Hirota-Satsuma coupled KdV (HS-cKdV) system based on the Riemann-Liouville derivative are obtained. Using the derived Lie point symmetries, we obtain similarity reductions and conservation laws of the considered system. Finally, some analytic solutions are furnished by means of the invariant subspace method in the Caputo sense.

Integrable discrete PT symmetric model.

PubMed

Ablowitz, Mark J; Musslimani, Ziad H

2014-09-01

An exactly solvable discrete PT invariant nonlinear Schrödinger-like model is introduced. It is an integrable Hamiltonian system that exhibits a nontrivial nonlinear PT symmetry. A discrete one-soliton solution is constructed using a left-right Riemann-Hilbert formulation. It is shown that this pure soliton exhibits unique features such as power oscillations and singularity formation. The proposed model can be viewed as a discretization of a recently obtained integrable nonlocal nonlinear Schrödinger equation.

Exact Schwarzschild-like solution in a bumblebee gravity model

NASA Astrophysics Data System (ADS)

Casana, R.; Cavalcante, A.; Poulis, F. P.; Santos, E. B.

2018-05-01

We obtain an exact vacuum solution from the gravity sector contained in the minimal standard-model extension. The theoretical model assumes a Riemann spacetime coupled to the bumblebee field which is responsible for the spontaneous Lorentz symmetry breaking. The solution achieved in a static and spherically symmetric scenario establishes a Schwarzschild-like black hole. In order to study the effects of the spontaneous Lorentz symmetry breaking we investigate some classic tests, including the advance of perihelion, the bending of light, and Shapiro's time delay. Furthermore, we compute some upper bounds, among which the most stringent associated with existing experimental data provides a sensitivity at the 10-15 level and that for future missions at the 10-19 level.

A class of traveling wave solutions for space-time fractional biological population model in mathematical physics

NASA Astrophysics Data System (ADS)

Akram, Ghazala; Batool, Fiza

2017-10-01

The (G'/G)-expansion method is utilized for a reliable treatment of space-time fractional biological population model. The method has been applied in the sense of the Jumarie's modified Riemann-Liouville derivative. Three classes of exact traveling wave solutions, hyperbolic, trigonometric and rational solutions of the associated equation are characterized with some free parameters. A generalized fractional complex transform is applied to convert the fractional equations to ordinary differential equations which subsequently resulted in number of exact solutions. It should be mentioned that the (G'/G)-expansion method is very effective and convenient for solving nonlinear partial differential equations of fractional order whose balancing number is a negative integer.

New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models

NASA Astrophysics Data System (ADS)

Toufik, Mekkaoui; Atangana, Abdon

2017-10-01

Recently a new concept of fractional differentiation with non-local and non-singular kernel was introduced in order to extend the limitations of the conventional Riemann-Liouville and Caputo fractional derivatives. A new numerical scheme has been developed, in this paper, for the newly established fractional differentiation. We present in general the error analysis. The new numerical scheme was applied to solve linear and non-linear fractional differential equations. We do not need a predictor-corrector to have an efficient algorithm, in this method. The comparison of approximate and exact solutions leaves no doubt believing that, the new numerical scheme is very efficient and converges toward exact solution very rapidly.

A new numerical approach for uniquely solvable exterior Riemann-Hilbert problem on region with corners

NASA Astrophysics Data System (ADS)

Zamzamir, Zamzana; Murid, Ali H. M.; Ismail, Munira

2014-06-01

Numerical solution for uniquely solvable exterior Riemann-Hilbert problem on region with corners at offcorner points has been explored by discretizing the related integral equation using Picard iteration method without any modifications to the left-hand side (LHS) and right-hand side (RHS) of the integral equation. Numerical errors for all iterations are converge to the required solution. However, for certain problems, it gives lower accuracy. Hence, this paper presents a new numerical approach for the problem by treating the generalized Neumann kernel at LHS and the function at RHS of the integral equation. Due to the existence of the corner points, Gaussian quadrature is employed which avoids the corner points during numerical integration. Numerical example on a test region is presented to demonstrate the effectiveness of this formulation.

Bernhard Riemann, a(rche)typical mathematical-physicist?

NASA Astrophysics Data System (ADS)

Elizalde, Emilio

2013-09-01

The work of Bernhard Riemann is discussed under the perspective of present day mathematics and physics, and with a prospective view towards the future, too. Against the (unfortunately rather widespread) trend---which predominantly dominated national scientific societies in Europe during the last Century---of strictly classifying the work of scientists with the aim to constrain them to separated domains of knowledge, without any possible interaction among those and often even fighting against each other (and which, no doubt, was in part responsible for the decline of European in favor of American science), it will be here argued, using Riemann as a model, archetypical example, that good research transcends any classification. Its uses and applications arguably permeate all domains, subjects and disciplines one can possibly define, to the point that it can be considered to be universally useful. After providing a very concise review of the main publications of Bernhard Riemann on physical problems, some connections between Riemann's papers and contemporary physics will be considered: (i) the uses of Riemann's work on the zeta function for devising applications to the regularization of quantum field theories in curved space-time, in particular, of quantum vacuum fluctuations; (ii) the uses of the Riemann tensor in general relativity and in recent generalizations of this theory, which aim at understanding the presently observed acceleration of the universe expansion (the dark energy issue). Finally, it will be argued that mathematical physics, which was yet not long ago a model paradigm for interdisciplinary activity---and had a very important pioneering role in this sense---is now quickly being surpassed by the extraordinarily fruitful interconnections which seem to pop up from nothing every day and simultaneously involve several disciplines, in the classical sense, including genetics, combinatorics, nanoelectronics, biochemistry, medicine, and even ps

A Godunov-like point-centered essentially Lagrangian hydrodynamic approach

DOE Office of Scientific and Technical Information (OSTI.GOV)

Morgan, Nathaniel R.; Waltz, Jacob I.; Burton, Donald E.

We present an essentially Lagrangian hydrodynamic scheme suitable for modeling complex compressible flows on tetrahedron meshes. The scheme reduces to a purely Lagrangian approach when the flow is linear or if the mesh size is equal to zero; as a result, we use the term essentially Lagrangian for the proposed approach. The motivation for developing a hydrodynamic method for tetrahedron meshes is because tetrahedron meshes have some advantages over other mesh topologies. Notable advantages include reduced complexity in generating conformal meshes, reduced complexity in mesh reconnection, and preserving tetrahedron cells with automatic mesh refinement. A challenge, however, is tetrahedron meshesmore » do not correctly deform with a lower order (i.e. piecewise constant) staggered-grid hydrodynamic scheme (SGH) or with a cell-centered hydrodynamic (CCH) scheme. The SGH and CCH approaches calculate the strain via the tetrahedron, which can cause artificial stiffness on large deformation problems. To resolve the stiffness problem, we adopt the point-centered hydrodynamic approach (PCH) and calculate the evolution of the flow via an integration path around the node. The PCH approach stores the conserved variables (mass, momentum, and total energy) at the node. The evolution equations for momentum and total energy are discretized using an edge-based finite element (FE) approach with linear basis functions. A multidirectional Riemann-like problem is introduced at the center of the tetrahedron to account for discontinuities in the flow such as a shock. Conservation is enforced at each tetrahedron center. The multidimensional Riemann-like problem used here is based on Lagrangian CCH work [8, 19, 37, 38, 44] and recent Lagrangian SGH work [33-35, 39, 45]. In addition, an approximate 1D Riemann problem is solved on each face of the nodal control volume to advect mass, momentum, and total energy. The 1D Riemann problem produces fluxes [18] that remove a volume error in the PCH discretization. A 2-stage Runge–Kutta method is used to evolve the solution in time. The details of the new hydrodynamic scheme are discussed; likewise, results from numerical test problems are presented.« less

A Godunov-like point-centered essentially Lagrangian hydrodynamic approach

DOE PAGES

Morgan, Nathaniel R.; Waltz, Jacob I.; Burton, Donald E.; ...

2014-10-28

We present an essentially Lagrangian hydrodynamic scheme suitable for modeling complex compressible flows on tetrahedron meshes. The scheme reduces to a purely Lagrangian approach when the flow is linear or if the mesh size is equal to zero; as a result, we use the term essentially Lagrangian for the proposed approach. The motivation for developing a hydrodynamic method for tetrahedron meshes is because tetrahedron meshes have some advantages over other mesh topologies. Notable advantages include reduced complexity in generating conformal meshes, reduced complexity in mesh reconnection, and preserving tetrahedron cells with automatic mesh refinement. A challenge, however, is tetrahedron meshesmore » do not correctly deform with a lower order (i.e. piecewise constant) staggered-grid hydrodynamic scheme (SGH) or with a cell-centered hydrodynamic (CCH) scheme. The SGH and CCH approaches calculate the strain via the tetrahedron, which can cause artificial stiffness on large deformation problems. To resolve the stiffness problem, we adopt the point-centered hydrodynamic approach (PCH) and calculate the evolution of the flow via an integration path around the node. The PCH approach stores the conserved variables (mass, momentum, and total energy) at the node. The evolution equations for momentum and total energy are discretized using an edge-based finite element (FE) approach with linear basis functions. A multidirectional Riemann-like problem is introduced at the center of the tetrahedron to account for discontinuities in the flow such as a shock. Conservation is enforced at each tetrahedron center. The multidimensional Riemann-like problem used here is based on Lagrangian CCH work [8, 19, 37, 38, 44] and recent Lagrangian SGH work [33-35, 39, 45]. In addition, an approximate 1D Riemann problem is solved on each face of the nodal control volume to advect mass, momentum, and total energy. The 1D Riemann problem produces fluxes [18] that remove a volume error in the PCH discretization. A 2-stage Runge–Kutta method is used to evolve the solution in time. The details of the new hydrodynamic scheme are discussed; likewise, results from numerical test problems are presented.« less

New analytical exact solutions of time fractional KdV-KZK equation by Kudryashov methods

NASA Astrophysics Data System (ADS)

S Saha, Ray

2016-04-01

In this paper, new exact solutions of the time fractional KdV-Khokhlov-Zabolotskaya-Kuznetsov (KdV-KZK) equation are obtained by the classical Kudryashov method and modified Kudryashov method respectively. For this purpose, the modified Riemann-Liouville derivative is used to convert the nonlinear time fractional KdV-KZK equation into the nonlinear ordinary differential equation. In the present analysis, the classical Kudryashov method and modified Kudryashov method are both used successively to compute the analytical solutions of the time fractional KdV-KZK equation. As a result, new exact solutions involving the symmetrical Fibonacci function, hyperbolic function and exponential function are obtained for the first time. The methods under consideration are reliable and efficient, and can be used as an alternative to establish new exact solutions of different types of fractional differential equations arising from mathematical physics. The obtained results are exhibited graphically in order to demonstrate the efficiencies and applicabilities of these proposed methods of solving the nonlinear time fractional KdV-KZK equation.

Prescribing the mixed scalar curvature of a foliated Riemann-Cartan manifold

NASA Astrophysics Data System (ADS)

Rovenski, Vladimir Y.; Zelenko, Leonid

2018-03-01

The mixed scalar curvature is the simplest curvature invariant of a foliated Riemannian manifold. We explore the problem of prescribing the leafwise constant mixed scalar curvature of a foliated Riemann-Cartan manifold by conformal change of the structure in tangent and normal to the leaves directions. Under certain geometrical assumptions and in two special cases: along a compact leaf and for a closed fibered manifold, we reduce the problem to solution of a nonlinear leafwise elliptic equation for the conformal factor. We are looking for its solutions that are stable stationary solutions of the associated parabolic equation. Our main tool is using of majorizing and minorizing nonlinear heat equations with constant coefficients and application of comparison theorems for solutions of Cauchy's problem for parabolic equations.

Three-point functions in duality-invariant higher-derivative gravity

DOE PAGES

Naseer, Usman; Zwiebach, Barton

2016-03-21

Here, doubled α'-geometry is the simplest higher-derivative gravitational theory with exact global duality symmetry. We use the double metric formulation of this theory to compute on-shell three-point functions to all orders in α'. A simple pattern emerges when comparing with the analogous bosonic and heterotic three-point functions. As in these theories, the amplitudes factorize. The theory has no Gauss-Bonnet term, but contains a Riemann-cubed interaction to second order in α'.

Andrei Andreevich Bolibrukh's works on the analytic theory of differential equations

NASA Astrophysics Data System (ADS)

Anosov, Dmitry V.; Leksin, Vladimir P.

2011-02-01

This paper contains an account of A.A. Bolibrukh's results obtained in the new directions of research that arose in the analytic theory of differential equations as a consequence of his sensational counterexample to the Riemann-Hilbert problem. A survey of results of his students in developing topics first considered by Bolibrukh is also presented. The main focus is on the role of the reducibility/irreducibility of systems of linear differential equations and their monodromy representations. A brief synopsis of results on the multidimensional Riemann-Hilbert problem and on isomonodromic deformations of Fuchsian systems is presented, and the main methods in the modern analytic theory of differential equations are sketched. Bibliography: 69 titles.

The Ostrovsky-Vakhnenko equation by a Riemann-Hilbert approach

NASA Astrophysics Data System (ADS)

Boutet de Monvel, Anne; Shepelsky, Dmitry

2015-01-01

We present an inverse scattering transform (IST) approach for the (differentiated) Ostrovsky-Vakhnenko equation This equation can also be viewed as the short wave model for the Degasperis-Procesi (sDP) equation. Our IST approach is based on an associated Riemann-Hilbert problem, which allows us to give a representation for the classical (smooth) solution, to get the principal term of its long time asymptotics, and also to describe loop soliton solutions. Dedicated to Johannes Sjöstrand with gratitude and admiration.

Colloquium: Physics of the Riemann hypothesis

NASA Astrophysics Data System (ADS)

Schumayer, Dániel; Hutchinson, David A. W.

2011-04-01

Physicists become acquainted with special functions early in their studies. Consider our perennial model, the harmonic oscillator, for which we need Hermite functions, or the Laguerre functions in quantum mechanics. Here a particular number-theoretical function is chosen, the Riemann zeta function, and its influence on the realm of physics is examined and also how physics may be suggestive for the resolution of one of mathematics’ most famous unconfirmed conjectures, the Riemann hypothesis. Does physics hold an essential key to the solution for this more than 100-year-old problem? In this work numerous models from different branches of physics are examined, from classical mechanics to statistical physics, where this function plays an integral role. This function is also shown to be related to quantum chaos and how its pole structure encodes when particles can undergo Bose-Einstein condensation at low temperature. Throughout these examinations light is shed on how the Riemann hypothesis can highlight physics. Naturally, the aim is not to be comprehensive, but rather focusing on the major models and aim to give an informed starting point for the interested reader.

On the modular structure of the genus-one Type II superstring low energy expansion

NASA Astrophysics Data System (ADS)

D'Hoker, Eric; Green, Michael B.; Vanhove, Pierre

2015-08-01

The analytic contribution to the low energy expansion of Type II string amplitudes at genus-one is a power series in space-time derivatives with coefficients that are determined by integrals of modular functions over the complex structure modulus of the world-sheet torus. These modular functions are associated with world-sheet vacuum Feynman diagrams and given by multiple sums over the discrete momenta on the torus. In this paper we exhibit exact differential and algebraic relations for a certain infinite class of such modular functions by showing that they satisfy Laplace eigenvalue equations with inhomogeneous terms that are polynomial in non-holomorphic Eisenstein series. Furthermore, we argue that the set of modular functions that contribute to the coefficients of interactions up to order are linear sums of functions in this class and quadratic polynomials in Eisenstein series and odd Riemann zeta values. Integration over the complex structure results in coefficients of the low energy expansion that are rational numbers multiplying monomials in odd Riemann zeta values.

A Riemann-Hilbert Approach to Complex Sharma-Tasso-Olver Equation on Half Line*

NASA Astrophysics Data System (ADS)

Zhang, Ning; Xia, Tie-Cheng; Hu, Bei-Bei

2017-11-01

In this paper, the Fokas unified method is used to analyze the initial-boundary value problem of a complex Sharma-Tasso-Olver (cSTO) equation on the half line. We show that the solution can be expressed in terms of the solution of a Riemann-Hilbert problem. The relevant jump matrices are explicitly given in terms of the matrix-value spectral functions spectral functions \\{a(λ ),b(λ )\\} and \\{A(λ ),B(λ )\\} , which depending on initial data {u}0(x)=u(x,0) and boundary data {g}0(y)=u(0,y), {g}1(y)={u}x(0,y), {g}2(y)={u}{xx}(0,y). These spectral functions are not independent, they satisfy a global relation.

A new Lagrangian method for three-dimensional steady supersonic flows

NASA Technical Reports Server (NTRS)

Loh, Ching-Yuen; Liou, Meng-Sing

1993-01-01

In this report, the new Lagrangian method introduced by Loh and Hui is extended for three-dimensional, steady supersonic flow computation. The derivation of the conservation form and the solution of the local Riemann solver using the Godunov and the high-resolution TVD (total variation diminished) scheme is presented. This new approach is accurate and robust, capable of handling complicated geometry and interactions between discontinuous waves. Test problems show that the extended Lagrangian method retains all the advantages of the two-dimensional method (e.g., crisp resolution of a slip-surface (contact discontinuity) and automatic grid generation). In this report, we also suggest a novel three dimensional Riemann problem in which interesting and intricate flow features are present.

A family of position- and orientation-independent embedded boundary methods for viscous flow and fluid-structure interaction problems

NASA Astrophysics Data System (ADS)

Huang, Daniel Z.; De Santis, Dante; Farhat, Charbel

2018-07-01

The Finite Volume method with Exact two-material Riemann Problems (FIVER) is both a computational framework for multi-material flows characterized by large density jumps, and an Embedded Boundary Method (EBM) for computational fluid dynamics and highly nonlinear Fluid-Structure Interaction (FSI) problems. This paper deals with the EBM aspect of FIVER. For FSI problems, this EBM has already demonstrated the ability to address viscous effects along wall boundaries, and large deformations and topological changes of such boundaries. However, like for most EBMs - also known as immersed boundary methods - the performance of FIVER in the vicinity of a wall boundary can be sensitive with respect to the position and orientation of this boundary relative to the embedding mesh. This is mainly due to ill-conditioning issues that arise when an embedded interface becomes too close to a node of the embedding mesh, which may lead to spurious oscillations in the computed solution gradients at the wall boundary. This paper resolves these issues by introducing an alternative definition of the active/inactive status of a mesh node that leads to the removal of all sources of potential ill-conditioning from all spatial approximations performed by FIVER in the vicinity of a fluid-structure interface. It also makes two additional contributions. The first one is a new procedure for constructing the fluid-structure half Riemann problem underlying the semi-discretization by FIVER of the convective fluxes. This procedure eliminates one extrapolation from the conventional treatment of the wall boundary conditions and replaces it by an interpolation, which improves robustness. The second contribution is a post-processing algorithm for computing quantities of interest at the wall that achieves smoothness in the computed solution and its gradients. Lessons learned from these enhancements and contributions that are triggered by the new definition of the status of a mesh node are then generalized and exploited to eliminate from the original version of the FIVER method its sensitivities with respect to both of the position and orientation of the wall boundary relative to the embedding mesh, while maintaining the original definition of the status of a mesh node. This leads to a family of second-generation FIVER methods whose performance is illustrated in this paper for several flow and FSI problems. These include a challenging flow problem over a bird wing characterized by a feather-induced surface roughness, and a complex flexible flapping wing problem for which experimental data is available.

The steady aerodynamics of aerofoils with porosity gradients.

PubMed

Hajian, Rozhin; Jaworski, Justin W

2017-09-01

This theoretical study determines the aerodynamic loads on an aerofoil with a prescribed porosity distribution in a steady incompressible flow. A Darcy porosity condition on the aerofoil surface furnishes a Fredholm integral equation for the pressure distribution, which is solved exactly and generally as a Riemann-Hilbert problem provided that the porosity distribution is Hölder-continuous. The Hölder condition includes as a subset any continuously differentiable porosity distributions that may be of practical interest. This formal restriction on the analysis is examined by a class of differentiable porosity distributions that approach a piecewise, discontinuous function in a certain parametric limit. The Hölder-continuous solution is verified in this limit against analytical results for partially porous aerofoils in the literature. Finally, a comparison made between the new theoretical predictions and experimental measurements of SD7003 aerofoils presented in the literature. Results from this analysis may be integrated into a theoretical framework to optimize turbulence noise suppression with minimal impact to aerodynamic performance.

The steady aerodynamics of aerofoils with porosity gradients

NASA Astrophysics Data System (ADS)

Hajian, Rozhin; Jaworski, Justin W.

2017-09-01

This theoretical study determines the aerodynamic loads on an aerofoil with a prescribed porosity distribution in a steady incompressible flow. A Darcy porosity condition on the aerofoil surface furnishes a Fredholm integral equation for the pressure distribution, which is solved exactly and generally as a Riemann-Hilbert problem provided that the porosity distribution is Hölder-continuous. The Hölder condition includes as a subset any continuously differentiable porosity distributions that may be of practical interest. This formal restriction on the analysis is examined by a class of differentiable porosity distributions that approach a piecewise, discontinuous function in a certain parametric limit. The Hölder-continuous solution is verified in this limit against analytical results for partially porous aerofoils in the literature. Finally, a comparison made between the new theoretical predictions and experimental measurements of SD7003 aerofoils presented in the literature. Results from this analysis may be integrated into a theoretical framework to optimize turbulence noise suppression with minimal impact to aerodynamic performance.

The Glimm scheme for perfect fluids on plane-symmetric Gowdy spacetimes

NASA Astrophysics Data System (ADS)

Barnes, A. P.; Lefloch, P. G.; Schmidt, B. G.; Stewart, J. M.

2004-11-01

We propose a new, augmented formulation of the coupled Euler Einstein equations for perfect fluids on plane-symmetric Gowdy spacetimes. The unknowns of the augmented system are the density and velocity of the fluid and the first- and second-order spacetime derivatives of the metric. We solve the Riemann problem for the augmented system, allowing propagating discontinuities in both the fluid variables and the first- and second-order derivatives of the geometry coefficients. Our main result, based on Glimm's random choice scheme, is the existence of solutions with bounded total variation of the Euler Einstein equations, up to the first time where a blow-up singularity (unbounded first-order derivatives of the geometry coefficients) occurs. We demonstrate the relevance of the augmented system for numerical relativity. We also consider general vacuum spacetimes and solve a Riemann problem, by relying on a theorem by Rendall on the characteristic value problem for the Einstein equations.

Riemann Solvers in Relativistic Hydrodynamics: Basics and Astrophysical Applications

NASA Astrophysics Data System (ADS)

Ibanez, Jose M.

2001-12-01

My contribution to these proceedings summarizes a general overview on t High Resolution Shock Capturing methods (HRSC) in the field of relativistic hydrodynamics with special emphasis on Riemann solvers. HRSC techniques achieve highly accurate numerical approximations (formally second order or better) in smooth regions of the flow, and capture the motion of unresolved steep gradients without creating spurious oscillations. In the first part I will show how these techniques have been extended to relativistic hydrodynamics, making it possible to explore some challenging astrophysical scenarios. I will review recent literature concerning the main properties of different special relativistic Riemann solvers, and discuss several 1D and 2D test problems which are commonly used to evaluate the performance of numerical methods in relativistic hydrodynamics. In the second part I will illustrate the use of HRSC methods in several astrophysical applications where special and general relativistic hydrodynamical processes play a crucial role.

Solitons of shallow-water models from energy-dependent spectral problems

NASA Astrophysics Data System (ADS)

Haberlin, Jack; Lyons, Tony

2018-01-01

The current work investigates the soliton solutions of the Kaup-Boussinesq equation using the inverse scattering transform method. We outline the construction of the Riemann-Hilbert problem for a pair of energy-dependent spectral problems for the system, which we then use to construct the solution of this hydrodynamic system.

Adaptive finite volume methods with well-balanced Riemann solvers for modeling floods in rugged terrain: Application to the Malpasset dam-break flood (France, 1959)

USGS Publications Warehouse

George, D.L.

2011-01-01

The simulation of advancing flood waves over rugged topography, by solving the shallow-water equations with well-balanced high-resolution finite volume methods and block-structured dynamic adaptive mesh refinement (AMR), is described and validated in this paper. The efficiency of block-structured AMR makes large-scale problems tractable, and allows the use of accurate and stable methods developed for solving general hyperbolic problems on quadrilateral grids. Features indicative of flooding in rugged terrain, such as advancing wet-dry fronts and non-stationary steady states due to balanced source terms from variable topography, present unique challenges and require modifications such as special Riemann solvers. A well-balanced Riemann solver for inundation and general (non-stationary) flow over topography is tested in this context. The difficulties of modeling floods in rugged terrain, and the rationale for and efficacy of using AMR and well-balanced methods, are presented. The algorithms are validated by simulating the Malpasset dam-break flood (France, 1959), which has served as a benchmark problem previously. Historical field data, laboratory model data and other numerical simulation results (computed on static fitted meshes) are shown for comparison. The methods are implemented in GEOCLAW, a subset of the open-source CLAWPACK software. All the software is freely available at. Published in 2010 by John Wiley & Sons, Ltd.

The existence of a real pole-free solution of the fourth order analogue of the Painlevé I equation

NASA Astrophysics Data System (ADS)

Claeys, T.; Vanlessen, M.

2007-05-01

We establish the existence of a real solution y(x, T) with no poles on the real line of the following fourth order analogue of the Painlevé I equation: \\[ \\begin{equation*}x=Ty-\\left(\\case 1 6 y^3+\\case{1}{24} (y_x^2+2yy_{xx}) +\\case {1}{240} y_{xxxx}\\right).\\end{equation*} \\] This proves the existence part of a conjecture posed by Dubrovin. We obtain our result by proving the solvability of an associated Riemann-Hilbert problem through the approach of a vanishing lemma. In addition, by applying the Deift/Zhou steepest-descent method to this Riemann-Hilbert problem, we obtain the asymptotics for y(x, T) as x → ±∞.

The Hurwitz Enumeration Problem of Branched Covers and Hodge Integrals

DOE Office of Scientific and Technical Information (OSTI.GOV)

Song, Yun S.

We use algebraic methods to compute the simple Hurwitz numbers for arbitrary source and target Riemann surfaces. For an elliptic curve target, we reproduce the results previously obtained by string theorists. Motivated by the Gromov-Witten potentials, we find a general generating function for the simple Hurwitz numbers in terms of the representation theory of the symmetric group S{sub n}. We also find a generating function for Hodge integrals on the moduli space {bar M}{sub g,2} of Riemann surfaces with two marked points, similar to that found by Faber and Pandharipande for the case of one marked point.

The Nonlinear Steepest Descent Method to Long-Time Asymptotics of the Coupled Nonlinear Schrödinger Equation

NASA Astrophysics Data System (ADS)

Geng, Xianguo; Liu, Huan

2018-04-01

The Riemann-Hilbert problem for the coupled nonlinear Schrödinger equation is formulated on the basis of the corresponding 3× 3 matrix spectral problem. Using the nonlinear steepest descent method, we obtain leading-order asymptotics for the Cauchy problem of the coupled nonlinear Schrödinger equation.

Use of Genetic Algorithms to solve Inverse Problems in Relativistic Hydrodynamics

NASA Astrophysics Data System (ADS)

Guzmán, F. S.; González, J. A.

2018-04-01

We present the use of Genetic Algorithms (GAs) as a strategy to solve inverse problems associated with models of relativistic hydrodynamics. The signal we consider to emulate an observation is the density of a relativistic gas, measured at a point where a shock is traveling. This shock is generated numerically out of a Riemann problem with mildly relativistic conditions. The inverse problem we propose is the prediction of the initial conditions of density, velocity and pressure of the Riemann problem that gave origin to that signal. For this we use the density, velocity and pressure of the gas at both sides of the discontinuity, as the six genes of an organism, initially with random values within a tolerance. We then prepare an initial population of N of these organisms and evolve them using methods based on GAs. In the end, the organism with the best fitness of each generation is compared to the signal and the process ends when the set of initial conditions of the organisms of a later generation fit the Signal within a tolerance.

A Riemann-Hilbert Approach for the Novikov Equation

NASA Astrophysics Data System (ADS)

Boutet de Monvel, Anne; Shepelsky, Dmitry; Zielinski, Lech

2016-09-01

We develop the inverse scattering transform method for the Novikov equation u_t-u_{txx}+4u^2u_x=3u u_xu_{xx}+u^2u_{xxx} considered on the line xin(-∞,∞) in the case of non-zero constant background. The approach is based on the analysis of an associated Riemann-Hilbert (RH) problem, which in this case is a 3× 3 matrix problem. The structure of this RH problem shares many common features with the case of the Degasperis-Procesi (DP) equation having quadratic nonlinear terms (see [Boutet de Monvel A., Shepelsky D., Nonlinearity 26 (2013), 2081-2107, arXiv:1107.5995]) and thus the Novikov equation can be viewed as a ''modified DP equation'', in analogy with the relationship between the Korteweg-de Vries (KdV) equation and the modified Korteweg-de Vries (mKdV) equation. We present parametric formulas giving the solution of the Cauchy problem for the Novikov equation in terms of the solution of the RH problem and discuss the possibilities to use the developed formalism for further studying of the Novikov equation.

A Riemann-Hilbert approach to asymptotic questions for orthogonal polynomials

NASA Astrophysics Data System (ADS)

Deift, P.; Kriecherbauer, T.; McLaughlin, K. T.-R.; Venakides, S.; Zhou, X.

2001-08-01

A few years ago the authors introduced a new approach to study asymptotic questions for orthogonal polynomials. In this paper we give an overview of our method and review the results which have been obtained in Deift et al. (Internat. Math. Res. Notices (1997) 759, Comm. Pure Appl. Math. 52 (1999) 1491, 1335), Deift (Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lecture Notes, Vol. 3, New York University, 1999), Kriecherbauer and McLaughlin (Internat. Math. Res. Notices (1999) 299) and Baik et al. (J. Amer. Math. Soc. 12 (1999) 1119). We mainly consider orthogonal polynomials with respect to weights on the real line which are either (1) Freud-type weights d[alpha](x)=e-Q(x) dx (Q polynomial or Q(x)=x[beta], [beta]>0), or (2) varying weights d[alpha]n(x)=e-nV(x) dx (V analytic, limx-->[infinity] V(x)/logx=[infinity]). We obtain Plancherel-Rotach-type asymptotics in the entire complex plane as well as asymptotic formulae with error estimates for the leading coefficients, for the recurrence coefficients, and for the zeros of the orthogonal polynomials. Our proof starts from an observation of Fokas et al. (Comm. Math. Phys. 142 (1991) 313) that the orthogonal polynomials can be determined as solutions of certain matrix valued Riemann-Hilbert problems. We analyze the Riemann-Hilbert problems by a steepest descent type method introduced by Deift and Zhou (Ann. Math. 137 (1993) 295) and further developed in Deift and Zhou (Comm. Pure Appl. Math. 48 (1995) 277) and Deift et al. (Proc. Nat. Acad. Sci. USA 95 (1998) 450). A crucial step in our analysis is the use of the well-known equilibrium measure which describes the asymptotic distribution of the zeros of the orthogonal polynomials.

The short pulse equation by a Riemann-Hilbert approach

NASA Astrophysics Data System (ADS)

Boutet de Monvel, Anne; Shepelsky, Dmitry; Zielinski, Lech

2017-07-01

We develop a Riemann-Hilbert approach to the inverse scattering transform method for the short pulse (SP) equation u_{xt}=u+{1/6}(u^3)_{xx} with zero boundary conditions (as |x|→ ∞). This approach is directly applied to a Lax pair for the SP equation. It allows us to give a parametric representation of the solution to the Cauchy problem. This representation is then used for studying the longtime behavior of the solution as well as for retrieving the soliton solutions. Finally, the analysis of the longtime behavior allows us to formulate, in spectral terms, a sufficient condition for the wave breaking.

High-order upwind schemes for the wave equation on overlapping grids: Maxwell's equations in second-order form

NASA Astrophysics Data System (ADS)

Angel, Jordan B.; Banks, Jeffrey W.; Henshaw, William D.

2018-01-01

High-order accurate upwind approximations for the wave equation in second-order form on overlapping grids are developed. Although upwind schemes are well established for first-order hyperbolic systems, it was only recently shown by Banks and Henshaw [1] how upwinding could be incorporated into the second-order form of the wave equation. This new upwind approach is extended here to solve the time-domain Maxwell's equations in second-order form; schemes of arbitrary order of accuracy are formulated for general curvilinear grids. Taylor time-stepping is used to develop single-step space-time schemes, and the upwind dissipation is incorporated by embedding the exact solution of a local Riemann problem into the discretization. Second-order and fourth-order accurate schemes are implemented for problems in two and three space dimensions, and overlapping grids are used to treat complex geometry and problems with multiple materials. Stability analysis of the upwind-scheme on overlapping grids is performed using normal mode theory. The stability analysis and computations confirm that the upwind scheme remains stable on overlapping grids, including the difficult case of thin boundary grids when the traditional non-dissipative scheme becomes unstable. The accuracy properties of the scheme are carefully evaluated on a series of classical scattering problems for both perfect conductors and dielectric materials in two and three space dimensions. The upwind scheme is shown to be robust and provide high-order accuracy.

Analytically solvable model of an electronic Mach-Zehnder interferometer

NASA Astrophysics Data System (ADS)

Ngo Dinh, Stéphane; Bagrets, Dmitry A.; Mirlin, Alexander D.

2013-05-01

We consider a class of models of nonequilibrium electronic Mach-Zehnder interferometers built on integer quantum Hall edges states. The models are characterized by the electron-electron interaction being restricted to the inner part of the interferometer and transmission coefficients of the quantum quantum point contacts, defining the interferometer, which may take arbitrary values from zero to one. We establish an exact solution of these models in terms of single-particle quantities, determinants and resolvents of Fredholm integral operators. In the general situation, the results can be obtained numerically. In the case of strong charging interaction, the operators acquire the block Toeplitz form. Analyzing the corresponding Riemann-Hilbert problem, we reduce the result to certain singular single-channel determinants (which are a generalization of Toeplitz determinants with Fisher-Hartwig singularities) and obtain an analytic result for the interference current (and, in particular, for the visibility of Aharonov-Bohm oscillations). Our results, which are in good agreement with experimental observations, show an intimate connection between the observed “lobe” structure in the visibility of Aharonov-Bohm oscillations and multiple branches in the asymptotics of singular integral determinants.

Essentially nonoscillatory postprocessing filtering methods

NASA Technical Reports Server (NTRS)

Lafon, F.; Osher, S.

1992-01-01

High order accurate centered flux approximations used in the computation of numerical solutions to nonlinear partial differential equations produce large oscillations in regions of sharp transitions. Here, we present a new class of filtering methods denoted by Essentially Nonoscillatory Least Squares (ENOLS), which constructs an upgraded filtered solution that is close to the physically correct weak solution of the original evolution equation. Our method relies on the evaluation of a least squares polynomial approximation to oscillatory data using a set of points which is determined via the ENO network. Numerical results are given in one and two space dimensions for both scalar and systems of hyperbolic conservation laws. Computational running time, efficiency, and robustness of method are illustrated in various examples such as Riemann initial data for both Burgers' and Euler's equations of gas dynamics. In all standard cases, the filtered solution appears to converge numerically to the correct solution of the original problem. Some interesting results based on nonstandard central difference schemes, which exactly preserve entropy, and have been recently shown generally not to be weakly convergent to a solution of the conservation law, are also obtained using our filters.

Gaussian quadrature and lattice discretization of the Fermi-Dirac distribution for graphene.

PubMed

Oettinger, D; Mendoza, M; Herrmann, H J

2013-07-01

We construct a lattice kinetic scheme to study electronic flow in graphene. For this purpose, we first derive a basis of orthogonal polynomials, using as the weight function the ultrarelativistic Fermi-Dirac distribution at rest. Later, we use these polynomials to expand the respective distribution in a moving frame, for both cases, undoped and doped graphene. In order to discretize the Boltzmann equation and make feasible the numerical implementation, we reduce the number of discrete points in momentum space to 18 by applying a Gaussian quadrature, finding that the family of representative wave (2+1)-vectors, which satisfies the quadrature, reconstructs a honeycomb lattice. The procedure and discrete model are validated by solving the Riemann problem, finding excellent agreement with other numerical models. In addition, we have extended the Riemann problem to the case of different dopings, finding that by increasing the chemical potential the electronic fluid behaves as if it increases its effective viscosity.

Solution of the Riemann problem for polarization waves in a two-component Bose-Einstein condensate

NASA Astrophysics Data System (ADS)

Ivanov, S. K.; Kamchatnov, A. M.; Congy, T.; Pavloff, N.

2017-12-01

We provide a classification of the possible flows of two-component Bose-Einstein condensates evolving from initially discontinuous profiles. We consider the situation where the dynamics can be reduced to the consideration of a single polarization mode (also denoted as "magnetic excitation") obeying a system of equations equivalent to the Landau-Lifshitz equation for an easy-plane ferromagnet. We present the full set of one-phase periodic solutions. The corresponding Whitham modulation equations are obtained together with formulas connecting their solutions with the Riemann invariants of the modulation equations. The problem is not genuinely nonlinear, and this results in a non-single-valued mapping of the solutions of the Whitham equations with physical wave patterns as well as the appearance of interesting elements—contact dispersive shock waves—that are absent in more standard, genuinely nonlinear situations. Our analytic results are confirmed by numerical simulations.

Khater method for nonlinear Sharma Tasso-Olever (STO) equation of fractional order

NASA Astrophysics Data System (ADS)

Bibi, Sadaf; Mohyud-Din, Syed Tauseef; Khan, Umar; Ahmed, Naveed

In this work, we have implemented a direct method, known as Khater method to establish exact solutions of nonlinear partial differential equations of fractional order. Number of solutions provided by this method is greater than other traditional methods. Exact solutions of nonlinear fractional order Sharma Tasso-Olever (STO) equation are expressed in terms of kink, travelling wave, periodic and solitary wave solutions. Modified Riemann-Liouville derivative and Fractional complex transform have been used for compatibility with fractional order sense. Solutions have been graphically simulated for understanding the physical aspects and importance of the method. A comparative discussion between our established results and the results obtained by existing ones is also presented. Our results clearly reveal that the proposed method is an effective, powerful and straightforward technique to work out new solutions of various types of differential equations of non-integer order in the fields of applied sciences and engineering.

Nonlinear Waves.

DTIC Science & Technology

1988-02-01

in Multi- dimensions II, P.M. Santini and A.S. Fokas, preprint INS#67, 1986. The Recursion Operator of the Kadomtsev - Petviashvili Equation and the...solitons, multidimensional inverse problems, Painleve equations , direct linearizations of certain nonlinear wave equations , DBAR problems, Riemann...the Navy is (a) the recent discovery that many of the equations describing ship hydrodynamics in channels of finite depth obey nonlinear equations

Efficient analytical implementation of the DOT Riemann solver for the de Saint Venant-Exner morphodynamic model

NASA Astrophysics Data System (ADS)

Carraro, F.; Valiani, A.; Caleffi, V.

2018-03-01

Within the framework of the de Saint Venant equations coupled with the Exner equation for morphodynamic evolution, this work presents a new efficient implementation of the Dumbser-Osher-Toro (DOT) scheme for non-conservative problems. The DOT path-conservative scheme is a robust upwind method based on a complete Riemann solver, but it has the drawback of requiring expensive numerical computations. Indeed, to compute the non-linear time evolution in each time step, the DOT scheme requires numerical computation of the flux matrix eigenstructure (the totality of eigenvalues and eigenvectors) several times at each cell edge. In this work, an analytical and compact formulation of the eigenstructure for the de Saint Venant-Exner (dSVE) model is introduced and tested in terms of numerical efficiency and stability. Using the original DOT and PRICE-C (a very efficient FORCE-type method) as reference methods, we present a convergence analysis (error against CPU time) to study the performance of the DOT method with our new analytical implementation of eigenstructure calculations (A-DOT). In particular, the numerical performance of the three methods is tested in three test cases: a movable bed Riemann problem with analytical solution; a problem with smooth analytical solution; a test in which the water flow is characterised by subcritical and supercritical regions. For a given target error, the A-DOT method is always the most efficient choice. Finally, two experimental data sets and different transport formulae are considered to test the A-DOT model in more practical case studies.

An insight into Newton's cooling law using fractional calculus

NASA Astrophysics Data System (ADS)

Mondol, Adreja; Gupta, Rivu; Das, Shantanu; Dutta, Tapati

2018-02-01

For small temperature differences between a heated body and its environment, Newton's law of cooling predicts that the instantaneous rate of change of temperature of any heated body with respect to time is proportional to the difference in temperature of the body with the ambient, time being measured in integer units. Our experiments on the cooling of different liquids (water, mustard oil, and mercury) did not fit the theoretical predictions of Newton's law of cooling in this form. The solution was done using both Caputo and Riemann-Liouville type fractional derivatives to check if natural phenomena showed any preference in mathematics. In both cases, we find that cooling of liquids has an identical value of the fractional derivative of time that increases with the viscosity of the liquid. On the other hand, the cooling studies on metal alloys could be fitted exactly by integer order time derivative equations. The proportionality constant between heat flux and temperature difference was examined with respect to variations in the depth of liquid and exposed surface area. A critical combination of these two parameters signals a change in the mode of heat transfer within liquids. The equivalence between the proportionality constants for the Caputo and Riemann-Liouville type derivatives is established.

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

DTIC Science & Technology

2014-01-01

with a Riemann flux !"#! (!! ,!!!,), where !!! denotes the solution outside the current element !. Various (approximate) Riemann solvers ...can be used to calculate the Riemann flux, and the Rusanov Riemann solver is adopted in this paper. Then Eq. (7) can be rewritten as !

Wave Riemann description of friction terms in unsteady shallow flows: Application to water and mud/debris floods

NASA Astrophysics Data System (ADS)

Murillo, J.; García-Navarro, P.

2012-02-01

In this work, the source term discretization in hyperbolic conservation laws with source terms is considered using an approximate augmented Riemann solver. The technique is applied to the shallow water equations with bed slope and friction terms with the focus on the friction discretization. The augmented Roe approximate Riemann solver provides a family of weak solutions for the shallow water equations, that are the basis of the upwind treatment of the source term. This has proved successful to explain and to avoid the appearance of instabilities and negative values of the thickness of the water layer in cases of variable bottom topography. Here, this strategy is extended to capture the peculiarities that may arise when defining more ambitious scenarios, that may include relevant stresses in cases of mud/debris flow. The conclusions of this analysis lead to the definition of an accurate and robust first order finite volume scheme, able to handle correctly transient problems considering frictional stresses in both clean water and debris flow, including in this last case a correct modelling of stopping conditions.

Advanced numerical methods for three dimensional two-phase flow calculations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Toumi, I.; Caruge, D.

1997-07-01

This paper is devoted to new numerical methods developed for both one and three dimensional two-phase flow calculations. These methods are finite volume numerical methods and are based on the use of Approximate Riemann Solvers concepts to define convective fluxes versus mean cell quantities. The first part of the paper presents the numerical method for a one dimensional hyperbolic two-fluid model including differential terms as added mass and interface pressure. This numerical solution scheme makes use of the Riemann problem solution to define backward and forward differencing to approximate spatial derivatives. The construction of this approximate Riemann solver uses anmore » extension of Roe`s method that has been successfully used to solve gas dynamic equations. As far as the two-fluid model is hyperbolic, this numerical method seems very efficient for the numerical solution of two-phase flow problems. The scheme was applied both to shock tube problems and to standard tests for two-fluid computer codes. The second part describes the numerical method in the three dimensional case. The authors discuss also some improvements performed to obtain a fully implicit solution method that provides fast running steady state calculations. Such a scheme is not implemented in a thermal-hydraulic computer code devoted to 3-D steady-state and transient computations. Some results obtained for Pressurised Water Reactors concerning upper plenum calculations and a steady state flow in the core with rod bow effect evaluation are presented. In practice these new numerical methods have proved to be stable on non staggered grids and capable of generating accurate non oscillating solutions for two-phase flow calculations.« less

Analyzing Lie symmetry and constructing conservation laws for time-fractional Benny-Lin equation

NASA Astrophysics Data System (ADS)

Rashidi, Saeede; Hejazi, S. Reza

This paper investigates the invariance properties of the time fractional Benny-Lin equation with Riemann-Liouville and Caputo derivatives. This equation can be reduced to the Kawahara equation, fifth-order Kdv equation, the Kuramoto-Sivashinsky equation and Navier-Stokes equation. By using the Lie group analysis method of fractional differential equations (FDEs), we derive Lie symmetries for the Benny-Lin equation. Conservation laws for this equation are obtained with the aid of the concept of nonlinear self-adjointness and the fractional generalization of the Noether’s operators. Furthermore, by means of the invariant subspace method, exact solutions of the equation are also constructed.

An added-mass partition algorithm for fluid–structure interactions of compressible fluids and nonlinear solids

DOE Office of Scientific and Technical Information (OSTI.GOV)

Banks, J.W., E-mail: banksj3@rpi.edu; Henshaw, W.D., E-mail: henshw@rpi.edu; Kapila, A.K., E-mail: kapila@rpi.edu

We describe an added-mass partitioned (AMP) algorithm for solving fluid–structure interaction (FSI) problems involving inviscid compressible fluids interacting with nonlinear solids that undergo large rotations and displacements. The computational approach is a mixed Eulerian–Lagrangian scheme that makes use of deforming composite grids (DCG) to treat large changes in the geometry in an accurate, flexible, and robust manner. The current work extends the AMP algorithm developed in Banks et al. [1] for linearly elasticity to the case of nonlinear solids. To ensure stability for the case of light solids, the new AMP algorithm embeds an approximate solution of a nonlinear fluid–solidmore » Riemann (FSR) problem into the interface treatment. The solution to the FSR problem is derived and shown to be of a similar form to that derived for linear solids: the state on the interface being fundamentally an impedance-weighted average of the fluid and solid states. Numerical simulations demonstrate that the AMP algorithm is stable even for light solids when added-mass effects are large. The accuracy and stability of the AMP scheme is verified by comparison to an exact solution using the method of analytical solutions and to a semi-analytical solution that is obtained for a rotating solid disk immersed in a fluid. The scheme is applied to the simulation of a planar shock impacting a light elliptical-shaped solid, and comparisons are made between solutions of the FSI problem for a neo-Hookean solid, a linearly elastic solid, and a rigid solid. The ability of the approach to handle large deformations is demonstrated for a problem of a high-speed flow past a light, thin, and flexible solid beam.« less

A Schrödinger equation for solving the Bender-Brody-Müller conjecture

NASA Astrophysics Data System (ADS)

Moxley, Frederick Ira

2017-11-01

The Hamiltonian of a quantum mechanical system has an affiliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the Riemann zeta function can be made. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. Such quantum Riemann zeta function analogies have led to the Bender-Brody-Müller (BBM) conjecture, which involves a non-Hermitian Hamiltonian that maps to the zeros of the Riemann zeta function. If the BBM Hamiltonian can be shown to be Hermitian, then the Riemann Hypothesis follows. As such, herein we perform a symmetrization procedure of the BBM Hamiltonian to obtain a unique Hermitian Hamiltonian that maps to the zeros of the analytic continuation of the Riemann zeta function, and discuss the eigenvalues of the results. Moreover, a second quantization of the resulting Schrödinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic continuation of the Riemann zeta function is obtained. Finally, the Hilbert-Pólya conjecture is discussed, and it is heuristically shown that the real part of every nontrivial zero of the Riemann zeta function converges at σ = 1/2.

A nodal discontinuous Galerkin approach to 3-D viscoelastic wave propagation in complex geological media

NASA Astrophysics Data System (ADS)

Lambrecht, L.; Lamert, A.; Friederich, W.; Möller, T.; Boxberg, M. S.

2018-03-01

A nodal discontinuous Galerkin (NDG) approach is developed and implemented for the computation of viscoelastic wavefields in complex geological media. The NDG approach combines unstructured tetrahedral meshes with an element-wise, high-order spatial interpolation of the wavefield based on Lagrange polynomials. Numerical fluxes are computed from an exact solution of the heterogeneous Riemann problem. Our implementation offers capabilities for modelling viscoelastic wave propagation in 1-D, 2-D and 3-D settings of very different spatial scale with little logistical overhead. It allows the import of external tetrahedral meshes provided by independent meshing software and can be run in a parallel computing environment. Computation of adjoint wavefields and an interface for the computation of waveform sensitivity kernels are offered. The method is validated in 2-D and 3-D by comparison to analytical solutions and results from a spectral element method. The capabilities of the NDG method are demonstrated through a 3-D example case taken from tunnel seismics which considers high-frequency elastic wave propagation around a curved underground tunnel cutting through inclined and faulted sedimentary strata. The NDG method was coded into the open-source software package NEXD and is available from GitHub.

Numerical algorithms for cold-relativistic plasma models in the presence of discontinuties

NASA Astrophysics Data System (ADS)

Hakim, Ammar; Cary, John; Bruhwiler, David; Geddes, Cameron; Leemans, Wim; Esarey, Eric

2006-10-01

A numerical algorithm is presented to solve cold-relativistic electron fluid equations in the presence of sharp gradients and discontinuities. The intended application is to laser wake-field accelerator simulations in which the laser induces accelerating fields thousands of times those achievable in conventional RF accelerators. The relativistic cold-fluid equations are formulated as non-classical system of hyperbolic balance laws. It is shown that the flux Jacobian for this system can not be diagonalized which causes numerical difficulties when developing shock-capturing algorithms. Further, the system is shown to admit generalized delta-shock solutions, first discovered in the context of sticky-particle dynamics (Bouchut, Ser. Adv. Math App. Sci., 22 (1994) pp. 171--190). A new approach, based on relaxation schemes proposed by Jin and Xin (Comm. Pure Appl. Math. 48 (1995) pp. 235--276) and LeVeque and Pelanti (J. Comput. Phys. 172 (2001) pp. 572--591) is developed to solve this system of equations. The method consists of finding an exact solution to a Riemann problem at each cell interface and coupling these to advance the solution in time. Applications to an intense laser propagating in an under-dense plasma are presented.

Compressible, multiphase semi-implicit method with moment of fluid interface representation

DOE PAGES

Jemison, Matthew; Sussman, Mark; Arienti, Marco

2014-09-16

A unified method for simulating multiphase flows using an exactly mass, momentum, and energy conserving Cell-Integrated Semi-Lagrangian advection algorithm is presented. The deforming material boundaries are represented using the moment-of-fluid method. Our new algorithm uses a semi-implicit pressure update scheme that asymptotically preserves the standard incompressible pressure projection method in the limit of infinite sound speed. The asymptotically preserving attribute makes the new method applicable to compressible and incompressible flows including stiff materials; enabling large time steps characteristic of incompressible flow algorithms rather than the small time steps required by explicit methods. Moreover, shocks are captured and material discontinuities aremore » tracked, without the aid of any approximate or exact Riemann solvers. As a result, wimulations of underwater explosions and fluid jetting in one, two, and three dimensions are presented which illustrate the effectiveness of the new algorithm at efficiently computing multiphase flows containing shock waves and material discontinuities with large “impedance mismatch.”« less

A point-centered arbitrary Lagrangian Eulerian hydrodynamic approach for tetrahedral meshes

DOE PAGES

Morgan, Nathaniel R.; Waltz, Jacob I.; Burton, Donald E.; ...

2015-02-24

We present a three dimensional (3D) arbitrary Lagrangian Eulerian (ALE) hydrodynamic scheme suitable for modeling complex compressible flows on tetrahedral meshes. The new approach stores the conserved variables (mass, momentum, and total energy) at the nodes of the mesh and solves the conservation equations on a control volume surrounding the point. This type of an approach is termed a point-centered hydrodynamic (PCH) method. The conservation equations are discretized using an edge-based finite element (FE) approach with linear basis functions. All fluxes in the new approach are calculated at the center of each tetrahedron. A multidirectional Riemann-like problem is solved atmore » the center of the tetrahedron. The advective fluxes are calculated by solving a 1D Riemann problem on each face of the nodal control volume. A 2-stage Runge–Kutta method is used to evolve the solution forward in time, where the advective fluxes are part of the temporal integration. The mesh velocity is smoothed by solving a Laplacian equation. The details of the new ALE hydrodynamic scheme are discussed. Results from a range of numerical test problems are presented.« less

Finite-volume application of high order ENO schemes to multi-dimensional boundary-value problems

NASA Technical Reports Server (NTRS)

Casper, Jay; Dorrepaal, J. Mark

1990-01-01

The finite volume approach in developing multi-dimensional, high-order accurate essentially non-oscillatory (ENO) schemes is considered. In particular, a two dimensional extension is proposed for the Euler equation of gas dynamics. This requires a spatial reconstruction operator that attains formal high order of accuracy in two dimensions by taking account of cross gradients. Given a set of cell averages in two spatial variables, polynomial interpolation of a two dimensional primitive function is employed in order to extract high-order pointwise values on cell interfaces. These points are appropriately chosen so that correspondingly high-order flux integrals are obtained through each interface by quadrature, at each point having calculated a flux contribution in an upwind fashion. The solution-in-the-small of Riemann's initial value problem (IVP) that is required for this pointwise flux computation is achieved using Roe's approximate Riemann solver. Issues to be considered in this two dimensional extension include the implementation of boundary conditions and application to general curvilinear coordinates. Results of numerical experiments are presented for qualitative and quantitative examination. These results contain the first successful application of ENO schemes to boundary value problems with solid walls.

Deformations of super Riemann surfaces

NASA Astrophysics Data System (ADS)

Ninnemann, Holger

1992-11-01

Two different approaches to (Kostant-Leites-) super Riemann surfaces are investigated. In the local approach, i.e. glueing open superdomains by superconformal transition functions, deformations of the superconformal structure are discussed. On the other hand, the representation of compact super Riemann surfaces of genus greater than one as a fundamental domain in the Poincaré upper half-plane provides a simple description of super Laplace operators acting on automorphic p-forms. Considering purely odd deformations of super Riemann surfaces, the number of linear independent holomorphic sections of arbitrary holomorphic line bundles will be shown to be independent of the odd moduli, leading to a simple proof of the Riemann-Roch theorem for compact super Riemann surfaces. As a further consequence, the explicit connections between determinants of super Laplacians and Selberg's super zeta functions can be determined, allowing to calculate at least the 2-loop contribution to the fermionic string partition function.

Upwind and symmetric shock-capturing schemes

NASA Technical Reports Server (NTRS)

Yee, H. C.

1987-01-01

The development of numerical methods for hyperbolic conservation laws has been a rapidly growing area for the last ten years. Many of the fundamental concepts and state-of-the-art developments can only be found in meeting proceedings or internal reports. This review paper attempts to give an overview and a unified formulation of a class of shock-capturing methods. Special emphasis is on the construction of the basic nonlinear scalar second-order schemes and the methods of extending these nonlinear scalar schemes to nonlinear systems via the extact Riemann solver, approximate Riemann solvers, and flux-vector splitting approaches. Generalization of these methods to efficiently include real gases and large systems of nonequilibrium flows is discussed. The performance of some of these schemes is illustrated by numerical examples for one-, two- and three-dimensional gas dynamics problems.

Topological degeneracy of non-Abelian states for dummies

NASA Astrophysics Data System (ADS)

Oshikawa, Masaki; Kim, Yong Baek; Shtengel, Kirill; Nayak, Chetan; Tewari, Sumanta

2007-06-01

We present a physical construction of degenerate groundstates of the Moore-Read Pfaffian states, which exhibits non-Abelian statistics, on general Riemann surface with genus g. The construction is given by a generalization of the recent argument [M.O., T. Senthil, Phys. Rev. Lett. 96 (2006) 060601] which relates fractionalization and topological order. The nontrivial groundstate degeneracy obtained by Read and Green [Phys. Rev. B 61 (2000) 10267] based on differential geometry is reproduced exactly. Some restrictions on the statistics, due to the fractional charge of the quasiparticle are also discussed. Furthermore, the groundstate degeneracy of the p + i p superconductor in two dimensions, which is closely related to the Pfaffian states, is discussed with a similar construction.

Noether symmetries in Gauss-Bonnet-teleparallel cosmology.

PubMed

Capozziello, Salvatore; De Laurentis, Mariafelicia; Dialektopoulos, Konstantinos F

2016-01-01

A generalized teleparallel cosmological model, [Formula: see text], containing the torsion scalar T and the teleparallel counterpart of the Gauss-Bonnet topological invariant [Formula: see text], is studied in the framework of the Noether symmetry approach. As [Formula: see text] gravity, where [Formula: see text] is the Gauss-Bonnet topological invariant and R is the Ricci curvature scalar, exhausts all the curvature information that one can construct from the Riemann tensor, in the same way, [Formula: see text] contains all the possible information directly related to the torsion tensor. In this paper, we discuss how the Noether symmetry approach allows one to fix the form of the function [Formula: see text] and to derive exact cosmological solutions.

Gas Evolution Dynamics in Godunov-Type Schemes and Analysis of Numerical Shock Instability

NASA Technical Reports Server (NTRS)

Xu, Kun

1999-01-01

In this paper we are going to study the gas evolution dynamics of the exact and approximate Riemann solvers, e.g., the Flux Vector Splitting (FVS) and the Flux Difference Splitting (FDS) schemes. Since the FVS scheme and the Kinetic Flux Vector Splitting (KFVS) scheme have the same physical mechanism and similar flux function, based on the analysis of the discretized KFVS scheme the weakness and advantage of the FVS scheme are closely observed. The subtle dissipative mechanism of the Godunov method in the 2D case is also analyzed, and the physical reason for shock instability, i.e., carbuncle phenomena and odd-even decoupling, is presented.

A new relativistic viscous hydrodynamics code and its application to the Kelvin-Helmholtz instability in high-energy heavy-ion collisions

NASA Astrophysics Data System (ADS)

Okamoto, Kazuhisa; Nonaka, Chiho

2017-06-01

We construct a new relativistic viscous hydrodynamics code optimized in the Milne coordinates. We split the conservation equations into an ideal part and a viscous part, using the Strang spitting method. In the code a Riemann solver based on the two-shock approximation is utilized for the ideal part and the Piecewise Exact Solution (PES) method is applied for the viscous part. We check the validity of our numerical calculations by comparing analytical solutions, the viscous Bjorken's flow and the Israel-Stewart theory in Gubser flow regime. Using the code, we discuss possible development of the Kelvin-Helmholtz instability in high-energy heavy-ion collisions.

Shock and Rarefaction Waves in a Heterogeneous Mantle

NASA Astrophysics Data System (ADS)

Jordan, J.; Hesse, M. A.

2012-12-01

We explore the effect of heterogeneities on partial melting and melt migration during active upwelling in the Earth's mantle. We have constructed simple, explicit nonlinear models in one dimension to examine heterogeneity and its dynamic affects on porosity, temperature and the magnesium number in a partially molten, porous medium comprised of olivine. The composition of the melt and solid are defined by a closed, binary phase diagram for a simplified, two-component olivine system. The two-component solid solution is represented by a phase loop where concentrations 0 and 1 to correspond to fayalite and forsterite, respectively. For analysis, we examine an advective system with a Riemann initial condition. Chromatographic tools and theory have primarily been used to track large, rare earth elements as tracers. In our case, we employ these theoretical tools to highlight the importance of the magnesium number, enthalpy and overall heterogeneity in the dynamics of melt migration. We calculate the eigenvectors and eigenvalues in the concentration-enthalpy space in order to glean the characteristics of the waves emerging the Riemann step. Analysis on Riemann problems of this nature shows us that the composition-enthalpy waves can be represented by self-similar solutions. The eigenvalues of the composition-enthalpy system represent the characteristic wave propagation speeds of the compositions and enthalpy through the domain. Furthermore, the corresponding eigenvectors are the directions of variation, or ``pathways," in concentration-enthalpy space that the characteristic waves follow. In the two-component system, the Riemann problem yields two waves connected by an intermediate concentration-enthalpy state determined by the intersections of the integral curves of the eigenvectors emanating from both the initial and boundary states. The first wave, ``slow path," and second wave, ``fast path," follow the aformentioned pathways set by the eigenvectors. The slow path wave has a zero eigenvalue, corresponding to a wave speed of zero, which preserves a residual imprint of the initial condition. Freezing fronts textemdash those that result in a negative change in porositytextemdash feature fast path waves that travel as shocks, whereas the fast path waves of melting fronts travel as spreading, rarefaction waves.

Using the Pottery Wheel to Explore Topics in Calculus

ERIC Educational Resources Information Center

Farnell, Elin; Snipes, Marie A.

2015-01-01

Students sometimes struggle with visualizing the three-dimensional solids encountered in certain integral problems in a calculus class. We present a project in which students create solids of revolution with clay on a pottery wheel and estimate the volumes of these objects using Riemann sums. In addition to giving students an opportunity for…

A high-order relativistic two-fluid electrodynamic scheme with consistent reconstruction of electromagnetic fields and a multidimensional Riemann solver for electromagnetism

DOE Office of Scientific and Technical Information (OSTI.GOV)

Balsara, Dinshaw S., E-mail: dbalsara@nd.edu; Amano, Takanobu, E-mail: amano@eps.s.u-tokyo.ac.jp; Garain, Sudip, E-mail: sgarain@nd.edu

In various astrophysics settings it is common to have a two-fluid relativistic plasma that interacts with the electromagnetic field. While it is common to ignore the displacement current in the ideal, classical magnetohydrodynamic limit, when the flows become relativistic this approximation is less than absolutely well-justified. In such a situation, it is more natural to consider a positively charged fluid made up of positrons or protons interacting with a negatively charged fluid made up of electrons. The two fluids interact collectively with the full set of Maxwell's equations. As a result, a solution strategy for that coupled system of equationsmore » is sought and found here. Our strategy extends to higher orders, providing increasing accuracy. The primary variables in the Maxwell solver are taken to be the facially-collocated components of the electric and magnetic fields. Consistent with such a collocation, three important innovations are reported here. The first two pertain to the Maxwell solver. In our first innovation, the magnetic field within each zone is reconstructed in a divergence-free fashion while the electric field within each zone is reconstructed in a form that is consistent with Gauss' law. In our second innovation, a multidimensionally upwinded strategy is presented which ensures that the magnetic field can be updated via a discrete interpretation of Faraday's law and the electric field can be updated via a discrete interpretation of the generalized Ampere's law. This multidimensional upwinding is achieved via a multidimensional Riemann solver. The multidimensional Riemann solver automatically provides edge-centered electric field components for the Stokes law-based update of the magnetic field. It also provides edge-centered magnetic field components for the Stokes law-based update of the electric field. The update strategy ensures that the electric field is always consistent with Gauss' law and the magnetic field is always divergence-free. This collocation also ensures that electromagnetic radiation that is propagating in a vacuum has both electric and magnetic fields that are exactly divergence-free. Coupled relativistic fluid dynamic equations are solved for the positively and negatively charged fluids. The fluids' numerical fluxes also provide a self-consistent current density for the update of the electric field. Our reconstruction strategy ensures that fluid velocities always remain sub-luminal. Our third innovation consists of an efficient design for several popular IMEX schemes so that they provide strong coupling between the finite-volume-based fluid solver and the electromagnetic fields at high order. This innovation makes it possible to efficiently utilize high order IMEX time update methods for stiff source terms in the update of high order finite-volume methods for hyperbolic conservation laws. We also show that this very general innovation should extend seamlessly to Runge–Kutta discontinuous Galerkin methods. The IMEX schemes enable us to use large CFL numbers even in the presence of stiff source terms. Several accuracy analyses are presented showing that our method meets its design accuracy in the MHD limit as well as in the limit of electromagnetic wave propagation. Several stringent test problems are also presented. We also present a relativistic version of the GEM problem, which shows that our algorithm can successfully adapt to challenging problems in high energy astrophysics.« less

A high-order relativistic two-fluid electrodynamic scheme with consistent reconstruction of electromagnetic fields and a multidimensional Riemann solver for electromagnetism

NASA Astrophysics Data System (ADS)

Balsara, Dinshaw S.; Amano, Takanobu; Garain, Sudip; Kim, Jinho

2016-08-01

In various astrophysics settings it is common to have a two-fluid relativistic plasma that interacts with the electromagnetic field. While it is common to ignore the displacement current in the ideal, classical magnetohydrodynamic limit, when the flows become relativistic this approximation is less than absolutely well-justified. In such a situation, it is more natural to consider a positively charged fluid made up of positrons or protons interacting with a negatively charged fluid made up of electrons. The two fluids interact collectively with the full set of Maxwell's equations. As a result, a solution strategy for that coupled system of equations is sought and found here. Our strategy extends to higher orders, providing increasing accuracy. The primary variables in the Maxwell solver are taken to be the facially-collocated components of the electric and magnetic fields. Consistent with such a collocation, three important innovations are reported here. The first two pertain to the Maxwell solver. In our first innovation, the magnetic field within each zone is reconstructed in a divergence-free fashion while the electric field within each zone is reconstructed in a form that is consistent with Gauss' law. In our second innovation, a multidimensionally upwinded strategy is presented which ensures that the magnetic field can be updated via a discrete interpretation of Faraday's law and the electric field can be updated via a discrete interpretation of the generalized Ampere's law. This multidimensional upwinding is achieved via a multidimensional Riemann solver. The multidimensional Riemann solver automatically provides edge-centered electric field components for the Stokes law-based update of the magnetic field. It also provides edge-centered magnetic field components for the Stokes law-based update of the electric field. The update strategy ensures that the electric field is always consistent with Gauss' law and the magnetic field is always divergence-free. This collocation also ensures that electromagnetic radiation that is propagating in a vacuum has both electric and magnetic fields that are exactly divergence-free. Coupled relativistic fluid dynamic equations are solved for the positively and negatively charged fluids. The fluids' numerical fluxes also provide a self-consistent current density for the update of the electric field. Our reconstruction strategy ensures that fluid velocities always remain sub-luminal. Our third innovation consists of an efficient design for several popular IMEX schemes so that they provide strong coupling between the finite-volume-based fluid solver and the electromagnetic fields at high order. This innovation makes it possible to efficiently utilize high order IMEX time update methods for stiff source terms in the update of high order finite-volume methods for hyperbolic conservation laws. We also show that this very general innovation should extend seamlessly to Runge-Kutta discontinuous Galerkin methods. The IMEX schemes enable us to use large CFL numbers even in the presence of stiff source terms. Several accuracy analyses are presented showing that our method meets its design accuracy in the MHD limit as well as in the limit of electromagnetic wave propagation. Several stringent test problems are also presented. We also present a relativistic version of the GEM problem, which shows that our algorithm can successfully adapt to challenging problems in high energy astrophysics.

Gasdynamic Inlet Isolation in Rotating Detonation Engine

DTIC Science & Technology

2010-12-01

2D Total Variation Diminishing (TVD): Continuous Riemann Solver Minimum Dissipation: LHS & RHS Activate pressure switch : Supersonic Activate...Total Variation Diminishing (TVD) limiter: Continuous Riemann Solver Minimum Dissipation: LHS & RHS Activate pressure switch : Supersonic Activate...Continuous 94 Riemann Solver Minimum Dissipation: LHS & RHS Activate pressure switch : Supersonic Activate pressure gradient switch: Normal

Initial-boundary value problems associated with the Ablowitz-Ladik system

NASA Astrophysics Data System (ADS)

Xia, Baoqiang; Fokas, A. S.

2018-02-01

We employ the Ablowitz-Ladik system as an illustrative example in order to demonstrate how to analyze initial-boundary value problems for integrable nonlinear differential-difference equations via the unified transform (Fokas method). In particular, we express the solutions of the integrable discrete nonlinear Schrödinger and integrable discrete modified Korteweg-de Vries equations in terms of the solutions of appropriate matrix Riemann-Hilbert problems. We also discuss in detail, for both the above discrete integrable equations, the associated global relations and the process of eliminating of the unknown boundary values.

Numerical simulation of the hydrodynamic instabilities of Richtmyer-Meshkov and Rayleigh-Taylor

NASA Astrophysics Data System (ADS)

Fortova, S. V.; Shepelev, V. V.; Troshkin, O. V.; Kozlov, S. A.

2017-09-01

The paper presents the results of numerical simulation of the development of hydrodynamic instabilities of Richtmyer-Meshkov and Rayleigh-Taylor encountered in experiments [1-3]. For the numerical solution used the TPS software package (Turbulence Problem Solver) that implements a generalized approach to constructing computer programs for a wide range of problems of hydrodynamics, described by the system of equations of hyperbolic type. As numerical methods are used the method of large particles and ENO-scheme of the second order with Roe solver for the approximate solution of the Riemann problem.

A pressure relaxation closure model for one-dimensional, two-material Lagrangian hydrodynamics based on the Riemann problem

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kamm, James R; Shashkov, Mikhail J

2009-01-01

Despite decades of development, Lagrangian hydrodynamics of strengthfree materials presents numerous open issues, even in one dimension. We focus on the problem of closing a system of equations for a two-material cell under the assumption of a single velocity model. There are several existing models and approaches, each possessing different levels of fidelity to the underlying physics and each exhibiting unique features in the computed solutions. We consider the case in which the change in heat in the constituent materials in the mixed cell is assumed equal. An instantaneous pressure equilibration model for a mixed cell can be cast asmore » four equations in four unknowns, comprised of the updated values of the specific internal energy and the specific volume for each of the two materials in the mixed cell. The unique contribution of our approach is a physics-inspired, geometry-based model in which the updated values of the sub-cell, relaxing-toward-equilibrium constituent pressures are related to a local Riemann problem through an optimization principle. This approach couples the modeling problem of assigning sub-cell pressures to the physics associated with the local, dynamic evolution. We package our approach in the framework of a standard predictor-corrector time integration scheme. We evaluate our model using idealized, two material problems using either ideal-gas or stiffened-gas equations of state and compare these results to those computed with the method of Tipton and with corresponding pure-material calculations.« less

Numerical Simulation of the Interaction of an Air Shock Wave with a Surface Gas-Dust Layer

NASA Astrophysics Data System (ADS)

Surov, V. S.

2018-05-01

Within the framework of the one-velocity and multivelocity models of a dust-laden gas with the use of the Godunov method with a linearized Riemann solver, the problem of the interaction of a shock wave with a dust-laden gas layer located along a solid plane surface has been studied.

Numerical Simulation of the Interaction of an Air Shock Wave with a Surface Gas-Dust Layer

NASA Astrophysics Data System (ADS)

Surov, V. S.

2018-03-01

Within the framework of the one-velocity and multivelocity models of a dust-laden gas with the use of the Godunov method with a linearized Riemann solver, the problem of the interaction of a shock wave with a dust-laden gas layer located along a solid plane surface has been studied.

A finite-volume HLLC-based scheme for compressible interfacial flows with surface tension

NASA Astrophysics Data System (ADS)

Garrick, Daniel P.; Owkes, Mark; Regele, Jonathan D.

2017-06-01

Shock waves are often used in experiments to create a shear flow across liquid droplets to study secondary atomization. Similar behavior occurs inside of supersonic combustors (scramjets) under startup conditions, but it is challenging to study these conditions experimentally. In order to investigate this phenomenon further, a numerical approach is developed to simulate compressible multiphase flows under the effects of surface tension forces. The flow field is solved via the compressible multicomponent Euler equations (i.e., the five equation model) discretized with the finite volume method on a uniform Cartesian grid. The solver utilizes a total variation diminishing (TVD) third-order Runge-Kutta method for time-marching and second order TVD spatial reconstruction. Surface tension is incorporated using the Continuum Surface Force (CSF) model. Fluxes are upwinded with a modified Harten-Lax-van Leer Contact (HLLC) approximate Riemann solver. An interface compression scheme is employed to counter numerical diffusion of the interface. The present work includes modifications to both the HLLC solver and the interface compression scheme to account for capillary force terms and the associated pressure jump across the gas-liquid interface. A simple method for numerically computing the interface curvature is developed and an acoustic scaling of the surface tension coefficient is proposed for the non-dimensionalization of the model. The model captures the surface tension induced pressure jump exactly if the exact curvature is known and is further verified with an oscillating elliptical droplet and Mach 1.47 and 3 shock-droplet interaction problems. The general characteristics of secondary atomization at a range of Weber numbers are also captured in a series of simulations.

A finite-volume HLLC-based scheme for compressible interfacial flows with surface tension

DOE Office of Scientific and Technical Information (OSTI.GOV)

Garrick, Daniel P.; Owkes, Mark; Regele, Jonathan D., E-mail: jregele@iastate.edu

Shock waves are often used in experiments to create a shear flow across liquid droplets to study secondary atomization. Similar behavior occurs inside of supersonic combustors (scramjets) under startup conditions, but it is challenging to study these conditions experimentally. In order to investigate this phenomenon further, a numerical approach is developed to simulate compressible multiphase flows under the effects of surface tension forces. The flow field is solved via the compressible multicomponent Euler equations (i.e., the five equation model) discretized with the finite volume method on a uniform Cartesian grid. The solver utilizes a total variation diminishing (TVD) third-order Runge–Kuttamore » method for time-marching and second order TVD spatial reconstruction. Surface tension is incorporated using the Continuum Surface Force (CSF) model. Fluxes are upwinded with a modified Harten–Lax–van Leer Contact (HLLC) approximate Riemann solver. An interface compression scheme is employed to counter numerical diffusion of the interface. The present work includes modifications to both the HLLC solver and the interface compression scheme to account for capillary force terms and the associated pressure jump across the gas–liquid interface. A simple method for numerically computing the interface curvature is developed and an acoustic scaling of the surface tension coefficient is proposed for the non-dimensionalization of the model. The model captures the surface tension induced pressure jump exactly if the exact curvature is known and is further verified with an oscillating elliptical droplet and Mach 1.47 and 3 shock-droplet interaction problems. The general characteristics of secondary atomization at a range of Weber numbers are also captured in a series of simulations.« less

c-Extremization from toric geometry

NASA Astrophysics Data System (ADS)

Amariti, Antonio; Cassia, Luca; Penati, Silvia

2018-04-01

We derive a geometric formulation of the 2d central charge cr from infinite families of 4d N = 1 superconformal field theories topologically twisted on constant curvature Riemann surfaces. They correspond to toric quiver gauge theories and are associated to D3 branes probing five dimensional Sasaki-Einstein geometries in the AdS/CFT correspondence. We show that cr can be expressed in terms of the areas of the toric diagram describing the moduli space of the 4d theory, both for toric geometries with smooth and singular horizons. We also study the relation between a-maximization in 4d and c-extremization in 2d, giving further evidences of the mixing of the baryonic symmetries with the exact R-current in two dimensions.

A new relativistic viscous hydrodynamics code and its application to the Kelvin–Helmholtz instability in high-energy heavy-ion collisions

DOE PAGES

Okamoto, Kazuhisa; Nonaka, Chiho

2017-06-09

Here, we construct a new relativistic viscous hydrodynamics code optimized in the Milne coordinates. We also split the conservation equations into an ideal part and a viscous part, using the Strang spitting method. In the code a Riemann solver based on the two-shock approximation is utilized for the ideal part and the Piecewise Exact Solution (PES) method is applied for the viscous part. Furthemore, we check the validity of our numerical calculations by comparing analytical solutions, the viscous Bjorken’s flow and the Israel–Stewart theory in Gubser flow regime. Using the code, we discuss possible development of the Kelvin–Helmholtz instability inmore » high-energy heavy-ion collisions.« less

The three-wave equation on the half-line

NASA Astrophysics Data System (ADS)

Xu, Jian; Fan, Engui

2014-01-01

The Fokas method is used to analyze the initial-boundary value problem for the three-wave equation p-{bi-bj}/{ai-aj}p+∑k ({bk-bj}/{ak-aj}-{bi-bk}/{ai-ak})pp=0, i,j,k=1,2,3, on the half-line. Assuming that the solution p(x,t) exists, we show that it can be recovered from its initial and boundary values via the solution of a Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ.

Lie Symmetry Analysis, Conservation Laws and Exact Power Series Solutions for Time-Fractional Fordy-Gibbons Equation

NASA Astrophysics Data System (ADS)

Feng, Lian-Li; Tian, Shou-Fu; Wang, Xiu-Bin; Zhang, Tian-Tian

2016-09-01

In this paper, the time fractional Fordy-Gibbons equation is investigated with Riemann-Liouville derivative. The equation can be reduced to the Caudrey-Dodd-Gibbon equation, Savada-Kotera equation and the Kaup-Kupershmidt equation, etc. By means of the Lie group analysis method, the invariance properties and symmetry reductions of the equation are derived. Furthermore, by means of the power series theory, its exact power series solutions of the equation are also constructed. Finally, two kinds of conservation laws of the equation are well obtained with aid of the self-adjoint method. Supported by the Fundamental Research Funds for Key Discipline Construction under Grant No. XZD201602, the Fundamental Research Funds for the Central Universities under Grant Nos. 2015QNA53 and 2015XKQY14, the Fundamental Research Funds for Postdoctoral at the Key Laboratory of Gas and Fire Control for Coal Mines, the General Financial Grant from the China Postdoctoral Science Foundation under Grant No. 2015M570498, and Natural Sciences Foundation of China under Grant No. 11301527

A Lagrangian meshfree method applied to linear and nonlinear elasticity.

PubMed

Walker, Wade A

2017-01-01

The repeated replacement method (RRM) is a Lagrangian meshfree method which we have previously applied to the Euler equations for compressible fluid flow. In this paper we present new enhancements to RRM, and we apply the enhanced method to both linear and nonlinear elasticity. We compare the results of ten test problems to those of analytic solvers, to demonstrate that RRM can successfully simulate these elastic systems without many of the requirements of traditional numerical methods such as numerical derivatives, equation system solvers, or Riemann solvers. We also show the relationship between error and computational effort for RRM on these systems, and compare RRM to other methods to highlight its strengths and weaknesses. And to further explain the two elastic equations used in the paper, we demonstrate the mathematical procedure used to create Riemann and Sedov-Taylor solvers for them, and detail the numerical techniques needed to embody those solvers in code.

A Lagrangian meshfree method applied to linear and nonlinear elasticity

PubMed Central

2017-01-01

The repeated replacement method (RRM) is a Lagrangian meshfree method which we have previously applied to the Euler equations for compressible fluid flow. In this paper we present new enhancements to RRM, and we apply the enhanced method to both linear and nonlinear elasticity. We compare the results of ten test problems to those of analytic solvers, to demonstrate that RRM can successfully simulate these elastic systems without many of the requirements of traditional numerical methods such as numerical derivatives, equation system solvers, or Riemann solvers. We also show the relationship between error and computational effort for RRM on these systems, and compare RRM to other methods to highlight its strengths and weaknesses. And to further explain the two elastic equations used in the paper, we demonstrate the mathematical procedure used to create Riemann and Sedov-Taylor solvers for them, and detail the numerical techniques needed to embody those solvers in code. PMID:29045443

Implicit and explicit subgrid-scale modeling in discontinuous Galerkin methods for large-eddy simulation

NASA Astrophysics Data System (ADS)

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

2017-11-01

Over the past few years, high-order discontinuous Galerkin (DG) methods for Large-Eddy Simulation (LES) have emerged as a promising approach to solve complex turbulent flows. Despite the significant research investment, the relation between the discretization scheme, the Riemann flux, the subgrid-scale (SGS) model and the accuracy of the resulting LES solver remains unclear. In this talk, we investigate the role of the Riemann solver and the SGS model in the ability to predict a variety of flow regimes, including transition to turbulence, wall-free turbulence, wall-bounded turbulence, and turbulence decay. The Taylor-Green vortex problem and the turbulent channel flow at various Reynolds numbers are considered. Numerical results show that DG methods implicitly introduce numerical dissipation in under-resolved turbulence simulations and, even in the high Reynolds number limit, this implicit dissipation provides a more accurate representation of the actual subgrid-scale dissipation than that by explicit models.

Adaptive Discontinuous Evolution Galerkin Method for Dry Atmospheric Flow

DTIC Science & Technology

2013-04-02

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

Numerical comparison of Riemann solvers for astrophysical hydrodynamics

NASA Astrophysics Data System (ADS)

Klingenberg, Christian; Schmidt, Wolfram; Waagan, Knut

2007-11-01

The idea of this work is to compare a new positive and entropy stable approximate Riemann solver by Francois Bouchut with a state-of the-art algorithm for astrophysical fluid dynamics. We implemented the new Riemann solver into an astrophysical PPM-code, the Prometheus code, and also made a version with a different, more theoretically grounded higher order algorithm than PPM. We present shock tube tests, two-dimensional instability tests and forced turbulence simulations in three dimensions. We find subtle differences between the codes in the shock tube tests, and in the statistics of the turbulence simulations. The new Riemann solver increases the computational speed without significant loss of accuracy.

Stratified flows with variable density: mathematical modelling and numerical challenges.

NASA Astrophysics Data System (ADS)

Murillo, Javier; Navas-Montilla, Adrian

2017-04-01

Stratified flows appear in a wide variety of fundamental problems in hydrological and geophysical sciences. They may involve from hyperconcentrated floods carrying sediment causing collapse, landslides and debris flows, to suspended material in turbidity currents where turbulence is a key process. Also, in stratified flows variable horizontal density is present. Depending on the case, density varies according to the volumetric concentration of different components or species that can represent transported or suspended materials or soluble substances. Multilayer approaches based on the shallow water equations provide suitable models but are not free from difficulties when moving to the numerical resolution of the governing equations. Considering the variety of temporal and spatial scales, transfer of mass and energy among layers may strongly differ from one case to another. As a consequence, in order to provide accurate solutions, very high order methods of proved quality are demanded. Under these complex scenarios it is necessary to observe that the numerical solution provides the expected order of accuracy but also converges to the physically based solution, which is not an easy task. To this purpose, this work will focus in the use of Energy balanced augmented solvers, in particular, the Augmented Roe Flux ADER scheme. References: J. Murillo , P. García-Navarro, Wave Riemann description of friction terms in unsteady shallow flows: Application to water and mud/debris floods. J. Comput. Phys. 231 (2012) 1963-2001. J. Murillo B. Latorre, P. García-Navarro. A Riemann solver for unsteady computation of 2D shallow flows with variable density. J. Comput. Phys.231 (2012) 4775-4807. A. Navas-Montilla, J. Murillo, Energy balanced numerical schemes with very high order. The Augmented Roe Flux ADER scheme. Application to the shallow water equations, J. Comput. Phys. 290 (2015) 188-218. A. Navas-Montilla, J. Murillo, Asymptotically and exactly energy balanced augmented flux-ADER schemes with application to hyperbolic conservation laws with geometric source terms, J. Comput. Phys. 317 (2016) 108-147. J. Murillo and A. Navas-Montilla, A comprehensive explanation and exercise of the source terms in hyperbolic systems using Roe type solutions. Application to the 1D-2D shallow water equations, Advances in Water Resources 98 (2016) 70-96.

A randomized trial of Rapid Rhino Riemann and Telfa nasal packs following endoscopic sinus surgery.

PubMed

Cruise, A S; Amonoo-Kuofi, K; Srouji, I; Kanagalingam, J; Georgalas, C; Patel, N N; Badia, L; Lund, V J

2006-02-01

To compare Telfa with the Rapid Rhino Riemann nasal pack for use following endoscopic sinus surgery. Prospective, randomized, double-blind, paired trial. Tertiary otolaryngology hospital. Forty-five adult patients undergoing bilateral endoscopic sinus surgery for either chronic rhinosinusitis or nasal polyps. A visual analogue scale was used to assess discomfort caused by the presence of the packs in the nose and by their removal. The amount of bleeding was noted with the packs in place and following their removal. Crusting and adhesions were assessed 2 and 6 weeks following surgery. Both packs performed well giving good haemostasis and causing little bleeding on removal. Both packs caused only mild discomfort while in the nose. On the visual analogue scale of 0-10 cm the mean visual analogue score for Rapid Rhino Riemann pack was 1.7 and for Telfa 2.0 (P = 0.371). The Rapid Rhino Riemann pack caused significantly less pain on removal compared with the Telfa pack with a mean visual analogue score of 2.0 in comparison with 3.7 for Telfa (P = 0.001). There were less adhesions with the Rapid Rhino Riemann than Telfa pack but this was not statistically significant (P = 0.102). Both Telfa and Rapid Rhino Riemann packs can be recommended as packs that control postoperative haemorrhage, do not cause bleeding on removal and cause little discomfort while in the nose. The Rapid Rhino Riemann pack has the advantage of causing significantly less pain on removal.

Accurate boundary conditions for exterior problems in gas dynamics

NASA Technical Reports Server (NTRS)

Hagstrom, Thomas; Hariharan, S. I.

1988-01-01

The numerical solution of exterior problems is typically accomplished by introducing an artificial, far field boundary and solving the equations on a truncated domain. For hyperbolic systems, boundary conditions at this boundary are often derived by imposing a principle of no reflection. However, waves with spherical symmetry in gas dynamics satisfy equations where incoming and outgoing Riemann variables are coupled. This suggests that natural reflections may be important. A reflecting boundary condition is proposed based on an asymptotic solution of the far field equations. Nonlinear energy estimates are obtained for the truncated problem and numerical experiments presented to validate the theory.

Accurate boundary conditions for exterior problems in gas dynamics

NASA Technical Reports Server (NTRS)

Hagstrom, Thomas; Hariharan, S. I.

1988-01-01

The numerical solution of exterior problems is typically accomplished by introducing an artificial, far-field boundary and solving the equations on a truncated domain. For hyperbolic systems, boundary conditions at this boundary are often derived by imposing a principle of no reflection. However, waves with spherical symmetry in gas dynamics satisfy equations where incoming and outgoing Riemann variables are coupled. This suggests that natural reflections may be important. A reflecting boundary condition is proposed based on an asymptotic solution of the far-field equations. Nonlinear energy estimates are obtained for the truncated problem and numerical experiments presented to validate the theory.

Numerical Inverse Scattering for the Toda Lattice

NASA Astrophysics Data System (ADS)

Bilman, Deniz; Trogdon, Thomas

2017-06-01

We present a method to compute the inverse scattering transform (IST) for the famed Toda lattice by solving the associated Riemann-Hilbert (RH) problem numerically. Deformations for the RH problem are incorporated so that the IST can be evaluated in O(1) operations for arbitrary points in the ( n, t)-domain, including short- and long-time regimes. No time-stepping is required to compute the solution because ( n, t) appear as parameters in the associated RH problem. The solution of the Toda lattice is computed in long-time asymptotic regions where the asymptotics are not known rigorously.

The first boundary-value problem for a fractional diffusion-wave equation in a non-cylindrical domain

NASA Astrophysics Data System (ADS)

Pskhu, A. V.

2017-12-01

We solve the first boundary-value problem in a non-cylindrical domain for a diffusion-wave equation with the Dzhrbashyan- Nersesyan operator of fractional differentiation with respect to the time variable. We prove an existence and uniqueness theorem for this problem, and construct a representation of the solution. We show that a sufficient condition for unique solubility is the condition of Hölder smoothness for the lateral boundary of the domain. The corresponding results for equations with Riemann- Liouville and Caputo derivatives are particular cases of results obtained here.

A new operational approach for solving fractional variational problems depending on indefinite integrals

NASA Astrophysics Data System (ADS)

Ezz-Eldien, S. S.; Doha, E. H.; Bhrawy, A. H.; El-Kalaawy, A. A.; Machado, J. A. T.

2018-04-01

In this paper, we propose a new accurate and robust numerical technique to approximate the solutions of fractional variational problems (FVPs) depending on indefinite integrals with a type of fixed Riemann-Liouville fractional integral. The proposed technique is based on the shifted Chebyshev polynomials as basis functions for the fractional integral operational matrix (FIOM). Together with the Lagrange multiplier method, these problems are then reduced to a system of algebraic equations, which greatly simplifies the solution process. Numerical examples are carried out to confirm the accuracy, efficiency and applicability of the proposed algorithm

Hyperbolic conservation laws and numerical methods

NASA Technical Reports Server (NTRS)

Leveque, Randall J.

1990-01-01

The mathematical structure of hyperbolic systems and the scalar equation case of conservation laws are discussed. Linear, nonlinear systems and the Riemann problem for the Euler equations are also studied. The numerical methods for conservation laws are presented in a nonstandard manner which leads to large time steps generalizations and computations on irregular grids. The solution of conservation laws with stiff source terms is examined.

Feature Detection and Curve Fitting Using Fast Walsh Transforms for Shock Tracking: Applications

NASA Technical Reports Server (NTRS)

Gnoffo, Peter A.

2017-01-01

Walsh functions form an orthonormal basis set consisting of square waves. Square waves make the system well suited for detecting and representing functions with discontinuities. Given a uniform distribution of 2p cells on a one-dimensional element, it has been proven that the inner product of the Walsh Root function for group p with every polynomial of degree < or = (p - 1) across the element is identically zero. It has also been proven that the magnitude and location of a discontinuous jump, as represented by a Heaviside function, are explicitly identified by its Fast Walsh Transform (FWT) coefficients. These two proofs enable an algorithm that quickly provides a Weighted Least Squares fit to distributions across the element that include a discontinuity. The detection of a discontinuity enables analytic relations to locally describe its evolution and provide increased accuracy. Time accurate examples are provided for advection, Burgers equation, and Riemann problems (diaphragm burst) in closed tubes and de Laval nozzles. New algorithms to detect up to two C0 and/or C1 discontinuities within a single element are developed for application to the Riemann problem, in which a contact discontinuity and shock wave form after the diaphragm bursts.

Potential profile near singularity point in kinetic Tonks-Langmuir discharges as a function of the ion sources temperature

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kos, L.; Tskhakaya, D. D.; Jelic, N.

2011-05-15

A plasma-sheath transition analysis requires a reliable mathematical expression for the plasma potential profile {Phi}(x) near the sheath edge x{sub s} in the limit {epsilon}{identical_to}{lambda}{sub D}/l=0 (where {lambda}{sub D} is the Debye length and l is a proper characteristic length of the discharge). Such expressions have been explicitly calculated for the fluid model and the singular (cold ion source) kinetic model, where exact analytic solutions for plasma equation ({epsilon}=0) are known, but not for the regular (warm ion source) kinetic model, where no analytic solution of the plasma equation has ever been obtained. For the latter case, Riemann [J. Phys.more » D: Appl. Phys. 24, 493 (1991)] only predicted a general formula assuming relatively high ion-source temperatures, i.e., much higher than the plasma-sheath potential drop. Riemann's formula, however, according to him, never was confirmed in explicit solutions of particular models (e.g., that of Bissell and Johnson [Phys. Fluids 30, 779 (1987)] and Scheuer and Emmert [Phys. Fluids 31, 3645 (1988)]) since ''the accuracy of the classical solutions is not sufficient to analyze the sheath vicinity''[Riemann, in Proceedings of the 62nd Annual Gaseous Electronic Conference, APS Meeting Abstracts, Vol. 54 (APS, 2009)]. Therefore, for many years, there has been a need for explicit calculation that might confirm the Riemann's general formula regarding the potential profile at the sheath edge in the cases of regular very warm ion sources. Fortunately, now we are able to achieve a very high accuracy of results [see, e.g., Kos et al., Phys. Plasmas 16, 093503 (2009)]. We perform this task by using both the analytic and the numerical method with explicit Maxwellian and ''water-bag'' ion source velocity distributions. We find the potential profile near the plasma-sheath edge in the whole range of ion source temperatures of general interest to plasma physics, from zero to ''practical infinity.'' While within limits of ''very low'' and ''relatively high'' ion source temperatures, the potential is proportional to the space coordinate powered by rational numbers {alpha}=1/2 and {alpha}=2/3, with medium ion source temperatures. We found {alpha} between these values being a non-rational number strongly dependent on the ion source temperature. The range of the non-rational power-law turns out to be a very narrow one, at the expense of the extension of {alpha}=2/3 region towards unexpectedly low ion source temperatures.« less

A new class of accurate, mesh-free hydrodynamic simulation methods

NASA Astrophysics Data System (ADS)

Hopkins, Philip F.

2015-06-01

We present two new Lagrangian methods for hydrodynamics, in a systematic comparison with moving-mesh, smoothed particle hydrodynamics (SPH), and stationary (non-moving) grid methods. The new methods are designed to simultaneously capture advantages of both SPH and grid-based/adaptive mesh refinement (AMR) schemes. They are based on a kernel discretization of the volume coupled to a high-order matrix gradient estimator and a Riemann solver acting over the volume `overlap'. We implement and test a parallel, second-order version of the method with self-gravity and cosmological integration, in the code GIZMO:1 this maintains exact mass, energy and momentum conservation; exhibits superior angular momentum conservation compared to all other methods we study; does not require `artificial diffusion' terms; and allows the fluid elements to move with the flow, so resolution is automatically adaptive. We consider a large suite of test problems, and find that on all problems the new methods appear competitive with moving-mesh schemes, with some advantages (particularly in angular momentum conservation), at the cost of enhanced noise. The new methods have many advantages versus SPH: proper convergence, good capturing of fluid-mixing instabilities, dramatically reduced `particle noise' and numerical viscosity, more accurate sub-sonic flow evolution, and sharp shock-capturing. Advantages versus non-moving meshes include: automatic adaptivity, dramatically reduced advection errors and numerical overmixing, velocity-independent errors, accurate coupling to gravity, good angular momentum conservation and elimination of `grid alignment' effects. We can, for example, follow hundreds of orbits of gaseous discs, while AMR and SPH methods break down in a few orbits. However, fixed meshes minimize `grid noise'. These differences are important for a range of astrophysical problems.

A Revelation: Quantum-Statistics and Classical-Statistics are Analytic-Geometry Conic-Sections and Numbers/Functions: Euler, Riemann, Bernoulli Generating-Functions: Conics to Numbers/Functions Deep Subtle Connections

NASA Astrophysics Data System (ADS)

Descartes, R.; Rota, G.-C.; Euler, L.; Bernoulli, J. D.; Siegel, Edward Carl-Ludwig

2011-03-01

Quantum-statistics Dichotomy: Fermi-Dirac(FDQS) Versus Bose-Einstein(BEQS), respectively with contact-repulsion/non-condensation(FDCR) versus attraction/ condensationBEC are manifestly-demonstrated by Taylor-expansion ONLY of their denominator exponential, identified BOTH as Descartes analytic-geometry conic-sections, FDQS as Elllipse (homotopy to rectangle FDQS distribution-function), VIA Maxwell-Boltzmann classical-statistics(MBCS) to Parabola MORPHISM, VS. BEQS to Hyperbola, Archimedes' HYPERBOLICITY INEVITABILITY, and as well generating-functions[Abramowitz-Stegun, Handbook Math.-Functions--p. 804!!!], respectively of Euler-numbers/functions, (via Riemann zeta-function(domination of quantum-statistics: [Pathria, Statistical-Mechanics; Huang, Statistical-Mechanics]) VS. Bernoulli-numbers/ functions. Much can be learned about statistical-physics from Euler-numbers/functions via Riemann zeta-function(s) VS. Bernoulli-numbers/functions [Conway-Guy, Book of Numbers] and about Euler-numbers/functions, via Riemann zeta-function(s) MORPHISM, VS. Bernoulli-numbers/ functions, visa versa!!! Ex.: Riemann-hypothesis PHYSICS proof PARTLY as BEQS BEC/BEA!!!

A shock-capturing SPH scheme based on adaptive kernel estimation

NASA Astrophysics Data System (ADS)

Sigalotti, Leonardo Di G.; López, Hender; Donoso, Arnaldo; Sira, Eloy; Klapp, Jaime

2006-02-01

Here we report a method that converts standard smoothed particle hydrodynamics (SPH) into a working shock-capturing scheme without relying on solutions to the Riemann problem. Unlike existing adaptive SPH simulations, the present scheme is based on an adaptive kernel estimation of the density, which combines intrinsic features of both the kernel and nearest neighbor approaches in a way that the amount of smoothing required in low-density regions is effectively controlled. Symmetrized SPH representations of the gas dynamic equations along with the usual kernel summation for the density are used to guarantee variational consistency. Implementation of the adaptive kernel estimation involves a very simple procedure and allows for a unique scheme that handles strong shocks and rarefactions the same way. Since it represents a general improvement of the integral interpolation on scattered data, it is also applicable to other fluid-dynamic models. When the method is applied to supersonic compressible flows with sharp discontinuities, as in the classical one-dimensional shock-tube problem and its variants, the accuracy of the results is comparable, and in most cases superior, to that obtained from high quality Godunov-type methods and SPH formulations based on Riemann solutions. The extension of the method to two- and three-space dimensions is straightforward. In particular, for the two-dimensional cylindrical Noh's shock implosion and Sedov point explosion problems the present scheme produces much better results than those obtained with conventional SPH codes.

Inverse scattering transform and soliton solutions for square matrix nonlinear Schrödinger equations with non-zero boundary conditions

NASA Astrophysics Data System (ADS)

Prinari, Barbara; Demontis, Francesco; Li, Sitai; Horikis, Theodoros P.

2018-04-01

The inverse scattering transform (IST) with non-zero boundary conditions at infinity is developed for an m × m matrix nonlinear Schrödinger-type equation which, in the case m = 2, has been proposed as a model to describe hyperfine spin F = 1 spinor Bose-Einstein condensates with either repulsive interatomic interactions and anti-ferromagnetic spin-exchange interactions (self-defocusing case), or attractive interatomic interactions and ferromagnetic spin-exchange interactions (self-focusing case). The IST for this system was first presented by Ieda et al. (2007) , using a different approach. In our formulation, both the direct and the inverse problems are posed in terms of a suitable uniformization variable which allows to develop the IST on the standard complex plane, instead of a two-sheeted Riemann surface or the cut plane with discontinuities along the cuts. Analyticity of the scattering eigenfunctions and scattering data, symmetries, properties of the discrete spectrum, and asymptotics are derived. The inverse problem is posed as a Riemann-Hilbert problem for the eigenfunctions, and the reconstruction formula of the potential in terms of eigenfunctions and scattering data is provided. In addition, the general behavior of the soliton solutions is analyzed in detail in the 2 × 2 self-focusing case, including some special solutions not previously discussed in the literature.

Hypergeometric Forms for Ising-Class Integrals

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bailey, David H.; Borwein, David; Borwein, Jonathan M.

2006-07-01

We apply experimental-mathematical principles to analyzecertain integrals relevant to the Ising theory of solid-state physics. Wefind representations of the these integrals in terms of MeijerG-functions and nested-Barnes integrals. Our investigations began bycomputing 500-digit numerical values of Cn,k,namely a 2-D array of Isingintegrals for all integers n, k where n is in [2,12]and k is in [0,25].We found that some Cn,k enjoy exact evaluations involving DirichletL-functions or the Riemann zeta function. In theprocess of analyzinghypergeometric representations, we found -- experimentally and strikingly-- that the Cn,k almost certainly satisfy certain inter-indicialrelations including discrete k-recursions. Using generating functions,differential theory, complex analysis, and Wilf-Zeilbergermore » algorithms weare able to prove some central cases of these relations.« less

Hydrodynamic simulations with the Godunov smoothed particle hydrodynamics

NASA Astrophysics Data System (ADS)

Murante, G.; Borgani, S.; Brunino, R.; Cha, S.-H.

2011-10-01

We present results based on an implementation of the Godunov smoothed particle hydrodynamics (GSPH), originally developed by Inutsuka, in the GADGET-3 hydrodynamic code. We first review the derivation of the GSPH discretization of the equations of moment and energy conservation, starting from the convolution of these equations with the interpolating kernel. The two most important aspects of the numerical implementation of these equations are (a) the appearance of fluid velocity and pressure obtained from the solution of the Riemann problem between each pair of particles, and (b) the absence of an artificial viscosity term. We carry out three different controlled hydrodynamical three-dimensional tests, namely the Sod shock tube, the development of Kelvin-Helmholtz instabilities in a shear-flow test and the 'blob' test describing the evolution of a cold cloud moving against a hot wind. The results of our tests confirm and extend in a number of aspects those recently obtained by Cha, Inutsuka & Nayakshin: (i) GSPH provides a much improved description of contact discontinuities, with respect to smoothed particle hydrodynamics (SPH), thus avoiding the appearance of spurious pressure forces; (ii) GSPH is able to follow the development of gas-dynamical instabilities, such as the Kevin-Helmholtz and the Rayleigh-Taylor ones; (iii) as a result, GSPH describes the development of curl structures in the shear-flow test and the dissolution of the cold cloud in the 'blob' test. Besides comparing the results of GSPH with those from standard SPH implementations, we also discuss in detail the effect on the performances of GSPH of changing different aspects of its implementation: choice of the number of neighbours, accuracy of the interpolation procedure to locate the interface between two fluid elements (particles) for the solution of the Riemann problem, order of the reconstruction for the assignment of variables at the interface, choice of the limiter to prevent oscillations of interpolated quantities in the solution of the Riemann Problem. The results of our tests demonstrate that GSPH is in fact a highly promising hydrodynamic scheme, also to be coupled to an N-body solver, for astrophysical and cosmological applications.

Giant graviton interactions and M2-branes ending on multiple M5-branes

NASA Astrophysics Data System (ADS)

Hirano, Shinji; Sato, Yuki

2018-05-01

We study splitting and joining interactions of giant gravitons with angular momenta N 1/2 ≪ J ≪ N in the type IIB string theory on AdS 5 × S 5 by describing them as instantons in the tiny graviton matrix model introduced by Sheikh-Jabbari. At large J the instanton equation can be mapped to the four-dimensional Laplace equation and the Coulomb potential for m point charges in an n-sheeted Riemann space corresponds to the m-to- n interaction process of giant gravitons. These instantons provide the holographic dual of correlators of all semi-heavy operators and the instanton amplitudes exactly agree with the pp-wave limit of Schur polynomial correlators in N = 4 SYM computed by Corley, Jevicki and Ramgoolam. By making a slight change of variables the same instanton equation is mathematically transformed into the Basu-Harvey equation which describes the system of M2-branes ending on M5-branes. As it turns out, the solutions to the sourceless Laplace equation on an n-sheeted Riemann space correspond to n M5-branes connected by M2-branes and we find general solutions representing M2-branes ending on multiple M5-branes. Among other solutions, the n = 3 case describes an M2-branes junction ending on three M5-branes. The effective theory on the moduli space of our solutions might shed light on the low energy effective theory of multiple M5-branes.

Dirac-Kähler particle in Riemann spherical space: boson interpretation

NASA Astrophysics Data System (ADS)

Ishkhanyan, A. M.; Florea, O.; Ovsiyuk, E. M.; Red'kov, V. M.

2015-11-01

In the context of the composite boson interpretation, we construct the exact general solution of the Dirac--K\\"ahler equation for the case of the spherical Riemann space of constant positive curvature, for which due to the geometry itself one may expect to have a discrete energy spectrum. In the case of the minimal value of the total angular momentum, $j=0$, the radial equations are reduced to second-order ordinary differential equations, which are straightforwardly solved in terms of the hypergeometric functions. For non-zero values of the total angular momentum, however, the radial equations are reduced to a pair of complicated fourth-order differential equations. Employing the factorization approach, we derive the general solution of these equations involving four independent fundamental solutions written in terms of combinations of the hypergeometric functions. The corresponding discrete energy spectrum is then determined via termination of the involved hypergeometric series, resulting in quasi-polynomial wave-functions. The constructed solutions lead to notable observations when compared with those for the ordinary Dirac particle. The energy spectrum for the Dirac-K\\"ahler particle in spherical space is much more complicated. Its structure substantially differs from that for the Dirac particle since it consists of two paralleled energy level series each of which is twofold degenerate. Besides, none of the two separate series coincides with the series for the Dirac particle. Thus, the Dirac--K\\"ahler field cannot be interpreted as a system of four Dirac fermions. Additional arguments supporting this conclusion are discussed.

Developing the Fundamental Theorem of Calculus. Applications of Calculus to Work, Area, and Distance Problems. [and] Atmospheric Pressure in Relation to Height and Temperature. Applications of Calculus to Atmospheric Pressure. [and] The Gradient and Some of Its Applications. Applications of Multivariate Calculus to Physics. [and] Kepler's Laws and the Inverse Square Law. Applications of Calculus to Physics. UMAP Units 323, 426, 431, 473.

ERIC Educational Resources Information Center

Lindstrom, Peter A.; And Others

This document consists of four units. The first of these views calculus applications to work, area, and distance problems. It is designed to help students gain experience in: 1) computing limits of Riemann sums; 2) computing definite integrals; and 3) solving elementary area, distance, and work problems by integration. The second module views…

Step-by-step integration for fractional operators

NASA Astrophysics Data System (ADS)

Colinas-Armijo, Natalia; Di Paola, Mario

2018-06-01

In this paper, an approach based on the definition of the Riemann-Liouville fractional operators is proposed in order to provide a different discretisation technique as alternative to the Grünwald-Letnikov operators. The proposed Riemann-Liouville discretisation consists of performing step-by-step integration based upon the discretisation of the function f(t). It has been shown that, as f(t) is discretised as stepwise or piecewise function, the Riemann-Liouville fractional integral and derivative are governing by operators very similar to the Grünwald-Letnikov operators. In order to show the accuracy and capabilities of the proposed Riemann-Liouville discretisation technique and the Grünwald-Letnikov discrete operators, both techniques have been applied to: unit step functions, exponential functions and sample functions of white noise.

A Few New 2+1-Dimensional Nonlinear Dynamics and the Representation of Riemann Curvature Tensors

NASA Astrophysics Data System (ADS)

Wang, Yan; Zhang, Yufeng; Zhang, Xiangzhi

2016-09-01

We first introduced a linear stationary equation with a quadratic operator in ∂x and ∂y, then a linear evolution equation is given by N-order polynomials of eigenfunctions. As applications, by taking N=2, we derived a (2+1)-dimensional generalized linear heat equation with two constant parameters associative with a symmetric space. When taking N=3, a pair of generalized Kadomtsev-Petviashvili equations with the same eigenvalues with the case of N=2 are generated. Similarly, a second-order flow associative with a homogeneous space is derived from the integrability condition of the two linear equations, which is a (2+1)-dimensional hyperbolic equation. When N=3, the third second flow associative with the homogeneous space is generated, which is a pair of new generalized Kadomtsev-Petviashvili equations. Finally, as an application of a Hermitian symmetric space, we established a pair of spectral problems to obtain a new (2+1)-dimensional generalized Schrödinger equation, which is expressed by the Riemann curvature tensors.

The Investigation of Ghost Fluid Method for Simulating the Compressible Two-Medium Flow

NASA Astrophysics Data System (ADS)

Lu, Hai Tian; Zhao, Ning; Wang, Donghong

2016-06-01

In this paper, we investigate the conservation error of the two-dimensional compressible two-medium flow simulated by the front tracking method. As the improved versions of the original ghost fluid method, the modified ghost fluid method and the real ghost fluid method are selected to define the interface boundary conditions, respectively, to show different effects on the conservation error. A Riemann problem is constructed along the normal direction of the interface in the front tracking method, with the goal of obtaining an efficient procedure to track the explicit sharp interface precisely. The corresponding Riemann solutions are also used directly in these improved ghost fluid methods. Extensive numerical examples including the sod tube and the shock-bubble interaction are tested to calculate the conservation error. It is found that these two ghost fluid methods have distinctive performances for different initial conditions of the flow field, and the related conclusions are made to suggest the best choice for the combination.

A fast Cauchy-Riemann solver. [differential equation solution for boundary conditions by finite difference approximation

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.

Methods for the computation of the multivalued Painlevé transcendents on their Riemann surfaces

NASA Astrophysics Data System (ADS)

Fasondini, Marco; Fornberg, Bengt; Weideman, J. A. C.

2017-09-01

We extend the numerical pole field solver (Fornberg and Weideman (2011) [12]) to enable the computation of the multivalued Painlevé transcendents, which are the solutions to the third, fifth and sixth Painlevé equations, on their Riemann surfaces. We display, for the first time, solutions to these equations on multiple Riemann sheets. We also provide numerical evidence for the existence of solutions to the sixth Painlevé equation that have pole-free sectors, known as tronquée solutions.

Dispersive shock waves in the Kadomtsev-Petviashvili and two dimensional Benjamin-Ono equations

NASA Astrophysics Data System (ADS)

Ablowitz, Mark J.; Demirci, Ali; Ma, Yi-Ping

2016-10-01

Dispersive shock waves (DSWs) in the Kadomtsev-Petviashvili (KP) equation and two dimensional Benjamin-Ono (2DBO) equation are considered using step like initial data along a parabolic front. Employing a parabolic similarity reduction exactly reduces the study of such DSWs in two space one time (2 + 1) dimensions to finding DSW solutions of (1 + 1) dimensional equations. With this ansatz, the KP and 2DBO equations can be exactly reduced to the cylindrical Korteweg-de Vries (cKdV) and cylindrical Benjamin-Ono (cBO) equations, respectively. Whitham modulation equations which describe DSW evolution in the cKdV and cBO equations are derived and Riemann type variables are introduced. DSWs obtained from the numerical solutions of the corresponding Whitham systems and direct numerical simulations of the cKdV and cBO equations are compared with very good agreement obtained. In turn, DSWs obtained from direct numerical simulations of the KP and 2DBO equations are compared with the cKdV and cBO equations, again with good agreement. It is concluded that the (2 + 1) DSW behavior along self similar parabolic fronts can be effectively described by the DSW solutions of the reduced (1 + 1) dimensional equations.

Inverse scattering transform for the time dependent Schrödinger equation with applications to the KPI equation

NASA Astrophysics Data System (ADS)

Zhou, Xin

1990-03-01

For the direct-inverse scattering transform of the time dependent Schrödinger equation, rigorous results are obtained based on an opertor-triangular-factorization approach. By viewing the equation as a first order operator equation, similar results as for the first order n x n matrix system are obtained. The nonlocal Riemann-Hilbert problem for inverse scattering is shown to have solution.

CAFE: A New Relativistic MHD Code

NASA Astrophysics Data System (ADS)

Lora-Clavijo, F. D.; Cruz-Osorio, A.; Guzmán, F. S.

2015-06-01

We introduce CAFE, a new independent code designed to solve the equations of relativistic ideal magnetohydrodynamics (RMHD) in three dimensions. We present the standard tests for an RMHD code and for the relativistic hydrodynamics regime because we have not reported them before. The tests include the one-dimensional Riemann problems related to blast waves, head-on collisions of streams, and states with transverse velocities, with and without magnetic field, which is aligned or transverse, constant or discontinuous across the initial discontinuity. Among the two-dimensional (2D) and 3D tests without magnetic field, we include the 2D Riemann problem, a one-dimensional shock tube along a diagonal, the high-speed Emery wind tunnel, the Kelvin-Helmholtz (KH) instability, a set of jets, and a 3D spherical blast wave, whereas in the presence of a magnetic field we show the magnetic rotor, the cylindrical explosion, a case of Kelvin-Helmholtz instability, and a 3D magnetic field advection loop. The code uses high-resolution shock-capturing methods, and we present the error analysis for a combination that uses the Harten, Lax, van Leer, and Einfeldt (HLLE) flux formula combined with a linear, piecewise parabolic method and fifth-order weighted essentially nonoscillatory reconstructors. We use the flux-constrained transport and the divergence cleaning methods to control the divergence-free magnetic field constraint.

Optimal control problem for linear fractional-order systems, described by equations with Hadamard-type derivative

NASA Astrophysics Data System (ADS)

Postnov, Sergey

2017-11-01

Two kinds of optimal control problem are investigated for linear time-invariant fractional-order systems with lumped parameters which dynamics described by equations with Hadamard-type derivative: the problem of control with minimal norm and the problem of control with minimal time at given restriction on control norm. The problem setting with nonlocal initial conditions studied. Admissible controls allowed to be the p-integrable functions (p > 1) at half-interval. The optimal control problem studied by moment method. The correctness and solvability conditions for the corresponding moment problem are derived. For several special cases the optimal control problems stated are solved analytically. Some analogies pointed for results obtained with the results which are known for integer-order systems and fractional-order systems describing by equations with Caputo- and Riemann-Liouville-type derivatives.

Loop Integrands for Scattering Amplitudes from the Riemann Sphere

NASA Astrophysics Data System (ADS)

Geyer, Yvonne; Mason, Lionel; Monteiro, Ricardo; Tourkine, Piotr

2015-09-01

The scattering equations on the Riemann sphere give rise to remarkable formulas for tree-level gauge theory and gravity amplitudes. Adamo, Casali, and Skinner conjectured a one-loop formula for supergravity amplitudes based on scattering equations on a torus. We use a residue theorem to transform this into a formula on the Riemann sphere. What emerges is a framework for loop integrands on the Riemann sphere that promises to have a wide application, based on off-shell scattering equations that depend on the loop momentum. We present new formulas, checked explicitly at low points, for supergravity and super-Yang-Mills amplitudes and for n -gon integrands at one loop. Finally, we show that the off-shell scattering equations naturally extend to arbitrary loop order, and we give a proposal for the all-loop integrands for supergravity and planar super-Yang-Mills theory.

The unstaggered extension to GFDL's FV3 dynamical core on the cubed-sphere

NASA Astrophysics Data System (ADS)

Chen, X.; Lin, S. J.; Harris, L.

2017-12-01

Finite-volume schemes have become popular for atmospheric transport since they provide intrinsic mass conservation to constituent species. Many CFD codes use unstaggered discretizations for finite volume methods with an approximate Riemann solver. However, this approach is inefficient for geophysical flows due to the complexity of the Riemann solver. We introduce a Low Mach number Approximate Riemann Solver (LMARS) simplified using assumptions appropriate for atmospheric flows: the wind speed is much slower than the sound speed, weak discontinuities, and locally uniform sound wave velocity. LMARS makes possible a Riemann-solver-based dynamical core comparable in computational efficiency to many current dynamical cores. We will present a 3D finite-volume dynamical core using LMARS in a cubed-sphere geometry with a vertically Lagrangian discretization. Results from standard idealized test cases will be discussed.

The High-Resolution Wave-Propagation Method Applied to Meso- and Micro-Scale Flows

NASA Technical Reports Server (NTRS)

Ahmad, Nashat N.; Proctor, Fred H.

2012-01-01

The high-resolution wave-propagation method for computing the nonhydrostatic atmospheric flows on meso- and micro-scales is described. The design and implementation of the Riemann solver used for computing the Godunov fluxes is discussed in detail. The method uses a flux-based wave decomposition in which the flux differences are written directly as the linear combination of the right eigenvectors of the hyperbolic system. The two advantages of the technique are: 1) the need for an explicit definition of the Roe matrix is eliminated and, 2) the inclusion of source term due to gravity does not result in discretization errors. The resulting flow solver is conservative and able to resolve regions of large gradients without introducing dispersion errors. The methodology is validated against exact analytical solutions and benchmark cases for non-hydrostatic atmospheric flows.

Exact solutions in 3D gravity with torsion

NASA Astrophysics Data System (ADS)

González, P. A.; Vásquez, Yerko

2011-08-01

We study the three-dimensional gravity with torsion given by the Mielke-Baekler (MB) model coupled to gravitational Chern-Simons term, and that possess electric charge described by Maxwell-Chern-Simons electrodynamics. We find and discuss this theory's charged black holes solutions and uncharged solutions. We find that for vanishing torsion our solutions by means of a coordinate transformation can be written as three-dimensional Chern-Simons black holes. We also discuss a special case of this theory, Topologically Massive Gravity (TMG) at chiral point, and we show that the logarithmic solution of TMG is also a solution of the MB model at a fixed point in the space of parameters. Furthermore, we show that our solutions generalize Gödel type solutions in a particular case. Also, we recover BTZ black hole in Riemann-Cartan spacetime for vanishing charge.

Initial-Boundary Value Problem for Two-Component Gerdjikov-Ivanov Equation with 3 × 3 Lax Pair on Half-Line

NASA Astrophysics Data System (ADS)

Zhu, Qiao-Zhen; Fan, En-Gui; Xu, Jian

2017-10-01

The Fokas unified method is used to analyze the initial-boundary value problem of two-component Gerdjikov-Ivanonv equation on the half-line. It is shown that the solution of the initial-boundary problem can be expressed in terms of the solution of a 3 × 3 Riemann-Hilbert problem. The Dirichlet to Neumann map is obtained through the global relation. Supported by grants from the National Science Foundation of China under Grant No. 11671095, National Science Foundation of China under Grant No. 11501365, Shanghai Sailing Program supported by Science and Technology Commission of Shanghai Municipality under Grant No 15YF1408100, and the Hujiang Foundation of China (B14005)

Uncertainty Propagation for Turbulent, Compressible Flow in a Quasi-1D Nozzle Using Stochastic Methods

NASA Technical Reports Server (NTRS)

Zang, Thomas A.; Mathelin, Lionel; Hussaini, M. Yousuff; Bataille, Francoise

2003-01-01

This paper describes a fully spectral, Polynomial Chaos method for the propagation of uncertainty in numerical simulations of compressible, turbulent flow, as well as a novel stochastic collocation algorithm for the same application. The stochastic collocation method is key to the efficient use of stochastic methods on problems with complex nonlinearities, such as those associated with the turbulence model equations in compressible flow and for CFD schemes requiring solution of a Riemann problem. Both methods are applied to compressible flow in a quasi-one-dimensional nozzle. The stochastic collocation method is roughly an order of magnitude faster than the fully Galerkin Polynomial Chaos method on the inviscid problem.

Relativistic Shock Waves in Viscous Gluon Matter

NASA Astrophysics Data System (ADS)

Bouras, I.; Molnár, E.; Niemi, H.; Xu, Z.; El, A.; Fochler, O.; Greiner, C.; Rischke, D. H.

2009-07-01

We solve the relativistic Riemann problem in viscous gluon matter employing a microscopic parton cascade. We demonstrate the transition from ideal to viscous shock waves by varying the shear viscosity to entropy density ratio η/s from zero to infinity. We show that an η/s ratio larger than 0.2 prevents the development of well-defined shock waves on time scales typical for ultrarelativistic heavy-ion collisions. Comparisons with viscous hydrodynamic calculations confirm our findings.

Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena

NASA Astrophysics Data System (ADS)

Atangana, Abdon; Gómez-Aguilar, J. F.

2018-04-01

To answer some issues raised about the concept of fractional differentiation and integration based on the exponential and Mittag-Leffler laws, we present, in this paper, fundamental differences between the power law, exponential decay, Mittag-Leffler law and their possible applications in nature. We demonstrate the failure of the semi-group principle in modeling real-world problems. We use natural phenomena to illustrate the importance of non-commutative and non-associative operators under which the Caputo-Fabrizio and Atangana-Baleanu fractional operators fall. We present statistical properties of generator for each fractional derivative, including Riemann-Liouville, Caputo-Fabrizio and Atangana-Baleanu ones. The Atangana-Baleanu and Caputo-Fabrizio fractional derivatives show crossover properties for the mean-square displacement, while the Riemann-Liouville is scale invariant. Their probability distributions are also a Gaussian to non-Gaussian crossover, with the difference that the Caputo Fabrizio kernel has a steady state between the transition. Only the Atangana-Baleanu kernel is a crossover for the waiting time distribution from stretched exponential to power law. A new criterion was suggested, namely the Atangana-Gómez fractional bracket, that helps describe the energy needed by a fractional derivative to characterize a 2-pletic manifold. Based on these properties, we classified fractional derivatives in three categories: weak, mild and strong fractional differential and integral operators. We presented some applications of fractional differential operators to describe real-world problems and we proved, with numerical simulations, that the Riemann-Liouville power-law derivative provides a description of real-world problems with much additional information, that can be seen as noise or error due to specific memory properties of its power-law kernel. The Caputo-Fabrizio derivative is less noisy while the Atangana-Baleanu fractional derivative provides an excellent description, due to its Mittag-Leffler memory, able to distinguish between dynamical systems taking place at different scales without steady state. The study suggests that the properties of associativity and commutativity or the semi-group principle are just irrelevant in fractional calculus. Properties of classical derivatives were established for the ordinary calculus with no memory effect and it is a failure of mathematical investigation to attempt to describe more complex natural phenomena using the same notions.

Divergence-free MHD on unstructured meshes using high order finite volume schemes based on multidimensional Riemann solvers

NASA Astrophysics Data System (ADS)

Balsara, Dinshaw S.; Dumbser, Michael

2015-10-01

Several advances have been reported in the recent literature on divergence-free finite volume schemes for Magnetohydrodynamics (MHD). Almost all of these advances are restricted to structured meshes. To retain full geometric versatility, however, it is also very important to make analogous advances in divergence-free schemes for MHD on unstructured meshes. Such schemes utilize a staggered Yee-type mesh, where all hydrodynamic quantities (mass, momentum and energy density) are cell-centered, while the magnetic fields are face-centered and the electric fields, which are so useful for the time update of the magnetic field, are centered at the edges. Three important advances are brought together in this paper in order to make it possible to have high order accurate finite volume schemes for the MHD equations on unstructured meshes. First, it is shown that a divergence-free WENO reconstruction of the magnetic field can be developed for unstructured meshes in two and three space dimensions using a classical cell-centered WENO algorithm, without the need to do a WENO reconstruction for the magnetic field on the faces. This is achieved via a novel constrained L2-projection operator that is used in each time step as a postprocessor of the cell-centered WENO reconstruction so that the magnetic field becomes locally and globally divergence free. Second, it is shown that recently-developed genuinely multidimensional Riemann solvers (called MuSIC Riemann solvers) can be used on unstructured meshes to obtain a multidimensionally upwinded representation of the electric field at each edge. Third, the above two innovations work well together with a high order accurate one-step ADER time stepping strategy, which requires the divergence-free nonlinear WENO reconstruction procedure to be carried out only once per time step. The resulting divergence-free ADER-WENO schemes with MuSIC Riemann solvers give us an efficient and easily-implemented strategy for divergence-free MHD on unstructured meshes. Several stringent two- and three-dimensional problems are shown to work well with the methods presented here.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Dumbser, Michael, E-mail: michael.dumbser@unitn.it; Balsara, Dinshaw S., E-mail: dbalsara@nd.edu

In this paper a new, simple and universal formulation of the HLLEM Riemann solver (RS) is proposed that works for general conservative and non-conservative systems of hyperbolic equations. For non-conservative PDE, a path-conservative formulation of the HLLEM RS is presented for the first time in this paper. The HLLEM Riemann solver is built on top of a novel and very robust path-conservative HLL method. It thus naturally inherits the positivity properties and the entropy enforcement of the underlying HLL scheme. However, with just the slight additional cost of evaluating eigenvectors and eigenvalues of intermediate characteristic fields, we can represent linearlymore » degenerate intermediate waves with a minimum of smearing. For conservative systems, our paper provides the easiest and most seamless path for taking a pre-existing HLL RS and quickly and effortlessly converting it to a RS that provides improved results, comparable with those of an HLLC, HLLD, Osher or Roe-type RS. This is done with minimal additional computational complexity, making our variant of the HLLEM RS also a very fast RS that can accurately represent linearly degenerate discontinuities. Our present HLLEM RS also transparently extends these advantages to non-conservative systems. For shallow water-type systems, the resulting method is proven to be well-balanced. Several test problems are presented for shallow water-type equations and two-phase flow models, as well as for gas dynamics with real equation of state, magnetohydrodynamics (MHD & RMHD), and nonlinear elasticity. Since our new formulation accommodates multiple intermediate waves and has a broader applicability than the original HLLEM method, it could alternatively be called the HLLI Riemann solver, where the “I” stands for the intermediate characteristic fields that can be accounted for. -- Highlights: •New simple and general path-conservative formulation of the HLLEM Riemann solver. •Application to general conservative and non-conservative hyperbolic systems. •Inclusion of sub-structure and resolution of intermediate characteristic fields. •Well-balanced for single- and two-layer shallow water equations and multi-phase flows. •Euler equations with real equation of state, MHD equations, nonlinear elasticity.« less

Toric Networks, Geometric R-Matrices and Generalized Discrete Toda Lattices

NASA Astrophysics Data System (ADS)

Inoue, Rei; Lam, Thomas; Pylyavskyy, Pavlo

2016-11-01

We use the combinatorics of toric networks and the double affine geometric R-matrix to define a three-parameter family of generalizations of the discrete Toda lattice. We construct the integrals of motion and a spectral map for this system. The family of commuting time evolutions arising from the action of the R-matrix is explicitly linearized on the Jacobian of the spectral curve. The solution to the initial value problem is constructed using Riemann theta functions.

Effect of interfacial stresses in an elastic body with a nanoinclusion

NASA Astrophysics Data System (ADS)

Vakaeva, Aleksandra B.; Grekov, Mikhail A.

2018-05-01

The 2-D problem of an infinite elastic solid with a nanoinclusion of a different from circular shape is solved. The interfacial stresses are acting at the interface. Contact of the inclusion with the matrix satisfies the ideal conditions of cohesion. The generalized Laplace - Young law defines conditions at the interface. To solve the problem, Gurtin - Murdoch surface elasticity model, Goursat - Kolosov complex potentials and the boundary perturbation method are used. The problem is reduced to the solution of two independent Riemann - Hilbert's boundary problems. For the circular inclusion, hypersingular integral equation in an unknown interfacial stress is derived. The algorithm of solving this equation is constructed. The influence of the interfacial stress and the dimension of the circular inclusion on the stress distribution and stress concentration at the interface are analyzed.

On the formation of Friedlander waves in a compressed-gas-driven shock tube

PubMed Central

Tasissa, Abiy F.; Hautefeuille, Martin; Fitek, John H.; Radovitzky, Raúl A.

2016-01-01

Compressed-gas-driven shock tubes have become popular as a laboratory-scale replacement for field blast tests. The well-known initial structure of the Riemann problem eventually evolves into a shock structure thought to resemble a Friedlander wave, although this remains to be demonstrated theoretically. In this paper, we develop a semi-analytical model to predict the key characteristics of pseudo blast waves forming in a shock tube: location where the wave first forms, peak over-pressure, decay time and impulse. The approach is based on combining the solutions of the two different types of wave interactions that arise in the shock tube after the family of rarefaction waves in the Riemann solution interacts with the closed end of the tube. The results of the analytical model are verified against numerical simulations obtained with a finite volume method. The model furnishes a rational approach to relate shock tube parameters to desired blast wave characteristics, and thus constitutes a useful tool for the design of shock tubes for blast testing. PMID:27118888

Riemann-Hypothesis Millennium-Problem(MP) Physics Proof via CATEGORY-SEMANTICS(C-S)/F=C Aristotle SQUARE-of-OPPOSITION(SoO) DEduction-LOGIC DichotomY

NASA Astrophysics Data System (ADS)

Baez, J.; Lapidaryus, M.; Siegel, Edward Carl-Ludwig

2011-03-01

Riemann-hypothesis physics-proof combines: Siegel-Antonoff-Smith[AMS Joint Mtg.(2002)-Abs.973-03-126] digits on-average statistics HIll[Am. J. Math 123, 3, 887(1996)] logarithm-function's (1,0)-fixed-point base=units=scale-invariance proven Newcomb[Am. J. Math. 4, 39(1881)]-Weyl[Goett. Nachr.(1914); Math. Ann. 7, 313(1916)]-Benford[Proc. Am. Phil. Soc. 78, 4, 51(1938)]-law [Kac, Math. of Stat.-Reasoning(1955); Raimi, Sci. Am. 221, 109(1969)] algebraic-inversion to ONLY Bose-Einstein quantum-statistics(BEQS) with digit d = 0 gapFUL Bose-Einstein Condensation(BEC) insight that digits are quanta are bosons were always digits, via Siegel-Baez category-semantics tabular list-format matrix truth-table analytics in Plato-Aristotle classic "square-of-opposition" : FUZZYICS=CATEGORYICS/Category-Semantics, with Goodkind Bose-Einstein condensation(BEC) ABOVE ground-state with/and Rayleigh(cut-limit of "short-cut method";1870)-Polya(1922)-"Anderson"(1958) localization [Doyle and Snell, Random-Walks and Electrical-Networks, MAA(1981)-p.99-100!!!].

Numerical 3+1 General Relativistic Magnetohydrodynamics: A Local Characteristic Approach

NASA Astrophysics Data System (ADS)

Antón, Luis; Zanotti, Olindo; Miralles, Juan A.; Martí, José M.; Ibáñez, José M.; Font, José A.; Pons, José A.

2006-01-01

We present a general procedure to solve numerically the general relativistic magnetohydrodynamics (GRMHD) equations within the framework of the 3+1 formalism. The work reported here extends our previous investigation in general relativistic hydrodynamics (Banyuls et al. 1997) where magnetic fields were not considered. The GRMHD equations are written in conservative form to exploit their hyperbolic character in the solution procedure. All theoretical ingredients necessary to build up high-resolution shock-capturing schemes based on the solution of local Riemann problems (i.e., Godunov-type schemes) are described. In particular, we use a renormalized set of regular eigenvectors of the flux Jacobians of the relativistic MHD equations. In addition, the paper describes a procedure based on the equivalence principle of general relativity that allows the use of Riemann solvers designed for special relativistic MHD in GRMHD. Our formulation and numerical methodology are assessed by performing various test simulations recently considered by different authors. These include magnetized shock tubes, spherical accretion onto a Schwarzschild black hole, equatorial accretion onto a Kerr black hole, and magnetized thick disks accreting onto a black hole and subject to the magnetorotational instability.

Central charge from adiabatic transport of cusp singularities in the quantum Hall effect

NASA Astrophysics Data System (ADS)

Can, Tankut

2017-04-01

We study quantum Hall (QH) states on a punctured Riemann sphere. We compute the Berry curvature under adiabatic motion in the moduli space in the large N limit. The Berry curvature is shown to be finite in the large N limit and controlled by the conformal dimension of the cusp singularity, a local property of the mean density. Utilizing exact sum rules obtained from a Ward identity, we show that for the Laughlin wave function, the dimension of a cusp singularity is given by the central charge, a robust geometric response coefficient in the QHE. Thus, adiabatic transport of curvature singularities can be used to determine the central charge of QH states. We also consider the effects of threaded fluxes and spin-deformed wave functions. Finally, we give a closed expression for all moments of the mean density in the integer QH state on a punctured disk.

Excitation basis for (3+1)d topological phases

NASA Astrophysics Data System (ADS)

Delcamp, Clement

2017-12-01

We consider an exactly solvable model in 3+1 dimensions, based on a finite group, which is a natural generalization of Kitaev's quantum double model. The corresponding lattice Hamiltonian yields excitations located at torus-boundaries. By cutting open the three-torus, we obtain a manifold bounded by two tori which supports states satisfying a higher-dimensional version of Ocneanu's tube algebra. This defines an algebraic structure extending the Drinfel'd double. Its irreducible representations, labeled by two fluxes and one charge, characterize the torus-excitations. The tensor product of such representations is introduced in order to construct a basis for (3+1)d gauge models which relies upon the fusion of the defect excitations. This basis is defined on manifolds of the form Σ × S_1 , with Σ a two-dimensional Riemann surface. As such, our construction is closely related to dimensional reduction from (3+1)d to (2+1)d topological orders.

Interaction of a conductive crack and of an electrode at a piezoelectric bimaterial interface

NASA Astrophysics Data System (ADS)

Onopriienko, Oleg; Loboda, Volodymyr; Sheveleva, Alla; Lapusta, Yuri

2018-06-01

The interaction of a conductive crack and an electrode at a piezoelectric bi-material interface is studied. The bimaterial is subjected to an in-plane electrical field parallel to the interface and an anti-plane mechanical loading. The problem is formulated and reduced, via the application of sectionally analytic vector functions, to a combined Dirichlet-Riemann boundary value problem. Simple analytical expressions for the stress, the electric field, and their intensity factors as well as for the crack faces' displacement jump are derived. Our numerical results illustrate the proposed approach and permit to draw some conclusions on the crack-electrode interaction.

An unstaggered central scheme on nonuniform grids for the simulation of a compressible two-phase flow model

DOE Office of Scientific and Technical Information (OSTI.GOV)

Touma, Rony; Zeidan, Dia

In this paper we extend a central finite volume method on nonuniform grids to the case of drift-flux two-phase flow problems. The numerical base scheme is an unstaggered, non oscillatory, second-order accurate finite volume scheme that evolves a piecewise linear numerical solution on a single grid and uses dual cells intermediately while updating the numerical solution to avoid the resolution of the Riemann problems arising at the cell interfaces. We then apply the numerical scheme and solve a classical drift-flux problem. The obtained results are in good agreement with corresponding ones appearing in the recent literature, thus confirming the potentialmore » of the proposed scheme.« less

Hidden algebra method (quasi-exact-solvability in quantum mechanics)

DOE Office of Scientific and Technical Information (OSTI.GOV)

Turbiner, Alexander; Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, Apartado, Postal 70-543, 04510 Mexico, D. F.

1996-02-20

A general introduction to quasi-exactly-solvable problems of quantum mechanics is presented. Main attention is given to multidimensional quasi-exactly-solvable and exactly-solvable Schroedinger operators. Exact-solvability of the Calogero and Sutherland N-body problems ass ociated with an existence of the hidden algebra slN is discussed extensively.

The solution of non-linear hyperbolic equation systems by the finite element method

NASA Technical Reports Server (NTRS)

Loehner, R.; Morgan, K.; Zienkiewicz, O. C.

1984-01-01

A finite-element method for the solution of nonlinear hyperbolic systems of equations, such as those encountered in non-self-adjoint problems of transient phenomena in convection-diffusion or in the mixed representation of wave problems, is developed and demonstrated. The problem is rewritten in moving coordinates and reinterpolated to the original mesh by a Taylor expansion prior to a standard Galerkin spatial discretization, and it is shown that this procedure is equivalent to the time-discretization approach of Donea (1984). Numerical results for sample problems are presented graphically, including such shallow-water problems as the breaking of a dam, the shoaling of a wave, and the outflow of a river; compressible flows such as the isothermal flow in a nozzle and the Riemann shock-tube problem; and the two-dimensional scalar-advection, nonlinear-shallow-water, and Euler equations.

Adaptively Refined Euler and Navier-Stokes Solutions with a Cartesian-Cell Based Scheme

NASA Technical Reports Server (NTRS)

Coirier, William J.; Powell, Kenneth G.

1995-01-01

A Cartesian-cell based scheme with adaptive mesh refinement for solving the Euler and Navier-Stokes equations in two dimensions has been developed and tested. Grids about geometrically complicated bodies were generated automatically, by recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, N-sided 'cut' cells were created using polygon-clipping algorithms. The grid was stored in a binary-tree data structure which provided a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive mesh refinement. The Euler and Navier-Stokes equations were solved on the resulting grids using an upwind, finite-volume formulation. The inviscid fluxes were found in an upwinded manner using a linear reconstruction of the cell primitives, providing the input states to an approximate Riemann solver. The viscous fluxes were formed using a Green-Gauss type of reconstruction upon a co-volume surrounding the cell interface. Data at the vertices of this co-volume were found in a linearly K-exact manner, which ensured linear K-exactness of the gradients. Adaptively-refined solutions for the inviscid flow about a four-element airfoil (test case 3) were compared to theory. Laminar, adaptively-refined solutions were compared to accepted computational, experimental and theoretical results.

Black holes in vector-tensor theories

DOE Office of Scientific and Technical Information (OSTI.GOV)

Heisenberg, Lavinia; Kase, Ryotaro; Tsujikawa, Shinji

We study static and spherically symmetric black hole (BH) solutions in second-order generalized Proca theories with nonminimal vector field derivative couplings to the Ricci scalar, the Einstein tensor, and the double dual Riemann tensor. We find concrete Lagrangians which give rise to exact BH solutions by imposing two conditions of the two identical metric components and the constant norm of the vector field. These exact solutions are described by either Reissner-Nordström (RN), stealth Schwarzschild, or extremal RN solutions with a non-trivial longitudinal mode of the vector field. We then numerically construct BH solutions without imposing these conditions. For cubic andmore » quartic Lagrangians with power-law couplings which encompass vector Galileons as the specific cases, we show the existence of BH solutions with the difference between two non-trivial metric components. The quintic-order power-law couplings do not give rise to non-trivial BH solutions regular throughout the horizon exterior. The sixth-order and intrinsic vector-mode couplings can lead to BH solutions with a secondary hair. For all the solutions, the vector field is regular at least at the future or past horizon. The deviation from General Relativity induced by the Proca hair can be potentially tested by future measurements of gravitational waves in the nonlinear regime of gravity.« less

Colliding holes in Riemann surfaces and quantum cluster algebras

NASA Astrophysics Data System (ADS)

Chekhov, Leonid; Mazzocco, Marta

2018-01-01

In this paper, we describe a new type of surgery for non-compact Riemann surfaces that naturally appears when colliding two holes or two sides of the same hole in an orientable Riemann surface with boundary (and possibly orbifold points). As a result of this surgery, bordered cusps appear on the boundary components of the Riemann surface. In Poincaré uniformization, these bordered cusps correspond to ideal triangles in the fundamental domain. We introduce the notion of bordered cusped Teichmüller space and endow it with a Poisson structure, quantization of which is achieved with a canonical quantum ordering. We give a complete combinatorial description of the bordered cusped Teichmüller space by introducing the notion of maximal cusped lamination, a lamination consisting of geodesic arcs between bordered cusps and closed geodesics homotopic to the boundaries such that it triangulates the Riemann surface. We show that each bordered cusp carries a natural decoration, i.e. a choice of a horocycle, so that the lengths of the arcs in the maximal cusped lamination are defined as λ-lengths in Thurston-Penner terminology. We compute the Goldman bracket explicitly in terms of these λ-lengths and show that the groupoid of flip morphisms acts as a generalized cluster algebra mutation. From the physical point of view, our construction provides an explicit coordinatization of moduli spaces of open/closed string worldsheets and their quantization.

The Riemann-Lanczos equations in general relativity and their integrability

NASA Astrophysics Data System (ADS)

Dolan, P.; Gerber, A.

2008-06-01

The aim of this paper is to examine the Riemann-Lanczos equations and how they can be made integrable. They consist of a system of linear first-order partial differential equations that arise in general relativity, whereby the Riemann curvature tensor is generated by an unknown third-order tensor potential field called the Lanczos tensor. Our approach is based on the theory of jet bundles, where all field variables and all their partial derivatives of all relevant orders are treated as independent variables alongside the local manifold coordinates (xa) on the given space-time manifold M. This approach is adopted in (a) Cartan's method of exterior differential systems, (b) Vessiot's dual method using vector field systems, and (c) the Janet-Riquier theory of systems of partial differential equations. All three methods allow for the most general situations under which integrability conditions can be found. They give equivalent results, namely, that involutivity is always achieved at all generic points of the jet manifold M after a finite number of prolongations. Two alternative methods that appear in the general relativity literature to find integrability conditions for the Riemann-Lanczos equations generate new partial differential equations for the Lanczos potential that introduce a source term, which is nonlinear in the components of the Riemann tensor. We show that such sources do not occur when either of method (a), (b), or (c) are used.

Development of relativistic shock waves in viscous gluon matter

NASA Astrophysics Data System (ADS)

Bouras, I.; Molnár, E.; Niemi, H.; Xu, Z.; El, A.; Fochler, O.; Greiner, C.; Rischke, D. H.

2009-11-01

To investigate the formation and the propagation of relativistic shock waves in viscous gluon matter we solve the relativistic Riemann problem using a microscopic parton cascade. We demonstrate the transition from ideal to viscous shock waves by varying the shear viscosity to entropy density ratio η/s. We show that an η/s ratio larger than 0.2 prevents the development of well-defined shock waves on time scales typical for ultrarelativistic heavy-ion collisions. These findings are confirmed by viscous hydrodynamic calculations.

An Expansion Formula with Higher-Order Derivatives for Fractional Operators of Variable Order

PubMed Central

Almeida, Ricardo

2013-01-01

We obtain approximation formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is given. The efficiency of the approximation method is illustrated with examples. As applications, we show how the obtained results are useful to solve differential equations, and problems of the calculus of variations that depend on fractional derivatives of Marchaud type. PMID:24319382

Eigenmode Analysis of Boundary Conditions for One-Dimensional Preconditioned Euler Equations

NASA Technical Reports Server (NTRS)

Darmofal, David L.

1998-01-01

An analysis of the effect of local preconditioning on boundary conditions for the subsonic, one-dimensional Euler equations is presented. Decay rates for the eigenmodes of the initial boundary value problem are determined for different boundary conditions. Riemann invariant boundary conditions based on the unpreconditioned Euler equations are shown to be reflective with preconditioning, and, at low Mach numbers, disturbances do not decay. Other boundary conditions are investigated which are non-reflective with preconditioning and numerical results are presented confirming the analysis.

Fractional Number Operator and Associated Fractional Diffusion Equations

NASA Astrophysics Data System (ADS)

Rguigui, Hafedh

2018-03-01

In this paper, we study the fractional number operator as an analog of the finite-dimensional fractional Laplacian. An important relation with the Ornstein-Uhlenbeck process is given. Using a semigroup approach, the solution of the Cauchy problem associated to the fractional number operator is presented. By means of the Mittag-Leffler function and the Laplace transform, we give the solution of the Caputo time fractional diffusion equation and Riemann-Liouville time fractional diffusion equation in infinite dimensions associated to the fractional number operator.

High-order centered difference methods with sharp shock resolution

NASA Technical Reports Server (NTRS)

Gustafsson, Bertil; Olsson, Pelle

1994-01-01

In this paper we consider high-order centered finite difference approximations of hyperbolic conservation laws. We propose different ways of adding artificial viscosity to obtain sharp shock resolution. For the Riemann problem we give simple explicit formulas for obtaining stationary one and two-point shocks. This can be done for any order of accuracy. It is shown that the addition of artificial viscosity is equivalent to ensuring the Lax k-shock condition. We also show numerical experiments that verify the theoretical results.

Multiphase fluid-solid coupled analysis of shock-bubble-stone interaction in shockwave lithotripsy.

PubMed

Wang, Kevin G

2017-10-01

A novel multiphase fluid-solid-coupled computational framework is applied to investigate the interaction of a kidney stone immersed in liquid with a lithotripsy shock wave (LSW) and a gas bubble near the stone. The main objective is to elucidate the effects of a bubble in the shock path to the elastic and fracture behaviors of the stone. The computational framework couples a finite volume 2-phase computational fluid dynamics solver with a finite element computational solid dynamics solver. The surface of the stone is represented as a dynamic embedded boundary in the computational fluid dynamics solver. The evolution of the bubble surface is captured by solving the level set equation. The interface conditions at the surfaces of the stone and the bubble are enforced through the construction and solution of local fluid-solid and 2-fluid Riemann problems. This computational framework is first verified for 3 example problems including a 1D multimaterial Riemann problem, a 3D shock-stone interaction problem, and a 3D shock-bubble interaction problem. Next, a series of shock-bubble-stone-coupled simulations are presented. This study suggests that the dynamic response of a bubble to LSW varies dramatically depending on its initial size. Bubbles with an initial radius smaller than a threshold collapse within 1 μs after the passage of LSW, whereas larger bubbles do not. For a typical LSW generated by an electrohydraulic lithotripter (p max  = 35.0MPa, p min  =- 10.1MPa), this threshold is approximately 0.12mm. Moreover, this study suggests that a noncollapsing bubble imposes a negative effect on stone fracture as it shields part of the LSW from the stone. On the other hand, a collapsing bubble may promote fracture on the proximal surface of the stone, yet hinder fracture from stone interior. Copyright © 2016 John Wiley & Sons, Ltd.

Simulating cosmic ray physics on a moving mesh

NASA Astrophysics Data System (ADS)

Pfrommer, C.; Pakmor, R.; Schaal, K.; Simpson, C. M.; Springel, V.

2017-03-01

We discuss new methods to integrate the cosmic ray (CR) evolution equations coupled to magnetohydrodynamics on an unstructured moving mesh, as realized in the massively parallel AREPO code for cosmological simulations. We account for diffusive shock acceleration of CRs at resolved shocks and at supernova remnants in the interstellar medium (ISM) and follow the advective CR transport within the magnetized plasma, as well as anisotropic diffusive transport of CRs along the local magnetic field. CR losses are included in terms of Coulomb and hadronic interactions with the thermal plasma. We demonstrate the accuracy of our formalism for CR acceleration at shocks through simulations of plane-parallel shock tubes that are compared to newly derived exact solutions of the Riemann shock-tube problem with CR acceleration. We find that the increased compressibility of the post-shock plasma due to the produced CRs decreases the shock speed. However, CR acceleration at spherically expanding blast waves does not significantly break the self-similarity of the Sedov-Taylor solution; the resulting modifications can be approximated by a suitably adjusted, but constant adiabatic index. In first applications of the new CR formalism to simulations of isolated galaxies and cosmic structure formation, we find that CRs add an important pressure component to the ISM that increases the vertical scaleheight of disc galaxies and thus reduces the star formation rate. Strong external structure formation shocks inject CRs into the gas, but the relative pressure of this component decreases towards halo centres as adiabatic compression favours the thermal over the CR pressure.

On the use of kinetic energy preserving DG-schemes for large eddy simulation

NASA Astrophysics Data System (ADS)

Flad, David; Gassner, Gregor

2017-12-01

Recently, element based high order methods such as Discontinuous Galerkin (DG) methods and the closely related flux reconstruction (FR) schemes have become popular for compressible large eddy simulation (LES). Element based high order methods with Riemann solver based interface numerical flux functions offer an interesting dispersion dissipation behavior for multi-scale problems: dispersion errors are very low for a broad range of scales, while dissipation errors are very low for well resolved scales and are very high for scales close to the Nyquist cutoff. In some sense, the inherent numerical dissipation caused by the interface Riemann solver acts as a filter of high frequency solution components. This observation motivates the trend that element based high order methods with Riemann solvers are used without an explicit LES model added. Only the high frequency type inherent dissipation caused by the Riemann solver at the element interfaces is used to account for the missing sub-grid scale dissipation. Due to under-resolution of vortical dominated structures typical for LES type setups, element based high order methods suffer from stability issues caused by aliasing errors of the non-linear flux terms. A very common strategy to fight these aliasing issues (and instabilities) is so-called polynomial de-aliasing, where interpolation is exchanged with projection based on an increased number of quadrature points. In this paper, we start with this common no-model or implicit LES (iLES) DG approach with polynomial de-aliasing and Riemann solver dissipation and review its capabilities and limitations. We find that the strategy gives excellent results, but only when the resolution is such, that about 40% of the dissipation is resolved. For more realistic, coarser resolutions used in classical LES e.g. of industrial applications, the iLES DG strategy becomes quite inaccurate. We show that there is no obvious fix to this strategy, as adding for instance a sub-grid-scale models on top doesn't change much or in worst case decreases the fidelity even more. Finally, the core of this work is a novel LES strategy based on split form DG methods that are kinetic energy preserving. The scheme offers excellent stability with full control over the amount and shape of the added artificial dissipation. This premise is the main idea of the work and we will assess the LES capabilities of the novel split form DG approach when applied to shock-free, moderate Mach number turbulence. We will demonstrate that the novel DG LES strategy offers similar accuracy as the iLES methodology for well resolved cases, but strongly increases fidelity in case of more realistic coarse resolutions.

Hidden algebra method (quasi-exact-solvability in quantum mechanics)

DOE Office of Scientific and Technical Information (OSTI.GOV)

Turbiner, A.

1996-02-01

A general introduction to quasi-exactly-solvable problems of quantum mechanics is presented. Main attention is given to multidimensional quasi-exactly-solvable and exactly-solvable Schroedinger operators. Exact-solvability of the Calogero and Sutherland {ital N}-body problems ass ociated with an existence of the hidden algebra {ital sl}{sub {ital N}} is discussed extensively. {copyright} {ital 1996 American Institute of Physics.}

The CRONOS Code for Astrophysical Magnetohydrodynamics

NASA Astrophysics Data System (ADS)

Kissmann, R.; Kleimann, J.; Krebl, B.; Wiengarten, T.

2018-06-01

We describe the magnetohydrodynamics (MHD) code CRONOS, which has been used in astrophysics and space-physics studies in recent years. CRONOS has been designed to be easily adaptable to the problem in hand, where the user can expand or exchange core modules or add new functionality to the code. This modularity comes about through its implementation using a C++ class structure. The core components of the code include solvers for both hydrodynamical (HD) and MHD problems. These problems are solved on different rectangular grids, which currently support Cartesian, spherical, and cylindrical coordinates. CRONOS uses a finite-volume description with different approximate Riemann solvers that can be chosen at runtime. Here, we describe the implementation of the code with a view toward its ongoing development. We illustrate the code’s potential through several (M)HD test problems and some astrophysical applications.

A Walsh Function Module Users' Manual

NASA Technical Reports Server (NTRS)

Gnoffo, Peter A.

2014-01-01

The solution of partial differential equations (PDEs) with Walsh functions offers new opportunities to simulate many challenging problems in mathematical physics. The approach was developed to better simulate hypersonic flows with shocks on unstructured grids. It is unique in that integrals and derivatives are computed using simple matrix multiplication of series representations of functions without the need for divided differences. The product of any two Walsh functions is another Walsh function - a feature that radically changes an algorithm for solving PDEs. A FORTRAN module for supporting Walsh function simulations is documented. A FORTRAN code is also documented with options for solving time-dependent problems: an advection equation, a Burgers equation, and a Riemann problem. The sample problems demonstrate the usage of the Walsh function module including such features as operator overloading, Fast Walsh Transforms in multi-dimensions, and a Fast Walsh reciprocal.

On Lovelock analogs of the Riemann tensor

NASA Astrophysics Data System (ADS)

Camanho, Xián O.; Dadhich, Naresh

2016-03-01

It is possible to define an analog of the Riemann tensor for Nth order Lovelock gravity, its characterizing property being that the trace of its Bianchi derivative yields the corresponding analog of the Einstein tensor. Interestingly there exist two parallel but distinct such analogs and the main purpose of this note is to reconcile both formulations. In addition we will introduce a simple tensor identity and use it to show that any pure Lovelock vacuum in odd d=2N+1 dimensions is Lovelock flat, i.e. any vacuum solution of the theory has vanishing Lovelock-Riemann tensor. Further, in the presence of cosmological constant it is the Lovelock-Weyl tensor that vanishes.

Numerical Integration with GeoGebra in High School

ERIC Educational Resources Information Center

Herceg, Dorde; Herceg, Dragoslav

2010-01-01

The concept of definite integral is almost always introduced as the Riemann integral, which is defined in terms of the Riemann sum, and its geometric interpretation. This definition is hard to understand for high school students. With the aid of mathematical software for visualisation and computation of approximate integrals, the notion of…

Wilson loops on Riemann surfaces, Liouville theory and covariantization of the conformal group

NASA Astrophysics Data System (ADS)

Matone, Marco; Pasti, Paolo

2015-06-01

The covariantization procedure is usually referred to the translation operator, that is the derivative. Here we introduce a general method to covariantize arbitrary differential operators, such as the ones defining the fundamental group of a given manifold. We focus on the differential operators representing the sl2(ℝ) generators, which in turn, generate, by exponentiation, the two-dimensional conformal transformations. A key point of our construction is the recent result on the closed forms of the Baker-Campbell-Hausdorff formula. In particular, our covariantization receipt is quite general. This has a deep consequence since it means that the covariantization of the conformal group is always definite. Our covariantization receipt is quite general and apply in general situations, including AdS/CFT. Here we focus on the projective unitary representations of the fundamental group of a Riemann surface, which may include elliptic points and punctures, introduced in the framework of noncommutative Riemann surfaces. It turns out that the covariantized conformal operators are built in terms of Wilson loops around Poincaré geodesics, implying a deep relationship between gauge theories on Riemann surfaces and Liouville theory.

An approximate Riemann solver for real gas parabolized Navier-Stokes equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Urbano, Annafederica, E-mail: annafederica.urbano@uniroma1.it; Nasuti, Francesco, E-mail: francesco.nasuti@uniroma1.it

2013-01-15

Under specific assumptions, parabolized Navier-Stokes equations are a suitable mean to study channel flows. A special case is that of high pressure flow of real gases in cooling channels where large crosswise gradients of thermophysical properties occur. To solve the parabolized Navier-Stokes equations by a space marching approach, the hyperbolicity of the system of governing equations is obtained, even for very low Mach number flow, by recasting equations such that the streamwise pressure gradient is considered as a source term. For this system of equations an approximate Roe's Riemann solver is developed as the core of a Godunov type finitemore » volume algorithm. The properties of the approximated Riemann solver, which is a modification of Roe's Riemann solver for the parabolized Navier-Stokes equations, are presented and discussed with emphasis given to its original features introduced to handle fluids governed by a generic real gas EoS. Sample solutions are obtained for low Mach number high compressible flows of transcritical methane, heated in straight long channels, to prove the solver ability to describe flows dominated by complex thermodynamic phenomena.« less

Special discontinuities in nonlinearly elastic media

NASA Astrophysics Data System (ADS)

Chugainova, A. P.

2017-06-01

Solutions of a nonlinear hyperbolic system of equations describing weakly nonlinear quasitransverse waves in a weakly anisotropic elastic medium are studied. The influence of small-scale processes of dissipation and dispersion is investigated. The small-scale processes determine the structure of discontinuities (shocks) and a set of discontinuities with a stationary structure. Among the discontinuities with a stationary structure, there are special ones that, in addition to relations following from conservation laws, satisfy additional relations required for the existence of their structure. In the phase plane, the structure of such discontinuities is represented by an integral curve joining two saddles. Special discontinuities lead to nonunique self-similar solutions of the Riemann problem. Asymptotics of non-self-similar problems for equations with dissipation and dispersion are found numerically. These asymptotics correspond to self-similar solutions of the problems.

The Osher scheme for non-equilibrium reacting flows

NASA Technical Reports Server (NTRS)

Suresh, Ambady; Liou, Meng-Sing

1992-01-01

An extension of the Osher upwind scheme to nonequilibrium reacting flows is presented. Owing to the presence of source terms, the Riemann problem is no longer self-similar and therefore its approximate solution becomes tedious. With simplicity in mind, a linearized approach which avoids an iterative solution is used to define the intermediate states and sonic points. The source terms are treated explicitly. Numerical computations are presented to demonstrate the feasibility, efficiency and accuracy of the proposed method. The test problems include a ZND (Zeldovich-Neumann-Doring) detonation problem for which spurious numerical solutions which propagate at mesh speed have been observed on coarse grids. With the present method, a change of limiter causes the solution to change from the physically correct CJ detonation solution to the spurious weak detonation solution.

AdS5 solutions from M5-branes on Riemann surface and D6-branes sources

DOE PAGES

Bah, Ibrahima

2015-09-24

Here, we describe the gravity duals of four-dimensional N = 1 superconformal field theories obtained by wrapping M5-branes on a punctured Riemann surface. The internal geometry, normal to the AdS 5 factor, generically preserves two U(1)s, with generators (J +, J –), that are fibered over the Riemann surface. The metric is governed by a single potential that satisfies a version of the Monge-Ampère equation. The spectrum of N = 1 punctures is given by the set of supersymmetric sources of the potential that are localized on the Riemann surface and lead to regular metrics near a puncture. We usemore » this system to study a class of punctures where the geometry near the sources corresponds to M-theory description of D6-branes. These carry a natural (p, q) label associated to the circle dual to the killing vector pJ + + qJ – which shrinks near the source. In the generic case the world volume of the D6-branes is AdS 5 × S 2 and they locally preserve N = 2 supersymmetry. When p = –q, the shrinking circle is dual to a flavor U(1). The metric in this case is non-degenerate only when there are co-dimension one sources obtained by smearing M5-branes that wrap the AdS 5 factor and the circle dual the superconformal R-symmetry. The D6-branes are extended along the AdS 5 and on cups that end on the co-dimension one branes. In the special case when the shrinking circle is dual to the R-symmetry, the D6-branes are extended along the AdS 5 and wrap an auxiliary Riemann surface with an arbitrary genus. When the Riemann surface is compact with constant curvature, the system is governed by a Monge-Ampère equation.« less

On the solutions of fractional order of evolution equations

NASA Astrophysics Data System (ADS)

Morales-Delgado, V. F.; Taneco-Hernández, M. A.; Gómez-Aguilar, J. F.

2017-01-01

In this paper we present a discussion of generalized Cauchy problems in a diffusion wave process, we consider bi-fractional-order evolution equations in the Riemann-Liouville, Liouville-Caputo, and Caputo-Fabrizio sense. Through Fourier transforms and Laplace transform we derive closed-form solutions to the Cauchy problems mentioned above. Similarly, we establish fundamental solutions. Finally, we give an application of the above results to the determination of decompositions of Dirac type for bi-fractional-order equations and write a formula for the moments for the fractional vibration of a beam equation. This type of decomposition allows us to speak of internal degrees of freedom in the vibration of a beam equation.

RELATIVISTIC MAGNETOHYDRODYNAMICS: RENORMALIZED EIGENVECTORS AND FULL WAVE DECOMPOSITION RIEMANN SOLVER

DOE Office of Scientific and Technical Information (OSTI.GOV)

Anton, Luis; MartI, Jose M; Ibanez, Jose M

2010-05-01

We obtain renormalized sets of right and left eigenvectors of the flux vector Jacobians of the relativistic MHD equations, which are regular and span a complete basis in any physical state including degenerate ones. The renormalization procedure relies on the characterization of the degeneracy types in terms of the normal and tangential components of the magnetic field to the wave front in the fluid rest frame. Proper expressions of the renormalized eigenvectors in conserved variables are obtained through the corresponding matrix transformations. Our work completes previous analysis that present different sets of right eigenvectors for non-degenerate and degenerate states, andmore » can be seen as a relativistic generalization of earlier work performed in classical MHD. Based on the full wave decomposition (FWD) provided by the renormalized set of eigenvectors in conserved variables, we have also developed a linearized (Roe-type) Riemann solver. Extensive testing against one- and two-dimensional standard numerical problems allows us to conclude that our solver is very robust. When compared with a family of simpler solvers that avoid the knowledge of the full characteristic structure of the equations in the computation of the numerical fluxes, our solver turns out to be less diffusive than HLL and HLLC, and comparable in accuracy to the HLLD solver. The amount of operations needed by the FWD solver makes it less efficient computationally than those of the HLL family in one-dimensional problems. However, its relative efficiency increases in multidimensional simulations.« less

Computational strategies for the Riemann zeta function

NASA Astrophysics Data System (ADS)

Borwein, Jonathan M.; Bradley, David M.; Crandall, Richard E.

2000-09-01

We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of the argument, the desired speed of computation, and the incidence of what we call "value recycling".

The Riemann Zeta Zeros from an Asymptotic Perspective

ERIC Educational Resources Information Center

Grant, Ken

2015-01-01

In 1859, on the occasion of being elected as a corresponding member of the Berlin Academy, Bernard Riemann (1826-66), a student of Carl Friedrich Gauss (1777-1855), presenteda lecture in which he presented a mathematics formula, derived from complex integration, which gave a precise count of the primes on the understanding that one of the terms in…

On small values of the Riemann zeta-function at Gram points

DOE Office of Scientific and Technical Information (OSTI.GOV)

Korolev, M A

In this paper, we prove the existence of a large set of Gram points t{sub n} such that the values ζ(0.5+it{sub n}) are 'anomalously' close to zero. A lower bound for the negative 'discrete' moment of the Riemann zeta-function on the critical line is also given. Bibliography: 13 titles.

Study Paths, Riemann Surfaces, and Strebel Differentials

ERIC Educational Resources Information Center

Buser, Peter; Semmler, Klaus-Dieter

2017-01-01

These pages aim to explain and interpret why the late Mika Seppälä, a conformal geometer, proposed to model student study behaviour using concepts from conformal geometry, such as Riemann surfaces and Strebel differentials. Over many years Mika Seppälä taught online calculus courses to students at Florida State University in the United States, as…

Optical Random Riemann Waves in Integrable Turbulence

NASA Astrophysics Data System (ADS)

Randoux, Stéphane; Gustave, François; Suret, Pierre; El, Gennady

2017-06-01

We examine integrable turbulence (IT) in the framework of the defocusing cubic one-dimensional nonlinear Schrödinger equation. This is done theoretically and experimentally, by realizing an optical fiber experiment in which the defocusing Kerr nonlinearity strongly dominates linear dispersive effects. Using a dispersive-hydrodynamic approach, we show that the development of IT can be divided into two distinct stages, the initial, prebreaking stage being described by a system of interacting random Riemann waves. We explain the low-tailed statistics of the wave intensity in IT and show that the Riemann invariants of the asymptotic nonlinear geometric optics system represent the observable quantities that provide new insight into statistical features of the initial stage of the IT development by exhibiting stationary probability density functions.

Averages of ratios of the Riemann zeta-function and correlations of divisor sums

NASA Astrophysics Data System (ADS)

Conrey, Brian; Keating, Jonathan P.

2017-10-01

Nonlinearity has published articles containing a significant number-theoretic component since the journal was first established. We examine one thread, concerning the statistics of the zeros of the Riemann zeta function. We extend this by establishing a connection between the ratios conjecture for the Riemann zeta-function and a conjecture concerning correlations of convolutions of Möbius and divisor functions. Specifically, we prove that the ratios conjecture and an arithmetic correlations conjecture imply the same result. This provides new support for the ratios conjecture, which previously had been motivated by analogy with formulae in random matrix theory and by a heuristic recipe. Our main theorem generalises a recent calculation pertaining to the special case of two-over-two ratios.

A numerical method for systems of conservation laws of mixed type admitting hyperbolic flux splitting

NASA Technical Reports Server (NTRS)

Shu, Chi-Wang

1992-01-01

The present treatment of elliptic regions via hyperbolic flux-splitting and high order methods proposes a flux splitting in which the corresponding Jacobians have real and positive/negative eigenvalues. While resembling the flux splitting used in hyperbolic systems, the present generalization of such splitting to elliptic regions allows the handling of mixed-type systems in a unified and heuristically stable fashion. The van der Waals fluid-dynamics equation is used. Convergence with good resolution to weak solutions for various Riemann problems are observed.

From nonlinear Schrödinger hierarchy to some (2+1)-dimensional nonlinear pseudodifferential equations

NASA Astrophysics Data System (ADS)

Yang, Xiao; Du, Dianlou

2010-08-01

The Poisson structure on CN×RN is introduced to give the Hamiltonian system associated with a spectral problem which yields the nonlinear Schrödinger (NLS) hierarchy. The Hamiltonian system is proven to be Liouville integrable. Some (2+1)-dimensional equations including NLS equation, Kadomtesev-Petviashvili I (KPI) equation, coupled KPI equation, and modified Kadomtesev-Petviashvili (mKP) equation, are decomposed into Hamilton flows via the NLS hierarchy. The algebraic curve, Abel-Jacobi coordinates, and Riemann-Jacobi inversion are used to obtain the algebrogeometric solutions of these equations.

Periodicity computation of generalized mathematical biology problems involving delay differential equations.

PubMed

Jasim Mohammed, M; Ibrahim, Rabha W; Ahmad, M Z

2017-03-01

In this paper, we consider a low initial population model. Our aim is to study the periodicity computation of this model by using neutral differential equations, which are recognized in various studies including biology. We generalize the neutral Rayleigh equation for the third-order by exploiting the model of fractional calculus, in particular the Riemann-Liouville differential operator. We establish the existence and uniqueness of a periodic computational outcome. The technique depends on the continuation theorem of the coincidence degree theory. Besides, an example is presented to demonstrate the finding.

The Great Gorilla Jump: An Introduction to Riemann Sums and Definite Integrals

ERIC Educational Resources Information Center

Sealey, Vicki; Engelke, Nicole

2012-01-01

The great gorilla jump is an activity designed to allow calculus students to construct an understanding of the structure of the Riemann sum and definite integral. The activity uses the ideas of position, velocity, and time to allow students to explore familiar ideas in a new way. Our research has shown that introducing the definite integral as…

A preconditioned formulation of the Cauchy-Riemann equations

NASA Technical Reports Server (NTRS)

Phillips, T. N.

1983-01-01

A preconditioning of the Cauchy-Riemann equations which results in a second-order system is described. This system is shown to have a unique solution if the boundary conditions are chosen carefully. This choice of boundary condition enables the solution of the first-order system to be retrieved. A numerical solution of the preconditioned equations is obtained by the multigrid method.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Amariti, Antonio; Toldo, Chiara

We consider 4d N = 1 SCFTs, topologically twisted on compact constant curvature Riemann surfaces, giving rise to 2d N = (0; 2) SCFTs. The exact R-current of these 2d SCFT extremizes the central charge c 2d, similarly to the 4d picture, where the exact R-current maximizes the central charge a 4d. There are global currents that do not mix with the R-current in 4d but their mixing becomes non trivial in 2d. In this paper we study the holographic dual of this process by analyzing a 5d N = 2 truncation of T 1,1 with one Betti vector multiplet,more » dual to the baryonic current on the CFT side. The holographic realization of the flow across dimensions connects AdS 5 to AdS 3 vacua in the supergravity picture. We verify the existence of the flow to AdS 3 solutions and we retrieve the field theory results for the mixing of the Betti vector with the graviphoton. Moreover, we extract the central charge from the Brown-Henneaux formula, matching with the results obtained in field theory. We develop a general formalism to obtain the central charge of a 2d SCFT from 5d N = 2 gauged supergravity with a generic number of vector multiplets, showing that its extremization corresponds to an attractor mechanism for the scalars in the supergravity picture.« less

Betti multiplets, flows across dimensions and c-extremization

DOE PAGES

Amariti, Antonio; Toldo, Chiara

2017-07-10

We consider 4d N = 1 SCFTs, topologically twisted on compact constant curvature Riemann surfaces, giving rise to 2d N = (0; 2) SCFTs. The exact R-current of these 2d SCFT extremizes the central charge c 2d, similarly to the 4d picture, where the exact R-current maximizes the central charge a 4d. There are global currents that do not mix with the R-current in 4d but their mixing becomes non trivial in 2d. In this paper we study the holographic dual of this process by analyzing a 5d N = 2 truncation of T 1,1 with one Betti vector multiplet,more » dual to the baryonic current on the CFT side. The holographic realization of the flow across dimensions connects AdS 5 to AdS 3 vacua in the supergravity picture. We verify the existence of the flow to AdS 3 solutions and we retrieve the field theory results for the mixing of the Betti vector with the graviphoton. Moreover, we extract the central charge from the Brown-Henneaux formula, matching with the results obtained in field theory. We develop a general formalism to obtain the central charge of a 2d SCFT from 5d N = 2 gauged supergravity with a generic number of vector multiplets, showing that its extremization corresponds to an attractor mechanism for the scalars in the supergravity picture.« less

Betti multiplets, flows across dimensions and c-extremization

NASA Astrophysics Data System (ADS)

Amariti, Antonio; Toldo, Chiara

2017-07-01

We consider 4d N = 1 SCFTs, topologically twisted on compact constant curvature Riemann surfaces, giving rise to 2d N = (0, 2) SCFTs. The exact R-current of these 2d SCFT extremizes the central charge c 2 d , similarly to the 4d picture, where the exact R-current maximizes the central charge a 4 d . There are global currents that do not mix with the R-current in 4d but their mixing becomes non trivial in 2d. In this paper we study the holographic dual of this process by analyzing a 5d N = 2 truncation of T 1,1 with one Betti vector multiplet, dual to the baryonic current on the CFT side. The holographic realization of the flow across dimensions connects AdS5 to AdS3 vacua in the supergravity picture. We verify the existence of the flow to AdS3 solutions and we retrieve the field theory results for the mixing of the Betti vector with the graviphoton. Moreover, we extract the central charge from the Brown-Henneaux formula, matching with the results obtained in field theory. We develop a general formalism to obtain the central charge of a 2d SCFT from 5d N = 2 gauged supergravity with a generic number of vector multiplets, showing that its extremization corresponds to an attractor mechanism for the scalars in the supergravity picture.

The role of fractional time-derivative operators on anomalous diffusion

NASA Astrophysics Data System (ADS)

Tateishi, Angel A.; Ribeiro, Haroldo V.; Lenzi, Ervin K.

2017-10-01

The generalized diffusion equations with fractional order derivatives have shown be quite efficient to describe the diffusion in complex systems, with the advantage of producing exact expressions for the underlying diffusive properties. Recently, researchers have proposed different fractional-time operators (namely: the Caputo-Fabrizio and Atangana-Baleanu) which, differently from the well-known Riemann-Liouville operator, are defined by non-singular memory kernels. Here we proposed to use these new operators to generalize the usual diffusion equation. By analyzing the corresponding fractional diffusion equations within the continuous time random walk framework, we obtained waiting time distributions characterized by exponential, stretched exponential, and power-law functions, as well as a crossover between two behaviors. For the mean square displacement, we found crossovers between usual and confined diffusion, and between usual and sub-diffusion. We obtained the exact expressions for the probability distributions, where non-Gaussian and stationary distributions emerged. This former feature is remarkable because the fractional diffusion equation is solved without external forces and subjected to the free diffusion boundary conditions. We have further shown that these new fractional diffusion equations are related to diffusive processes with stochastic resetting, and to fractional diffusion equations with derivatives of distributed order. Thus, our results suggest that these new operators may be a simple and efficient way for incorporating different structural aspects into the system, opening new possibilities for modeling and investigating anomalous diffusive processes.

Computational approach to compact Riemann surfaces

NASA Astrophysics Data System (ADS)

Frauendiener, Jörg; Klein, Christian

2017-01-01

A purely numerical approach to compact Riemann surfaces starting from plane algebraic curves is presented. The critical points of the algebraic curve are computed via a two-dimensional Newton iteration. The starting values for this iteration are obtained from the resultants with respect to both coordinates of the algebraic curve and a suitable pairing of their zeros. A set of generators of the fundamental group for the complement of these critical points in the complex plane is constructed from circles around these points and connecting lines obtained from a minimal spanning tree. The monodromies are computed by solving the defining equation of the algebraic curve on collocation points along these contours and by analytically continuing the roots. The collocation points are chosen to correspond to Chebychev collocation points for an ensuing Clenshaw-Curtis integration of the holomorphic differentials which gives the periods of the Riemann surface with spectral accuracy. At the singularities of the algebraic curve, Puiseux expansions computed by contour integration on the circles around the singularities are used to identify the holomorphic differentials. The Abel map is also computed with the Clenshaw-Curtis algorithm and contour integrals. As an application of the code, solutions to the Kadomtsev-Petviashvili equation are computed on non-hyperelliptic Riemann surfaces.

Rational Solutions of the Painlevé-II Equation Revisited

NASA Astrophysics Data System (ADS)

Miller, Peter D.; Sheng, Yue

2017-08-01

The rational solutions of the Painlevé-II equation appear in several applications and are known to have many remarkable algebraic and analytic properties. They also have several different representations, useful in different ways for establishing these properties. In particular, Riemann-Hilbert representations have proven to be useful for extracting the asymptotic behavior of the rational solutions in the limit of large degree (equivalently the large-parameter limit). We review the elementary properties of the rational Painlevé-II functions, and then we describe three different Riemann-Hilbert representations of them that have appeared in the literature: a representation by means of the isomonodromy theory of the Flaschka-Newell Lax pair, a second representation by means of the isomonodromy theory of the Jimbo-Miwa Lax pair, and a third representation found by Bertola and Bothner related to pseudo-orthogonal polynomials. We prove that the Flaschka-Newell and Bertola-Bothner Riemann-Hilbert representations of the rational Painlevé-II functions are explicitly connected to each other. Finally, we review recent results describing the asymptotic behavior of the rational Painlevé-II functions obtained from these Riemann-Hilbert representations by means of the steepest descent method.

Baker-Akhiezer Spinor Kernel and Tau-functions on Moduli Spaces of Meromorphic Differentials

NASA Astrophysics Data System (ADS)

Kalla, C.; Korotkin, D.

2014-11-01

In this paper we study the Baker-Akhiezer spinor kernel on moduli spaces of meromorphic differentials on Riemann surfaces. We introduce the Baker-Akhiezer tau-function which is related to both the Bergman tau-function (which was studied before in the context of Hurwitz spaces and spaces of holomorphic Abelian and quadratic differentials) and the KP tau-function on such spaces. In particular, we derive variational formulas of Rauch-Ahlfors type on moduli spaces of meromorphic differentials with prescribed singularities: we use the system of homological coordinates, consisting of absolute and relative periods of the meromorphic differential, and show how to vary the fundamental objects associated to a Riemann surface (the matrix of b-periods, normalized Abelian differentials, the Bergman bidifferential, the Szegö kernel and the Baker-Akhiezer spinor kernel) with respect to these coordinates. The variational formulas encode dependence both on the moduli of the Riemann surface and on the choice of meromorphic differential (variation of the meromorphic differential while keeping the Riemann surface fixed corresponds to flows of KP type). Analyzing the global properties of the Bergman and Baker-Akhiezer tau-functions, we establish relationships between various divisor classes on the moduli spaces.

Nonminimal coupling for the gravitational and electromagnetic fields: Black hole solutions and solitons

NASA Astrophysics Data System (ADS)

Balakin, Alexander B.; Bochkarev, Vladimir V.; Lemos, José P. S.

2008-04-01

Using a Lagrangian formalism, a three-parameter nonminimal Einstein-Maxwell theory is established. The three parameters q1, q2, and q3 characterize the cross-terms in the Lagrangian, between the Maxwell field and terms linear in the Ricci scalar, Ricci tensor, and Riemann tensor, respectively. Static spherically symmetric equations are set up, and the three parameters are interrelated and chosen so that effectively the system reduces to a one parameter only, q. Specific black hole and other type of one-parameter solutions are studied. First, as a preparation, the Reissner-Nordström solution, with q1=q2=q3=0, is displayed. Then, we search for solutions in which the electric field is regular everywhere as well as asymptotically Coulombian, and the metric potentials are regular at the center as well as asymptotically flat. In this context, the one-parameter model with q1≡-q, q2=2q, q3=-q, called the Gauss-Bonnet model, is analyzed in detail. The study is done through the solution of the Abel equation (the key equation), and the dynamical system associated with the model. There is extra focus on an exact solution of the model and its critical properties. Finally, an exactly integrable one-parameter model, with q1≡-q, q2=q, q3=0, is considered also in detail. A special submodel, in which the Fibonacci number appears naturally, of this one-parameter model is shown, and the corresponding exact solution is presented. Interestingly enough, it is a soliton of the theory, the Fibonacci soliton, without horizons and with a mild conical singularity at the center.

Exact image theory for the problem of dielectric/magnetic slab

NASA Technical Reports Server (NTRS)

Lindell, I. V.

1987-01-01

Exact image method, recently introduced for the exact solution of electromagnetic field problems involving homogeneous half spaces and microstrip-like geometries, is developed for the problem of homogeneous slab of dielectric and/or magnetic material in free space. Expressions for image sources, creating the exact reflected and transmitted fields, are given and their numerical evaluation is demonstrated. Nonradiating modes, guided by the slab and responsible for the loss of convergence of the image functions, are considered and extracted. The theory allows, for example, an analysis of finite ground planes in microstrip antenna structures.

Polymeric quantum mechanics and the zeros of the Riemann zeta function

NASA Astrophysics Data System (ADS)

Berra-Montiel, Jasel; Molgado, Alberto

We analyze the Berry-Keating model and the Sierra and Rodríguez-Laguna Hamiltonian within the polymeric quantization formalism. By using the polymer representation, we obtain for both models, the associated polymeric quantum Hamiltonians and the corresponding stationary wave functions. The self-adjointness condition provides a proper domain for the Hamiltonian operator and the energy spectrum, which turned out to be dependent on an introduced scale parameter. By performing a counting of semiclassical states, we prove that the polymer representation reproduces the smooth part of the Riemann-von Mangoldt formula, and also introduces a correction depending on the energy and the scale parameter. This may shed some light on the understanding of the fluctuation behavior of the zeros of the Riemann function from a purely quantum point of view.

A spectral hybridizable discontinuous Galerkin method for elastic-acoustic wave propagation

NASA Astrophysics Data System (ADS)

Terrana, S.; Vilotte, J. P.; Guillot, L.

2018-04-01

We introduce a time-domain, high-order in space, hybridizable discontinuous Galerkin (DG) spectral element method (HDG-SEM) for wave equations in coupled elastic-acoustic media. The method is based on a first-order hyperbolic velocity-strain formulation of the wave equations written in conservative form. This method follows the HDG approach by introducing a hybrid unknown, which is the approximation of the velocity on the elements boundaries, as the only globally (i.e. interelement) coupled degrees of freedom. In this paper, we first present a hybridized formulation of the exact Riemann solver at the element boundaries, taking into account elastic-elastic, acoustic-acoustic and elastic-acoustic interfaces. We then use this Riemann solver to derive an explicit construction of the HDG stabilization function τ for all the above-mentioned interfaces. We thus obtain an HDG scheme for coupled elastic-acoustic problems. This scheme is then discretized in space on quadrangular/hexahedral meshes using arbitrary high-order polynomial basis for both volumetric and hybrid fields, using an approach similar to the spectral element methods. This leads to a semi-discrete system of algebraic differential equations (ADEs), which thanks to the structure of the global conservativity condition can be reformulated easily as a classical system of first-order ordinary differential equations in time, allowing the use of classical explicit or implicit time integration schemes. When an explicit time scheme is used, the HDG method can be seen as a reformulation of a DG with upwind fluxes. The introduction of the velocity hybrid unknown leads to relatively simple computations at the element boundaries which, in turn, makes the HDG approach competitive with the DG-upwind methods. Extensive numerical results are provided to illustrate and assess the accuracy and convergence properties of this HDG-SEM. The approximate velocity is shown to converge with the optimal order of k + 1 in the L2-norm, when element polynomials of order k are used, and to exhibit the classical spectral convergence of SEM. Additional inexpensive local post-processing in both the elastic and the acoustic case allow to achieve higher convergence orders. The HDG scheme provides a natural framework for coupling classical, continuous Galerkin SEM with HDG-SEM in the same simulation, and it is shown numerically in this paper. As such, the proposed HDG-SEM can combine the efficiency of the continuous SEM with the flexibility of the HDG approaches. Finally, more complex numerical results, inspired from real geophysical applications, are presented to illustrate the capabilities of the method for wave propagation in heterogeneous elastic-acoustic media with complex geometries.

A fast numerical method for ideal fluid flow in domains with multiple stirrers

NASA Astrophysics Data System (ADS)

Nasser, Mohamed M. S.; Green, Christopher C.

2018-03-01

A collection of arbitrarily-shaped solid objects, each moving at a constant speed, can be used to mix or stir ideal fluid, and can give rise to interesting flow patterns. Assuming these systems of fluid stirrers are two-dimensional, the mathematical problem of resolving the flow field—given a particular distribution of any finite number of stirrers of specified shape and speed—can be formulated as a Riemann-Hilbert (R-H) problem. We show that this R-H problem can be solved numerically using a fast and accurate algorithm for any finite number of stirrers based around a boundary integral equation with the generalized Neumann kernel. Various systems of fluid stirrers are considered, and our numerical scheme is shown to handle highly multiply connected domains (i.e. systems of many fluid stirrers) with minimal computational expense.

Snyder-like modified gravity in Newton's spacetime

NASA Astrophysics Data System (ADS)

Leiva, Carlos

This work is focused on searching a geodesic interpretation of the dynamics of a particle under the effects of a Snyder-like deformation in the background of the Kepler problem. In order to accomplish that task, a Newtonian spacetime is used. Newtonian spacetime is not a metric manifold, but allows to introduce a torsion-free connection in order to interpret the dynamic equations of the deformed Kepler problem as geodesics in a curved spacetime. These geodesics and the curvature terms of the Riemann and Ricci tensors show a mass and a fundamental length dependence as expected, but are velocity-independent that is a feature present in other classical approaches to the problem. In this sense, the effect of introducing a deformed algebra is examined and the corresponding curvature terms calculated, as well as the modifications of the integrals of motion.

Symmetry-Resolved Entanglement in Many-Body Systems.

PubMed

Goldstein, Moshe; Sela, Eran

2018-05-18

Similarly to the system Hamiltonian, a subsystem's reduced density matrix is composed of blocks characterized by symmetry quantum numbers (charge sectors). We present a geometric approach for extracting the contribution of individual charge sectors to the subsystem's entanglement measures within the replica trick method, via threading appropriate conjugate Aharonov-Bohm fluxes through a multisheet Riemann surface. Specializing to the case of 1+1D conformal field theory, we obtain general exact results for the entanglement entropies and spectrum, and apply them to a variety of systems, ranging from free and interacting fermions to spin and parafermion chains, and verify them numerically. We find that the total entanglement entropy, which scales as lnL, is composed of sqrt[lnL] contributions of individual subsystem charge sectors for interacting fermion chains, or even O(L^{0}) contributions when total spin conservation is also accounted for. We also explain how measurements of the contribution to the entanglement from separate charge sectors can be performed experimentally with existing techniques.

Kinetic Equation for a Soliton Gas and Its Hydrodynamic Reductions

NASA Astrophysics Data System (ADS)

El, G. A.; Kamchatnov, A. M.; Pavlov, M. V.; Zykov, S. A.

2011-04-01

We introduce and study a new class of kinetic equations, which arise in the description of nonequilibrium macroscopic dynamics of soliton gases with elastic collisions between solitons. These equations represent nonlinear integro-differential systems and have a novel structure, which we investigate by studying in detail the class of N-component `cold-gas' hydrodynamic reductions. We prove that these reductions represent integrable linearly degenerate hydrodynamic type systems for arbitrary N which is a strong evidence in favour of integrability of the full kinetic equation. We derive compact explicit representations for the Riemann invariants and characteristic velocities of the hydrodynamic reductions in terms of the `cold-gas' component densities and construct a number of exact solutions having special properties (quasiperiodic, self-similar). Hydrodynamic symmetries are then derived and investigated. The obtained results shed light on the structure of a continuum limit for a large class of integrable systems of hydrodynamic type and are also relevant to the description of turbulent motion in conservative compressible flows.

An approximate Riemann solver for thermal and chemical nonequilibrium flows

NASA Technical Reports Server (NTRS)

Prabhu, Ramadas K.

1994-01-01

Among the many methods available for the determination of inviscid fluxes across a surface of discontinuity, the flux-difference-splitting technique that employs Roe-averaged variables has been used extensively by the CFD community because of its simplicity and its ability to capture shocks exactly. This method, originally developed for perfect gas flows, has since been extended to equilibrium as well as nonequilibrium flows. Determination of the Roe-averaged variables for the case of a perfect gas flow is a simple task; however, for thermal and chemical nonequilibrium flows, some of the variables are not uniquely defined. Methods available in the literature to determine these variables seem to lack sound bases. The present paper describes a simple, yet accurate, method to determine all the variables for nonequilibrium flows in the Roe-average state. The basis for this method is the requirement that the Roe-averaged variables form a consistent set of thermodynamic variables. The present method satisfies the requirement that the square of the speed of sound be positive.

Rotating solutions in critical Lovelock gravities

DOE PAGES

Cvetič, M.; Feng, Xing -Hui; Lü, H.; ...

2016-12-12

For appropriate choices of the coupling constants, the equations of motion of Lovelock gravities up to order n in the Riemann tensor can be factorized such that the theories admits a single (A)dS vacuum. In this paper we construct two classes of exact rotating metrics in such critical Lovelock gravities of order n in d = 2n + 1 dimensions. In one class, the n angular momenta in the n orthogonal spatial 2-planes are equal, and hence the metric is of cohomogeneity one. We construct these metrics in a Kerr-Schild form, but they can then be recast in terms ofmore » Boyer-Lindquist coordinates. The other class involves metrics with only a single non-vanishing angular momentum. Again we construct them in a Kerr-Schild form, but in this case it does not seem to be possible to recast them in Boyer-Lindquist form. Furthermore, both classes of solutions have naked curvature singularities, arising because of the over rotation of the configurations.« less

Rotating solutions in critical Lovelock gravities

NASA Astrophysics Data System (ADS)

Cvetič, M.; Feng, Xing-Hui; Lü, H.; Pope, C. N.

2017-02-01

For appropriate choices of the coupling constants, the equations of motion of Lovelock gravities up to order n in the Riemann tensor can be factorized such that the theories admit a single (A)dS vacuum. In this paper we construct two classes of exact rotating metrics in such critical Lovelock gravities of order n in d = 2 n + 1 dimensions. In one class, the n angular momenta in the n orthogonal spatial 2-planes are equal, and hence the metric is of cohomogeneity one. We construct these metrics in a Kerr-Schild form, but they can then be recast in terms of Boyer-Lindquist coordinates. The other class involves metrics with only a single non-vanishing angular momentum. Again we construct them in a Kerr-Schild form, but in this case it does not seem to be possible to recast them in Boyer-Lindquist form. Both classes of solutions have naked curvature singularities, arising because of the over rotation of the configurations.

Rotating solutions in critical Lovelock gravities

DOE Office of Scientific and Technical Information (OSTI.GOV)

Cvetič, M.; Feng, Xing -Hui; Lü, H.

For appropriate choices of the coupling constants, the equations of motion of Lovelock gravities up to order n in the Riemann tensor can be factorized such that the theories admits a single (A)dS vacuum. In this paper we construct two classes of exact rotating metrics in such critical Lovelock gravities of order n in d = 2n + 1 dimensions. In one class, the n angular momenta in the n orthogonal spatial 2-planes are equal, and hence the metric is of cohomogeneity one. We construct these metrics in a Kerr-Schild form, but they can then be recast in terms ofmore » Boyer-Lindquist coordinates. The other class involves metrics with only a single non-vanishing angular momentum. Again we construct them in a Kerr-Schild form, but in this case it does not seem to be possible to recast them in Boyer-Lindquist form. Furthermore, both classes of solutions have naked curvature singularities, arising because of the over rotation of the configurations.« less

Intersecting surface defects and two-dimensional CFT

NASA Astrophysics Data System (ADS)

Gomis, Jaume; Le Floch, Bruno; Pan, Yiwen; Peelaers, Wolfger

2017-08-01

We initiate the study of intersecting surface operators/defects in 4D quantum field theories (QFTs). We characterize these defects by coupled 4D/2D/0D theories constructed by coupling the degrees of freedom localized at a point and on intersecting surfaces in spacetime to each other and to the 4D QFT. We construct supersymmetric intersecting surface defects preserving just two supercharges in N =2 gauge theories. These defects are amenable to exact analysis by localization of the partition function of the underlying 4D/2D/0D QFT. We identify the 4D/2D/0D QFTs that describe intersecting surface operators in N =2 gauge theories realized by intersecting M2 branes ending on N M5 branes wrapping a Riemann surface. We conjecture and provide evidence for an explicit equivalence between the squashed four-sphere partition function of these intersecting defects and correlation functions in Liouville/Toda CFT with the insertion of arbitrary degenerate vertex operators, which are labeled by two representations of S U (N ).

Symmetry-Resolved Entanglement in Many-Body Systems

NASA Astrophysics Data System (ADS)

Goldstein, Moshe; Sela, Eran

2018-05-01

Similarly to the system Hamiltonian, a subsystem's reduced density matrix is composed of blocks characterized by symmetry quantum numbers (charge sectors). We present a geometric approach for extracting the contribution of individual charge sectors to the subsystem's entanglement measures within the replica trick method, via threading appropriate conjugate Aharonov-Bohm fluxes through a multisheet Riemann surface. Specializing to the case of 1 +1 D conformal field theory, we obtain general exact results for the entanglement entropies and spectrum, and apply them to a variety of systems, ranging from free and interacting fermions to spin and parafermion chains, and verify them numerically. We find that the total entanglement entropy, which scales as ln L , is composed of √{ln L } contributions of individual subsystem charge sectors for interacting fermion chains, or even O (L0) contributions when total spin conservation is also accounted for. We also explain how measurements of the contribution to the entanglement from separate charge sectors can be performed experimentally with existing techniques.

The Osher scheme for real gases

NASA Technical Reports Server (NTRS)

Suresh, Ambady; Liou, Meng-Sing

1990-01-01

An extension of Osher's approximate Riemann solver to include gases with an arbitrary equation of state is presented. By a judicious choice of thermodynamic variables, the Riemann invariats are reduced to quadratures which are then approximated numerically. The extension is rigorous and does not involve any further assumptions or approximations over the ideal gas case. Numerical results are presented to demonstrate the feasibility and accuracy of the proposed method.

Linearizable quantum supersymmetric σ models

NASA Astrophysics Data System (ADS)

Haba, Z.

1988-07-01

Euclidean quantization of superfields with values in a Hermitian manifold and defined on a super-Riemann surface is discussed. It is shown that stochastic differential equations relating an interacting σ superfield to the free one become linear if the field takes values in a generalized Poincaré upper half-plane. A renormalized perturbative solution is obtained. Fields with values in a Riemann surface are discussed in brief.

Teaching Ideas and Activities for Classroom: Integrating Technology into the Pedagogy of Integral Calculus and the Approximation of Definite Integrals

ERIC Educational Resources Information Center

Caglayan, Gunhan

2016-01-01

The purpose of this article is to offer teaching ideas in the treatment of the definite integral concept and the Riemann sums in a technology-supported environment. Specifically, the article offers teaching ideas and activities for classroom for the numerical methods of approximating a definite integral via left- and right-hand Riemann sums, along…

A contribution to the great Riemann solver debate

NASA Technical Reports Server (NTRS)

Quirk, James J.

1992-01-01

The aims of this paper are threefold: to increase the level of awareness within the shock capturing community to the fact that many Godunov-type methods contain subtle flaws that can cause spurious solutions to be computed; to identify one mechanism that might thwart attempts to produce very high resolution simulations; and to proffer a simple strategy for overcoming the specific failings of individual Riemann solvers.

The escape of high explosive products: An exact-solution problem for verification of hydrodynamics codes

DOE PAGES

Doebling, Scott William

2016-10-22

This paper documents the escape of high explosive (HE) products problem. The problem, first presented by Fickett & Rivard, tests the implementation and numerical behavior of a high explosive detonation and energy release model and its interaction with an associated compressible hydrodynamics simulation code. The problem simulates the detonation of a finite-length, one-dimensional piece of HE that is driven by a piston from one end and adjacent to a void at the other end. The HE equation of state is modeled as a polytropic ideal gas. The HE detonation is assumed to be instantaneous with an infinitesimal reaction zone. Viamore » judicious selection of the material specific heat ratio, the problem has an exact solution with linear characteristics, enabling a straightforward calculation of the physical variables as a function of time and space. Lastly, implementation of the exact solution in the Python code ExactPack is discussed, as are verification cases for the exact solution code.« less

On the global Casimir effect in the Schwarzschild spacetime

NASA Astrophysics Data System (ADS)

Muniz, C. R.; Tahim, M. O.; Cunha, M. S.; Vieira, H. S.

2018-01-01

In this paper we study the vacuum quantum fluctuations of the stationary modes of an uncharged scalar field with mass m around a Schwarzschild black hole with mass M, at zero and non-zero temperatures. The procedure consists of calculating the energy eigenvalues starting from the exact solutions found for the dynamics of the scalar field, considering a frequency cutoff in which the particle is not absorbed by the black hole. From this result, we obtain the exterior contributions for the vacuum energy associated to the stationary states of the scalar field, by considering the half-summing of the levels of energy and taking into account the respective degeneracies, in order to better capture the nontrivial topology of the black hole spacetime. Then we use the Riemann's zeta function to regularize the vacuum energy thus found. Such a regularized quantity is the Casimir energy, whose analytic computation we show to yield a convergent series. The Casimir energy obtained does not take into account any boundaries artificially imposed on the system, just the nontrivial spacetime topology associated to the source and its singularity. We suggest that this latter manifests itself through the vacuum tension calculated on the event horizon. We also investigate the problem by considering the thermal corrections via Helmholtz free energy calculation, computing the Casimir internal energy, the corresponding tension on the event horizon, the Casimir entropy, and the thermal capacity of the regularized quantum vacuum, analyzing their behavior at low and high temperatures, pointing out the thermodynamic instability of the system in the considered regime, i.e. mMll 1.

Renormalization in Quantum Field Theory and the Riemann-Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem

NASA Astrophysics Data System (ADS)

Connes, Alain; Kreimer, Dirk

This paper gives a complete selfcontained proof of our result announced in [6] showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the Riemann-Hilbert problem. We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra which is commutative as an algebra. It is the dual Hopf algebra of the enveloping algebra of a Lie algebra whose basis is labelled by the one particle irreducible Feynman graphs. The Lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa. The corresponding Lie group G is the group of characters of . We shall then show that, using dimensional regularization, the bare (unrenormalized) theory gives rise to a loop where C is a small circle of complex dimensions around the integer dimension D of space-time. Our main result is that the renormalized theory is just the evaluation at z=D of the holomorphic part γ+ of the Birkhoff decomposition of γ. We begin to analyse the group G and show that it is a semi-direct product of an easily understood abelian group by a highly non-trivial group closely tied up with groups of diffeomorphisms. The analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title.

A nominally second-order cell-centered Lagrangian scheme for simulating elastic-plastic flows on two-dimensional unstructured grids

NASA Astrophysics Data System (ADS)

Maire, Pierre-Henri; Abgrall, Rémi; Breil, Jérôme; Loubère, Raphaël; Rebourcet, Bernard

2013-02-01

In this paper, we describe a cell-centered Lagrangian scheme devoted to the numerical simulation of solid dynamics on two-dimensional unstructured grids in planar geometry. This numerical method, utilizes the classical elastic-perfectly plastic material model initially proposed by Wilkins [M.L. Wilkins, Calculation of elastic-plastic flow, Meth. Comput. Phys. (1964)]. In this model, the Cauchy stress tensor is decomposed into the sum of its deviatoric part and the thermodynamic pressure which is defined by means of an equation of state. Regarding the deviatoric stress, its time evolution is governed by a classical constitutive law for isotropic material. The plasticity model employs the von Mises yield criterion and is implemented by means of the radial return algorithm. The numerical scheme relies on a finite volume cell-centered method wherein numerical fluxes are expressed in terms of sub-cell force. The generic form of the sub-cell force is obtained by requiring the scheme to satisfy a semi-discrete dissipation inequality. Sub-cell force and nodal velocity to move the grid are computed consistently with cell volume variation by means of a node-centered solver, which results from total energy conservation. The nominally second-order extension is achieved by developing a two-dimensional extension in the Lagrangian framework of the Generalized Riemann Problem methodology, introduced by Ben-Artzi and Falcovitz [M. Ben-Artzi, J. Falcovitz, Generalized Riemann Problems in Computational Fluid Dynamics, Cambridge Monogr. Appl. Comput. Math. (2003)]. Finally, the robustness and the accuracy of the numerical scheme are assessed through the computation of several test cases.

Addition of Improved Shock-Capturing Schemes to OVERFLOW 2.1

NASA Technical Reports Server (NTRS)

Burning, Pieter G.; Nichols, Robert H.; Tramel, Robert W.

2009-01-01

Existing approximate Riemann solvers do not perform well when the grid is not aligned with strong shocks in the flow field. Three new approximate Riemann algorithms are investigated to improve solution accuracy and stability in the vicinity of strong shocks. The new algorithms are compared to the existing upwind algorithms in OVERFLOW 2.1. The new algorithms use a multidimensional pressure gradient based switch to transition to a more numerically dissipative algorithm in the vicinity of strong shocks. One new algorithm also attempts to artificially thicken captured shocks in order to alleviate the errors in the solution introduced by "stair-stepping" of the shock resulting from the approximate Riemann solver. This algorithm performed well for all the example cases and produced results that were almost insensitive to the alignment of the grid and the shock.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Singleton, Jr., Robert; Israel, Daniel M.; Doebling, Scott William

For code verification, one compares the code output against known exact solutions. There are many standard test problems used in this capacity, such as the Noh and Sedov problems. ExactPack is a utility that integrates many of these exact solution codes into a common API (application program interface), and can be used as a stand-alone code or as a python package. ExactPack consists of python driver scripts that access a library of exact solutions written in Fortran or Python. The spatial profiles of the relevant physical quantities, such as the density, fluid velocity, sound speed, or internal energy, are returnedmore » at a time specified by the user. The solution profiles can be viewed and examined by a command line interface or a graphical user interface, and a number of analysis tools and unit tests are also provided. We have documented the physics of each problem in the solution library, and provided complete documentation on how to extend the library to include additional exact solutions. ExactPack’s code architecture makes it easy to extend the solution-code library to include additional exact solutions in a robust, reliable, and maintainable manner.« less

A Depth-Averaged 2-D Simulation for Coastal Barrier Breaching Processes

DTIC Science & Technology

2011-05-01

including bed change and variable flow density in the flow continuity and momentum equations. The model adopts the HLL approximate Riemann solver to handle...flow density in the flow continuity and momentum equations. The model adopts the HLL approximate Riemann solver to handle the mixed-regime flows near...18 547 Keulegan equation or the Bernoulli equation, and the breach morphological change is determined using simplified sediment transport models

A compressible two-layer model for transient gas-liquid flows in pipes

NASA Astrophysics Data System (ADS)

Demay, Charles; Hérard, Jean-Marc

2017-03-01

This work is dedicated to the modeling of gas-liquid flows in pipes. As a first step, a new two-layer model is proposed to deal with the stratified regime. The starting point is the isentropic Euler set of equations for each phase where the classical hydrostatic assumption is made for the liquid. The main difference with the models issued from the classical literature is that the liquid as well as the gas is assumed compressible. In that framework, an averaging process results in a five-equation system where the hydrostatic constraint has been used to define the interfacial pressure. Closure laws for the interfacial velocity and source terms such as mass and momentum transfer are provided following an entropy inequality. The resulting model is hyperbolic with non-conservative terms. Therefore, regarding the homogeneous part of the system, the definition and uniqueness of jump conditions is studied carefully and acquired. The nature of characteristic fields and the corresponding Riemann invariants are also detailed. Thus, one may build analytical solutions for the Riemann problem. In addition, positivity is obtained for heights and densities. The overall derivation deals with gas-liquid flows through rectangular channels, circular pipes with variable cross section and includes vapor-liquid flows.

Covariant path integrals on hyperbolic surfaces

NASA Astrophysics Data System (ADS)

Schaefer, Joe

1997-11-01

DeWitt's covariant formulation of path integration [B. De Witt, "Dynamical theory in curved spaces. I. A review of the classical and quantum action principles," Rev. Mod. Phys. 29, 377-397 (1957)] has two practical advantages over the traditional methods of "lattice approximations;" there is no ordering problem, and classical symmetries are manifestly preserved at the quantum level. Applying the spectral theorem for unbounded self-adjoint operators, we provide a rigorous proof of the convergence of certain path integrals on Riemann surfaces of constant curvature -1. The Pauli-DeWitt curvature correction term arises, as in DeWitt's work. Introducing a Fuchsian group Γ of the first kind, and a continuous, bounded, Γ-automorphic potential V, we obtain a Feynman-Kac formula for the automorphic Schrödinger equation on the Riemann surface ΓH. We analyze the Wick rotation and prove the strong convergence of the so-called Feynman maps [K. D. Elworthy, Path Integration on Manifolds, Mathematical Aspects of Superspace, edited by Seifert, Clarke, and Rosenblum (Reidel, Boston, 1983), pp. 47-90] on a dense set of states. Finally, we give a new proof of some results in C. Grosche and F. Steiner, "The path integral on the Poincare upper half plane and for Liouville quantum mechanics," Phys. Lett. A 123, 319-328 (1987).

A Legendre tau-spectral method for solving time-fractional heat equation with nonlocal conditions.

PubMed

Bhrawy, A H; Alghamdi, M A

2014-01-01

We develop the tau-spectral method to solve the time-fractional heat equation (T-FHE) with nonlocal condition. In order to achieve highly accurate solution of this problem, the operational matrix of fractional integration (described in the Riemann-Liouville sense) for shifted Legendre polynomials is investigated in conjunction with tau-spectral scheme and the Legendre operational polynomials are used as the base function. The main advantage in using the presented scheme is that it converts the T-FHE with nonlocal condition to a system of algebraic equations that simplifies the problem. For demonstrating the validity and applicability of the developed spectral scheme, two numerical examples are presented. The logarithmic graphs of the maximum absolute errors is presented to achieve the exponential convergence of the proposed method. Comparing between our spectral method and other methods ensures that our method is more accurate than those solved similar problem.

A Legendre tau-Spectral Method for Solving Time-Fractional Heat Equation with Nonlocal Conditions

PubMed Central

Bhrawy, A. H.; Alghamdi, M. A.

2014-01-01

We develop the tau-spectral method to solve the time-fractional heat equation (T-FHE) with nonlocal condition. In order to achieve highly accurate solution of this problem, the operational matrix of fractional integration (described in the Riemann-Liouville sense) for shifted Legendre polynomials is investigated in conjunction with tau-spectral scheme and the Legendre operational polynomials are used as the base function. The main advantage in using the presented scheme is that it converts the T-FHE with nonlocal condition to a system of algebraic equations that simplifies the problem. For demonstrating the validity and applicability of the developed spectral scheme, two numerical examples are presented. The logarithmic graphs of the maximum absolute errors is presented to achieve the exponential convergence of the proposed method. Comparing between our spectral method and other methods ensures that our method is more accurate than those solved similar problem. PMID:25057507

The piecewise-linear predictor-corrector code - A Lagrangian-remap method for astrophysical flows

NASA Technical Reports Server (NTRS)

Lufkin, Eric A.; Hawley, John F.

1993-01-01

We describe a time-explicit finite-difference algorithm for solving the nonlinear fluid equations. The method is similar to existing Eulerian schemes in its use of operator-splitting and artificial viscosity, except that we solve the Lagrangian equations of motion with a predictor-corrector and then remap onto a fixed Eulerian grid. The remap is formulated to eliminate errors associated with coordinate singularities, with a general prescription for remaps of arbitrary order. We perform a comprehensive series of tests on standard problems. Self-convergence tests show that the code has a second-order rate of convergence in smooth, two-dimensional flow, with pressure forces, gravity, and curvilinear geometry included. While not as accurate on idealized problems as high-order Riemann-solving schemes, the predictor-corrector Lagrangian-remap code has great flexibility for application to a variety of astrophysical problems.

An asymptotic formula for polynomials orthonormal with respect to a varying weight. II

DOE Office of Scientific and Technical Information (OSTI.GOV)

Komlov, A V; Suetin, S P

2014-09-30

This paper gives a proof of the theorem announced by the authors in the preceding paper with the same title. The theorem states that asymptotically the behaviour of the polynomials which are orthonormal with respect to the varying weight e{sup −2nQ(x)}p{sub g}(x)/√(∏{sub j=1}{sup 2p}(x−e{sub j})) coincides with the asymptotic behaviour of the Nuttall psi-function, which solves a special boundary-value problem on the relevant hyperelliptic Riemann surface of genus g=p−1. Here e{sub 1}

The high hall ventilation with the simplified simulation of the fan

NASA Astrophysics Data System (ADS)

Kyncl, Martin; Pelant, Jaroslav

2018-06-01

Here we work with the system of equations describing the non-stationary compressible turbulent multi-component flow in the gravitational field. We focus on the numerical simulation of the fan situated inside the high hall. The RANS equations are discretized with the use of the finite volume method. The original modification of the Riemann problem and its solution is used at the boundaries. The combination of specific boundary conditions is used for the simulation of the fan. The presented computational results are computed with own-developed code (C, FORTRAN, multiprocessor, unstructured meshes in general).

Exploration of Toeplitz-like matrices with unbounded symbols is not a purely academic journey

NASA Astrophysics Data System (ADS)

Böttcher, A.; Garoni, C.; Serra-Capizzano, S.

2017-11-01

It is often asked why Toeplitz-like matrices with unbounded symbols are worth studying. This paper gives an answer by presenting several concrete problems that motivate such studies. It surveys the central results of the theory of Generalized Locally Toeplitz (GLT) sequences in a self-contained tool-kit fashion, and gives a new extension from bounded Riemann integrable functions to unbounded almost everywhere continuous functions. The emergence of unbounded symbols is illustrated by local grid refinements in finite difference and finite element discretizations and also by preconditioning strategies. Bibliography: 40 titles.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Mendl, Christian B.; Spohn, Herbert

The nonequilibrium dynamics of anharmonic chains is studied by imposing an initial domain-wall state, in which the two half lattices are prepared in equilibrium with distinct parameters. Here, we analyse the Riemann problem for the corresponding Euler equations and, in specific cases, compare with molecular dynamics. Additionally, the fluctuations of time-integrated currents are investigated. In analogy with the KPZ equation, their typical fluctuations should be of size t 1/3 and have a Tracy–Widom GUE distributed amplitude. The proper extension to anharmonic chains is explained and tested through molecular dynamics. Our results are calibrated against the stochastic LeRoux lattice gas.

Standardized Definitions for Code Verification Test Problems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Doebling, Scott William

This document contains standardized definitions for several commonly used code verification test problems. These definitions are intended to contain sufficient information to set up the test problem in a computational physics code. These definitions are intended to be used in conjunction with exact solutions to these problems generated using Exact- Pack, www.github.com/lanl/exactpack.

Assimilation of Wave and Current Data for Prediction of Inlet and River Mouth Dynamics

DTIC Science & Technology

2013-07-01

onto the Delft3D computational grid and the specification of Riemann -type boundary conditions for the boundary-normal velocity and surface elevation...conditions from time- history data from in situ tide gages. The corrections are applied to the surface-elevation contribution to the Riemann boundary...The algorithms described above are all of the strong-constraint variational variety, and make use of adjoint solvers corresponding to the various

Exploration and extension of an improved Riemann track fitting algorithm

NASA Astrophysics Data System (ADS)

Strandlie, A.; Frühwirth, R.

2017-09-01

Recently, a new Riemann track fit which operates on translated and scaled measurements has been proposed. This study shows that the new Riemann fit is virtually as precise as popular approaches such as the Kalman filter or an iterative non-linear track fitting procedure, and significantly more precise than other, non-iterative circular track fitting approaches over a large range of measurement uncertainties. The fit is then extended in two directions: first, the measurements are allowed to lie on plane sensors of arbitrary orientation; second, the full error propagation from the measurements to the estimated circle parameters is computed. The covariance matrix of the estimated track parameters can therefore be computed without recourse to asymptotic properties, and is consequently valid for any number of observation. It does, however, assume normally distributed measurement errors. The calculations are validated on a simulated track sample and show excellent agreement with the theoretical expectations.

A distinguishing gravitational property for gravitational equation in higher dimensions

NASA Astrophysics Data System (ADS)

Dadhich, Naresh

2016-03-01

It is well known that Einstein gravity is kinematic (meaning that there is no non-trivial vacuum solution; i.e. the Riemann tensor vanishes whenever the Ricci tensor does so) in 3 dimension because the Riemann tensor is entirely given in terms of the Ricci tensor. Could this property be universalized for all odd dimensions in a generalized theory? The answer is yes, and this property uniquely singles out pure Lovelock (it has only one Nth order term in the action) gravity for which the Nth order Lovelock-Riemann tensor is indeed given in terms of the corresponding Ricci tensor for all odd, d=2N+1, dimensions. This feature of gravity is realized only in higher dimensions and it uniquely picks out pure Lovelock gravity from all other generalizations of Einstein gravity. It serves as a good distinguishing and guiding criterion for the gravitational equation in higher dimensions.

Exact parallel algorithms for some members of the traveling salesman problem family

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pekny, J.F.

1989-01-01

The traveling salesman problem and its many generalizations comprise one of the best known combinatorial optimization problem families. Most members of the family are NP-complete problems so that exact algorithms require an unpredictable and sometimes large computational effort. Parallel computers offer hope for providing the power required to meet these demands. A major barrier to applying parallel computers is the lack of parallel algorithms. The contributions presented in this thesis center around new exact parallel algorithms for the asymmetric traveling salesman problem (ATSP), prize collecting traveling salesman problem (PCTSP), and resource constrained traveling salesman problem (RCTSP). The RCTSP is amore » particularly difficult member of the family since finding a feasible solution is an NP-complete problem. An exact sequential algorithm is also presented for the directed hamiltonian cycle problem (DHCP). The DHCP algorithm is superior to current heuristic approaches and represents the first exact method applicable to large graphs. Computational results presented for each of the algorithms demonstrates the effectiveness of combining efficient algorithms with parallel computing methods. Performance statistics are reported for randomly generated ATSPs with 7,500 cities, PCTSPs with 200 cities, RCTSPs with 200 cities, DHCPs with 3,500 vertices, and assignment problems of size 10,000. Sequential results were collected on a Sun 4/260 engineering workstation, while parallel results were collected using a 14 and 100 processor BBN Butterfly Plus computer. The computational results represent the largest instances ever solved to optimality on any type of computer.« less

A non-oscillatory energy-splitting method for the computation of compressible multi-fluid flows

NASA Astrophysics Data System (ADS)

Lei, Xin; Li, Jiequan

2018-04-01

This paper proposes a new non-oscillatory energy-splitting conservative algorithm for computing multi-fluid flows in the Eulerian framework. In comparison with existing multi-fluid algorithms in the literature, it is shown that the mass fraction model with isobaric hypothesis is a plausible choice for designing numerical methods for multi-fluid flows. Then we construct a conservative Godunov-based scheme with the high order accurate extension by using the generalized Riemann problem solver, through the detailed analysis of kinetic energy exchange when fluids are mixed under the hypothesis of isobaric equilibrium. Numerical experiments are carried out for the shock-interface interaction and shock-bubble interaction problems, which display the excellent performance of this type of schemes and demonstrate that nonphysical oscillations are suppressed around material interfaces substantially.

Energy dissipation in a friction-controlled slide of a body excited by random motions of the foundation

NASA Astrophysics Data System (ADS)

Berezin, Sergey; Zayats, Oleg

2018-01-01

We study a friction-controlled slide of a body excited by random motions of the foundation it is placed on. Specifically, we are interested in such quantities as displacement, traveled distance, and energy loss due to friction. We assume that the random excitation is switched off at some time (possibly infinite) and show that the problem can be treated in an analytic, explicit, manner. Particularly, we derive formulas for the moments of the displacement and distance, and also for the average energy loss. To accomplish that we use the Pugachev-Sveshnikov equation for the characteristic function of a continuous random process given by a system of SDEs. This equation is solved by reduction to a parametric Riemann boundary value problem of complex analysis.

A Class of High-Resolution Explicit and Implicit Shock-Capturing Methods

NASA Technical Reports Server (NTRS)

Yee, H. C.

1994-01-01

The development of shock-capturing finite difference methods for hyperbolic conservation laws has been a rapidly growing area for the last decade. Many of the fundamental concepts, state-of-the-art developments and applications to fluid dynamics problems can only be found in meeting proceedings, scientific journals and internal reports. This paper attempts to give a unified and generalized formulation of a class of high-resolution, explicit and implicit shock capturing methods, and to illustrate their versatility in various steady and unsteady complex shock waves, perfect gases, equilibrium real gases and nonequilibrium flow computations. These numerical methods are formulated for the purpose of ease and efficient implementation into a practical computer code. The various constructions of high-resolution shock-capturing methods fall nicely into the present framework and a computer code can be implemented with the various methods as separate modules. Included is a systematic overview of the basic design principle of the various related numerical methods. Special emphasis will be on the construction of the basic nonlinear, spatially second and third-order schemes for nonlinear scalar hyperbolic conservation laws and the methods of extending these nonlinear scalar schemes to nonlinear systems via the approximate Riemann solvers and flux-vector splitting approaches. Generalization of these methods to efficiently include real gases and large systems of nonequilibrium flows will be discussed. Some perbolic conservation laws to problems containing stiff source terms and terms and shock waves are also included. The performance of some of these schemes is illustrated by numerical examples for one-, two- and three-dimensional gas-dynamics problems. The use of the Lax-Friedrichs numerical flux to obtain high-resolution shock-capturing schemes is generalized. This method can be extended to nonlinear systems of equations without the use of Riemann solvers or flux-vector splitting approaches and thus provides a large savings for multidimensional, equilibrium real gases and nonequilibrium flow computations.

On the Critical Behaviour, Crossover Point and Complexity of the Exact Cover Problem

NASA Technical Reports Server (NTRS)

Morris, Robin D.; Smelyanskiy, Vadim N.; Shumow, Daniel; Koga, Dennis (Technical Monitor)

2003-01-01

Research into quantum algorithms for NP-complete problems has rekindled interest in the detailed study a broad class of combinatorial problems. A recent paper applied the quantum adiabatic evolution algorithm to the Exact Cover problem for 3-sets (EC3), and provided an empirical evidence that the algorithm was polynomial. In this paper we provide a detailed study of the characteristics of the exact cover problem. We present the annealing approximation applied to EC3, which gives an over-estimate of the phase transition point. We also identify empirically the phase transition point. We also study the complexity of two classical algorithms on this problem: Davis-Putnam and Simulated Annealing. For these algorithms, EC3 is significantly easier than 3-SAT.

Riemann-Liouville Fractional Calculus of Certain Finite Class of Classical Orthogonal Polynomials

NASA Astrophysics Data System (ADS)

Malik, Pradeep; Swaminathan, A.

2010-11-01

In this work we consider certain class of classical orthogonal polynomials defined on the positive real line. These polynomials have their weight function related to the probability density function of F distribution and are finite in number up to orthogonality. We generalize these polynomials for fractional order by considering the Riemann-Liouville type operator on these polynomials. Various properties like explicit representation in terms of hypergeometric functions, differential equations, recurrence relations are derived.

Aspects of general higher-order gravities

NASA Astrophysics Data System (ADS)

Bueno, Pablo; Cano, Pablo A.; Min, Vincent S.; Visser, Manus R.

2017-02-01

We study several aspects of higher-order gravities constructed from general contractions of the Riemann tensor and the metric in arbitrary dimensions. First, we use the fast-linearization procedure presented in [P. Bueno and P. A. Cano, arXiv:1607.06463] to obtain the equations satisfied by the metric perturbation modes on a maximally symmetric background in the presence of matter and to classify L (Riemann ) theories according to their spectrum. Then, we linearize all theories up to quartic order in curvature and use this result to construct quartic versions of Einsteinian cubic gravity. In addition, we show that the most general cubic gravity constructed in a dimension-independent way and which does not propagate the ghostlike spin-2 mode (but can propagate the scalar) is a linear combination of f (Lovelock ) invariants, plus the Einsteinian cubic gravity term, plus a new ghost-free gravity term. Next, we construct the generalized Newton potential and the post-Newtonian parameter γ for general L (Riemann ) gravities in arbitrary dimensions, unveiling some interesting differences with respect to the four-dimensional case. We also study the emission and propagation of gravitational radiation from sources for these theories in four dimensions, providing a generalized formula for the power emitted. Finally, we review Wald's formalism for general L (Riemann ) theories and construct new explicit expressions for the relevant quantities involved. Many examples illustrate our calculations.

Structure of the reconnection layer and the associated slow shocks: Two-dimensional simulations of a Riemann problem

NASA Astrophysics Data System (ADS)

Cremer, Michael; Scholer, Manfred

2000-12-01

The kinetic structure of the reconnection layer in the magnetotail is investigated by two-dimensional hybrid simulations. As a proxy, the solution of the Riemann problem of the collapse of a current sheet with a normal magnetic field component is considered for two cases of the plasma beta (particle to magnetic field pressure): β=0.02 and β=0.002. The collapse results in an expanding layer of compressed and heated plasma, which is accelerated up to the Alfvén speed vA. The boundary layer separating this hot reconnection like layer from the cold lobe plasma is characterized by a beam of back-streaming ions with a field-aligned bulk speed of ~=2vA relative to the cold lobe ion population at rest. As a consequence, obliquely propagating waves are excited via the electromagnetic ion/ion cyclotron instability, which led to perpendicular heating of the ions in the boundary layer as well as further outside the layer in the lobe. In both regions, waves are found which propagate almost parallel to the magnetic field and which are identified as Alfvén ion cyclotron (AIC) waves. These waves are excited by the temperature anisotropy instability. The temperature anisotropy increases with decreasing plasma beta. Thus the anisotropy threshold of the instability is exceeded even in the case of a rather small beta value. The AIC waves, when convected downstream of what can be defined as the the slow shock, make an important contribution to the ion thermalization process. More detailed information on the dissipation process in the slow shocks is gained by analyzing individual ion trajectories.

A nominally second-order cell-centered Lagrangian scheme for simulating elastic–plastic flows on two-dimensional unstructured grids

DOE Office of Scientific and Technical Information (OSTI.GOV)

Maire, Pierre-Henri, E-mail: maire@celia.u-bordeaux1.fr; Abgrall, Rémi, E-mail: remi.abgrall@math.u-bordeau1.fr; Breil, Jérôme, E-mail: breil@celia.u-bordeaux1.fr

2013-02-15

In this paper, we describe a cell-centered Lagrangian scheme devoted to the numerical simulation of solid dynamics on two-dimensional unstructured grids in planar geometry. This numerical method, utilizes the classical elastic-perfectly plastic material model initially proposed by Wilkins [M.L. Wilkins, Calculation of elastic–plastic flow, Meth. Comput. Phys. (1964)]. In this model, the Cauchy stress tensor is decomposed into the sum of its deviatoric part and the thermodynamic pressure which is defined by means of an equation of state. Regarding the deviatoric stress, its time evolution is governed by a classical constitutive law for isotropic material. The plasticity model employs themore » von Mises yield criterion and is implemented by means of the radial return algorithm. The numerical scheme relies on a finite volume cell-centered method wherein numerical fluxes are expressed in terms of sub-cell force. The generic form of the sub-cell force is obtained by requiring the scheme to satisfy a semi-discrete dissipation inequality. Sub-cell force and nodal velocity to move the grid are computed consistently with cell volume variation by means of a node-centered solver, which results from total energy conservation. The nominally second-order extension is achieved by developing a two-dimensional extension in the Lagrangian framework of the Generalized Riemann Problem methodology, introduced by Ben-Artzi and Falcovitz [M. Ben-Artzi, J. Falcovitz, Generalized Riemann Problems in Computational Fluid Dynamics, Cambridge Monogr. Appl. Comput. Math. (2003)]. Finally, the robustness and the accuracy of the numerical scheme are assessed through the computation of several test cases.« less

Faster than classical quantum algorithm for dense formulas of exact satisfiability and occupation problems

NASA Astrophysics Data System (ADS)

Mandrà, Salvatore; Giacomo Guerreschi, Gian; Aspuru-Guzik, Alán

2016-07-01

We present an exact quantum algorithm for solving the Exact Satisfiability problem, which belongs to the important NP-complete complexity class. The algorithm is based on an intuitive approach that can be divided into two parts: the first step consists in the identification and efficient characterization of a restricted subspace that contains all the valid assignments of the Exact Satisfiability; while the second part performs a quantum search in such restricted subspace. The quantum algorithm can be used either to find a valid assignment (or to certify that no solution exists) or to count the total number of valid assignments. The query complexities for the worst-case are respectively bounded by O(\\sqrt{{2}n-{M\\prime }}) and O({2}n-{M\\prime }), where n is the number of variables and {M}\\prime the number of linearly independent clauses. Remarkably, the proposed quantum algorithm results to be faster than any known exact classical algorithm to solve dense formulas of Exact Satisfiability. As a concrete application, we provide the worst-case complexity for the Hamiltonian cycle problem obtained after mapping it to a suitable Occupation problem. Specifically, we show that the time complexity for the proposed quantum algorithm is bounded by O({2}n/4) for 3-regular undirected graphs, where n is the number of nodes. The same worst-case complexity holds for (3,3)-regular bipartite graphs. As a reference, the current best classical algorithm has a (worst-case) running time bounded by O({2}31n/96). Finally, when compared to heuristic techniques for Exact Satisfiability problems, the proposed quantum algorithm is faster than the classical WalkSAT and Adiabatic Quantum Optimization for random instances with a density of constraints close to the satisfiability threshold, the regime in which instances are typically the hardest to solve. The proposed quantum algorithm can be straightforwardly extended to the generalized version of the Exact Satisfiability known as Occupation problem. The general version of the algorithm is presented and analyzed.

On metrics and super-Riemann surfaces

NASA Astrophysics Data System (ADS)

Hodgkin, Luke

1987-08-01

It is shown that any super-Riemann surface M admits a large space of metrics (in a rather basic sense); while if M is of compact genus g type, g>1, M admits a unique metric whose lift to the universal cover is superconformally equivalent to the standard (Baranov-Shvarts) metric on the super-half plane. This explains the relation between the different methods of calculation of the upper Teichmüller space by the author (using arbitrary superconformal transformations) and Crane and Rabin (using only isometries).

New gravitational solutions via a Riemann-Hilbert approach

NASA Astrophysics Data System (ADS)

Cardoso, G. L.; Serra, J. C.

2018-03-01

We consider the Riemann-Hilbert factorization approach to solving the field equations of dimensionally reduced gravity theories. First we prove that functions belonging to a certain class possess a canonical factorization due to properties of the underlying spectral curve. Then we use this result, together with appropriate matricial decompositions, to study the canonical factorization of non-meromorphic monodromy matrices that describe deformations of seed monodromy matrices associated with known solutions. This results in new solutions, with unusual features, to the field equations.

Benchmark problems and solutions

NASA Technical Reports Server (NTRS)

Tam, Christopher K. W.

1995-01-01

The scientific committee, after careful consideration, adopted six categories of benchmark problems for the workshop. These problems do not cover all the important computational issues relevant to Computational Aeroacoustics (CAA). The deciding factor to limit the number of categories to six was the amount of effort needed to solve these problems. For reference purpose, the benchmark problems are provided here. They are followed by the exact or approximate analytical solutions. At present, an exact solution for the Category 6 problem is not available.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Doebling, Scott William

This paper documents the escape of high explosive (HE) products problem. The problem, first presented by Fickett & Rivard, tests the implementation and numerical behavior of a high explosive detonation and energy release model and its interaction with an associated compressible hydrodynamics simulation code. The problem simulates the detonation of a finite-length, one-dimensional piece of HE that is driven by a piston from one end and adjacent to a void at the other end. The HE equation of state is modeled as a polytropic ideal gas. The HE detonation is assumed to be instantaneous with an infinitesimal reaction zone. Viamore » judicious selection of the material specific heat ratio, the problem has an exact solution with linear characteristics, enabling a straightforward calculation of the physical variables as a function of time and space. Lastly, implementation of the exact solution in the Python code ExactPack is discussed, as are verification cases for the exact solution code.« less

Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis

NASA Astrophysics Data System (ADS)

Jiao, Yujian; Wang, Li-Lian; Huang, Can

2016-01-01

The purpose of this paper is twofold. Firstly, we provide explicit and compact formulas for computing both Caputo and (modified) Riemann-Liouville (RL) fractional pseudospectral differentiation matrices (F-PSDMs) of any order at general Jacobi-Gauss-Lobatto (JGL) points. We show that in the Caputo case, it suffices to compute F-PSDM of order μ ∈ (0 , 1) to compute that of any order k + μ with integer k ≥ 0, while in the modified RL case, it is only necessary to evaluate a fractional integral matrix of order μ ∈ (0 , 1). Secondly, we introduce suitable fractional JGL Birkhoff interpolation problems leading to new interpolation polynomial basis functions with remarkable properties: (i) the matrix generated from the new basis yields the exact inverse of F-PSDM at "interior" JGL points; (ii) the matrix of the highest fractional derivative in a collocation scheme under the new basis is diagonal; and (iii) the resulted linear system is well-conditioned in the Caputo case, while in the modified RL case, the eigenvalues of the coefficient matrix are highly concentrated. In both cases, the linear systems of the collocation schemes using the new basis can be solved by an iterative solver within a few iterations. Notably, the inverse can be computed in a very stable manner, so this offers optimal preconditioners for usual fractional collocation methods for fractional differential equations (FDEs). It is also noteworthy that the choice of certain special JGL points with parameters related to the order of the equations can ease the implementation. We highlight that the use of the Bateman's fractional integral formulas and fast transforms between Jacobi polynomials with different parameters, is essential for our algorithm development.

An entropy maximization problem related to optical communication

NASA Technical Reports Server (NTRS)

Mceliece, R. J.; Rodemich, E. R.; Swanson, L.

1986-01-01

In relation to a problem in optical communication, the paper considers the general problem of maximizing the entropy of a stationary radom process that is subject to an average transition cost constraint. By using a recent result of Justesen and Hoholdt, an exact solution to the problem is presented and a class of finite state encoders that give a good approximation to the exact solution is suggested.

Exact Magnetic Diffusion Solutions for Magnetohydrodynamic Code Verification

DOE Office of Scientific and Technical Information (OSTI.GOV)

Miller, D S

In this paper, the authors present several new exact analytic space and time dependent solutions to the problem of magnetic diffusion in R-Z geometry. These problems serve to verify several different elements of an MHD implementation: magnetic diffusion, external circuit time integration, current and voltage energy sources, spatially dependent conductivities, and ohmic heating. The exact solutions are shown in comparison with 2D simulation results from the Ares code.

Elementary solutions of coupled model equations in the kinetic theory of gases

NASA Technical Reports Server (NTRS)

Kriese, J. T.; Siewert, C. E.; Chang, T. S.

1974-01-01

The method of elementary solutions is employed to solve two coupled integrodifferential equations sufficient for determining temperature-density effects in a linearized BGK model in the kinetic theory of gases. Full-range completeness and orthogonality theorems are proved for the developed normal modes and the infinite-medium Green's function is constructed as an illustration of the full-range formalism. The appropriate homogeneous matrix Riemann problem is discussed, and half-range completeness and orthogonality theorems are proved for a certain subset of the normal modes. The required existence and uniqueness theorems relevant to the H matrix, basic to the half-range analysis, are proved, and an accurate and efficient computational method is discussed. The half-space temperature-slip problem is solved analytically, and a highly accurate value of the temperature-slip coefficient is reported.

Numerical methods for systems of conservation laws of mixed type using flux splitting

NASA Technical Reports Server (NTRS)

Shu, Chi-Wang

1990-01-01

The essentially non-oscillatory (ENO) finite difference scheme is applied to systems of conservation laws of mixed hyperbolic-elliptic type. A flux splitting, with the corresponding Jacobi matrices having real and positive/negative eigenvalues, is used. The hyperbolic ENO operator is applied separately. The scheme is numerically tested on the van der Waals equation in fluid dynamics. Convergence was observed with good resolution to weak solutions for various Riemann problems, which are then numerically checked to be admissible as the viscosity-capillarity limits. The interesting phenomena of the shrinking of elliptic regions if they are present in the initial conditions were also observed.

A new approach for solving the three-dimensional steady Euler equations. I - General theory

NASA Technical Reports Server (NTRS)

Chang, S.-C.; Adamczyk, J. J.

1986-01-01

The present iterative procedure combines the Clebsch potentials and the Munk-Prim (1947) substitution principle with an extension of a semidirect Cauchy-Riemann solver to three dimensions, in order to solve steady, inviscid three-dimensional rotational flow problems in either subsonic or incompressible flow regimes. This solution procedure can be used, upon discretization, to obtain inviscid subsonic flow solutions in a 180-deg turning channel. In addition to accurately predicting the behavior of weak secondary flows, the algorithm can generate solutions for strong secondary flows and will yield acceptable flow solutions after only 10-20 outer loop iterations.

Numerical solution of 3D Navier-Stokes equations with upwind implicit schemes

NASA Technical Reports Server (NTRS)

Marx, Yves P.

1990-01-01

An upwind MUSCL type implicit scheme for the three-dimensional Navier-Stokes equations is presented. Comparison between different approximate Riemann solvers (Roe and Osher) are performed and the influence of the reconstructions schemes on the accuracy of the solution as well as on the convergence of the method is studied. A new limiter is introduced in order to remove the problems usually associated with non-linear upwind schemes. The implementation of a diagonal upwind implicit operator for the three-dimensional Navier-Stokes equations is also discussed. Finally the turbulence modeling is assessed. Good prediction of separated flows are demonstrated if a non-equilibrium turbulence model is used.

A new approach for solving the three-dimensional steady Euler equations. I - General theory

NASA Astrophysics Data System (ADS)

Chang, S.-C.; Adamczyk, J. J.

1986-08-01

The present iterative procedure combines the Clebsch potentials and the Munk-Prim (1947) substitution principle with an extension of a semidirect Cauchy-Riemann solver to three dimensions, in order to solve steady, inviscid three-dimensional rotational flow problems in either subsonic or incompressible flow regimes. This solution procedure can be used, upon discretization, to obtain inviscid subsonic flow solutions in a 180-deg turning channel. In addition to accurately predicting the behavior of weak secondary flows, the algorithm can generate solutions for strong secondary flows and will yield acceptable flow solutions after only 10-20 outer loop iterations.

General phase transition models for vehicular traffic with point constraints on the flow

NASA Astrophysics Data System (ADS)

Dal Santo, E.; Rosini, M. D.; Dymski, N.; Benyahia, M.

2017-12-01

We generalize the phase transition model studied in [R. Colombo. Hyperbolic phase transition in traffic flow.\\ SIAM J.\\ Appl.\\ Math., 63(2):708-721, 2002], that describes the evolution of vehicular traffic along a one-lane road. Two different phases are taken into account, according to whether the traffic is low or heavy. The model is given by a scalar conservation law in the \\emph{free-flow} phase and by a system of two conservation laws in the \\emph{congested} phase. In particular, we study the resulting Riemann problems in the case a local point constraint on the flux of the solutions is enforced.

Shocks, Rarefaction Waves, and Current Fluctuations for Anharmonic Chains

DOE PAGES

Mendl, Christian B.; Spohn, Herbert

2016-10-04

The nonequilibrium dynamics of anharmonic chains is studied by imposing an initial domain-wall state, in which the two half lattices are prepared in equilibrium with distinct parameters. Here, we analyse the Riemann problem for the corresponding Euler equations and, in specific cases, compare with molecular dynamics. Additionally, the fluctuations of time-integrated currents are investigated. In analogy with the KPZ equation, their typical fluctuations should be of size t 1/3 and have a Tracy–Widom GUE distributed amplitude. The proper extension to anharmonic chains is explained and tested through molecular dynamics. Our results are calibrated against the stochastic LeRoux lattice gas.

Sign changes in sums of the Liouville function

NASA Astrophysics Data System (ADS)

Borwein, Peter; Ferguson, Ron; Mossinghoff, Michael J.

2008-09-01

The Liouville function λ(n) is the completely multiplicative function whose value is -1 at each prime. We develop some algorithms for computing the sum T(n)Dsum_{kD1}^n λ(k)/k , and use these methods to determine the smallest positive integer n where T(n)<0 . This answers a question originating in some work of Turan, who linked the behavior of T(n) to questions about the Riemann zeta function. We also study the problem of evaluating Polya's sum L(n)Dsum_{kD1}^nλ(k) , and we determine some new local extrema for this function, including some new positive values.

Quasi-periodic solutions to the hierarchy of four-component Toda lattices

NASA Astrophysics Data System (ADS)

Wei, Jiao; Geng, Xianguo; Zeng, Xin

2016-08-01

Starting from a discrete 3×3 matrix spectral problem, the hierarchy of four-component Toda lattices is derived by using the stationary discrete zero-curvature equation. Resorting to the characteristic polynomial of the Lax matrix for the hierarchy, we introduce a trigonal curve Km-2 of genus m - 2 and present the related Baker-Akhiezer function and meromorphic function on it. Asymptotic expansions for the Baker-Akhiezer function and meromorphic function are given near three infinite points on the trigonal curve, from which explicit quasi-periodic solutions for the hierarchy of four-component Toda lattices are obtained in terms of the Riemann theta function.

Letter: Modeling reactive shock waves in heterogeneous solids at the continuum level with stochastic differential equations

NASA Astrophysics Data System (ADS)

Kittell, D. E.; Yarrington, C. D.; Lechman, J. B.; Baer, M. R.

2018-05-01

A new paradigm is introduced for modeling reactive shock waves in heterogeneous solids at the continuum level. Inspired by the probability density function methods from turbulent reactive flows, it is hypothesized that the unreacted material microstructures lead to a distribution of heat release rates from chemical reaction. Fluctuations in heat release, rather than velocity, are coupled to the reactive Euler equations which are then solved via the Riemann problem. A numerically efficient, one-dimensional hydrocode is used to demonstrate this new approach, and simulation results of a representative impact calculation (inert flyer into explosive target) are discussed.

From free fields to AdS space. II

NASA Astrophysics Data System (ADS)

Gopakumar, Rajesh

2004-07-01

We continue with the program of paper I [Phys. Rev. D 70, 025009 (2004)] to implement open-closed string duality on free gauge field theory (in the large-N limit). In this paper we consider correlators such as <∏ni=1TrΦJi(xi)>. The Schwinger parametrization of this n-point function exhibits a partial gluing up into a set of basic skeleton graphs. We argue that the moduli space of the planar skeleton graphs is exactly the same as the moduli space of genus zero Riemann surfaces with n holes. In other words, we can explicitly rewrite the n-point (planar) free-field correlator as an integral over the moduli space of a sphere with n holes. A preliminary study of the integrand also indicates compatibility with a string theory on AdS space. The details of our argument are quite insensitive to the specific form of the operators and generalize to diagrams of a higher genus as well. We take this as evidence of the field theory’s ability to reorganize itself into a string theory.

Closed solutions to a differential-difference equation and an associated plate solidification problem.

PubMed

Layeni, Olawanle P; Akinola, Adegbola P; Johnson, Jesse V

2016-01-01

Two distinct and novel formalisms for deriving exact closed solutions of a class of variable-coefficient differential-difference equations arising from a plate solidification problem are introduced. Thereupon, exact closed traveling wave and similarity solutions to the plate solidification problem are obtained for some special cases of time-varying plate surface temperature.

Solving large-scale PDE-constrained Bayesian inverse problems with Riemann manifold Hamiltonian Monte Carlo

NASA Astrophysics Data System (ADS)

Bui-Thanh, T.; Girolami, M.

2014-11-01

We consider the Riemann manifold Hamiltonian Monte Carlo (RMHMC) method for solving statistical inverse problems governed by partial differential equations (PDEs). The Bayesian framework is employed to cast the inverse problem into the task of statistical inference whose solution is the posterior distribution in infinite dimensional parameter space conditional upon observation data and Gaussian prior measure. We discretize both the likelihood and the prior using the H1-conforming finite element method together with a matrix transfer technique. The power of the RMHMC method is that it exploits the geometric structure induced by the PDE constraints of the underlying inverse problem. Consequently, each RMHMC posterior sample is almost uncorrelated/independent from the others providing statistically efficient Markov chain simulation. However this statistical efficiency comes at a computational cost. This motivates us to consider computationally more efficient strategies for RMHMC. At the heart of our construction is the fact that for Gaussian error structures the Fisher information matrix coincides with the Gauss-Newton Hessian. We exploit this fact in considering a computationally simplified RMHMC method combining state-of-the-art adjoint techniques and the superiority of the RMHMC method. Specifically, we first form the Gauss-Newton Hessian at the maximum a posteriori point and then use it as a fixed constant metric tensor throughout RMHMC simulation. This eliminates the need for the computationally costly differential geometric Christoffel symbols, which in turn greatly reduces computational effort at a corresponding loss of sampling efficiency. We further reduce the cost of forming the Fisher information matrix by using a low rank approximation via a randomized singular value decomposition technique. This is efficient since a small number of Hessian-vector products are required. The Hessian-vector product in turn requires only two extra PDE solves using the adjoint technique. Various numerical results up to 1025 parameters are presented to demonstrate the ability of the RMHMC method in exploring the geometric structure of the problem to propose (almost) uncorrelated/independent samples that are far away from each other, and yet the acceptance rate is almost unity. The results also suggest that for the PDE models considered the proposed fixed metric RMHMC can attain almost as high a quality performance as the original RMHMC, i.e. generating (almost) uncorrelated/independent samples, while being two orders of magnitude less computationally expensive.

Statistical Physics on the Eve of the 21st Century: in Honour of J B McGuire on the Occasion of His 65th Birthday

NASA Astrophysics Data System (ADS)

Batchelor, Murray T.; Wille, Luc T.

The Table of Contents for the book is as follows: * Preface * Modelling the Immune System - An Example of the Simulation of Complex Biological Systems * Brief Overview of Quantum Computation * Quantal Information in Statistical Physics * Modeling Economic Randomness: Statistical Mechanics of Market Phenomena * Essentially Singular Solutions of Feigenbaum- Type Functional Equations * Spatiotemporal Chaotic Dynamics in Coupled Map Lattices * Approach to Equilibrium of Chaotic Systems * From Level to Level in Brain and Behavior * Linear and Entropic Transformations of the Hydrophobic Free Energy Sequence Help Characterize a Novel Brain Polyprotein: CART's Protein * Dynamical Systems Response to Pulsed High-Frequency Fields * Bose-Einstein Condensates in the Light of Nonlinear Physics * Markov Superposition Expansion for the Entropy and Correlation Functions in Two and Three Dimensions * Calculation of Wave Center Deflection and Multifractal Analysis of Directed Waves Through the Study of su(1,1)Ferromagnets * Spectral Properties and Phases in Hierarchical Master Equations * Universality of the Distribution Functions of Random Matrix Theory * The Universal Chiral Partition Function for Exclusion Statistics * Continuous Space-Time Symmetries in a Lattice Field Theory * Quelques Cas Limites du Problème à N Corps Unidimensionnel * Integrable Models of Correlated Electrons * On the Riemann Surface of the Three-State Chiral Potts Model * Two Exactly Soluble Lattice Models in Three Dimensions * Competition of Ferromagnetic and Antiferromagnetic Order in the Spin-l/2 XXZ Chain at Finite Temperature * Extended Vertex Operator Algebras and Monomial Bases * Parity and Charge Conjugation Symmetries and S Matrix of the XXZ Chain * An Exactly Solvable Constrained XXZ Chain * Integrable Mixed Vertex Models Ftom the Braid-Monoid Algebra * From Yang-Baxter Equations to Dynamical Zeta Functions for Birational Tlansformations * Hexagonal Lattice Directed Site Animals * Direction in the Star-Triangle Relations * A Self-Avoiding Walk Through Exactly Solved Lattice Models in Statistical Mechanics

Matrix theory interpretation of discrete light cone quantization string worldsheets

PubMed

Grignani; Orland; Paniak; Semenoff

2000-10-16

We study the null compactification of type-IIA string perturbation theory at finite temperature. We prove a theorem about Riemann surfaces establishing that the moduli spaces of infinite-momentum-frame superstring worldsheets are identical to those of branched-cover instantons in the matrix-string model conjectured to describe M theory. This means that the identification of string degrees of freedom in the matrix model proposed by Dijkgraaf, Verlinde, and Verlinde is correct and that its natural generalization produces the moduli space of Riemann surfaces at all orders in the genus expansion.

Causality and a -theorem constraints on Ricci polynomial and Riemann cubic gravities

NASA Astrophysics Data System (ADS)

Li, Yue-Zhou; Lü, H.; Wu, Jun-Bao

2018-01-01

In this paper, we study Einstein gravity extended with Ricci polynomials and derive the constraints on the coupling constants from the considerations of being ghost-free, exhibiting an a -theorem and maintaining causality. The salient feature is that Einstein metrics with appropriate effective cosmological constants continue to be solutions with the inclusion of such Ricci polynomials and the causality constraint is automatically satisfied. The ghost-free and a -theorem conditions can only be both met starting at the quartic order. We also study these constraints on general Riemann cubic gravities.

Isomonodromy for the Degenerate Fifth Painlevé Equation

NASA Astrophysics Data System (ADS)

Acosta-Humánez, Primitivo B.; van der Put, Marius; Top, Jaap

2017-05-01

This is a sequel to papers by the last two authors making the Riemann-Hilbert correspondence and isomonodromy explicit. For the degenerate fifth Painlevé equation, the moduli spaces for connections and for monodromy are explicitly computed. It is proven that the extended Riemann-Hilbert morphism is an isomorphism. As a consequence these equations have the Painlevé property and the Okamoto-Painlevé space is identified with a moduli space of connections. Using MAPLE computations, one obtains formulas for the degenerate fifth Painlevé equation, for the Bäcklund transformations.

Solving Integer Programs from Dependence and Synchronization Problems

DTIC Science & Technology

1993-03-01

DEFF.NSNE Solving Integer Programs from Dependence and Synchronization Problems Jaspal Subhlok March 1993 CMU-CS-93-130 School of Computer ScienceT IC...method Is an exact and efficient way of solving integer programming problems arising in dependence and synchronization analysis of parallel programs...7/;- p Keywords: Exact dependence tesing, integer programming. parallelilzng compilers, parallel program analysis, synchronization analysis Solving

Exact solution of large asymmetric traveling salesman problems.

PubMed

Miller, D L; Pekny, J F

1991-02-15

The traveling salesman problem is one of a class of difficult problems in combinatorial optimization that is representative of a large number of important scientific and engineering problems. A survey is given of recent applications and methods for solving large problems. In addition, an algorithm for the exact solution of the asymmetric traveling salesman problem is presented along with computational results for several classes of problems. The results show that the algorithm performs remarkably well for some classes of problems, determining an optimal solution even for problems with large numbers of cities, yet for other classes, even small problems thwart determination of a provably optimal solution.

Can superhorizon cosmological perturbations explain the acceleration of the universe?

NASA Astrophysics Data System (ADS)

Hirata, Christopher M.; Seljak, Uroš

2005-10-01

We investigate the recent suggestions by Barausse et al. and Kolb et al. that the acceleration of the universe could be explained by large superhorizon fluctuations generated by inflation. We show that no acceleration can be produced by this mechanism. We begin by showing how the application of Raychaudhuri equation to inhomogeneous cosmologies results in several “no go” theorems for accelerated expansion. Next we derive an exact solution for a specific case of initial perturbations, for which application of the Kolb et al. expressions leads to an acceleration, while the exact solution reveals that no acceleration is present. We show that the discrepancy can be traced to higher-order terms that were dropped in the Kolb et al. analysis. We proceed with the analysis of initial value formulation of general relativity to argue that causality severely limits what observable effects can be derived from superhorizon perturbations. By constructing a Riemann normal coordinate system on initial slice we show that no infrared divergence terms arise in this coordinate system. Thus any divergences found previously can be eliminated by a local rescaling of coordinates and are unobservable. We perform an explicit analysis of the variance of the deceleration parameter for the case of single-field inflation using usual coordinates and show that the infrared-divergent terms found by Barausse et al. and Kolb et al. cancel against several additional terms not considered in their analysis. Finally, we argue that introducing isocurvature perturbations does not alter our conclusion that the accelerating expansion of the universe cannot be explained by superhorizon modes.

Exact analytical solution of a classical Josephson tunnel junction problem

NASA Astrophysics Data System (ADS)

Kuplevakhsky, S. V.; Glukhov, A. M.

2010-10-01

We give an exact and complete analytical solution of the classical problem of a Josephson tunnel junction of arbitrary length W ɛ(0,∞) in the presence of external magnetic fields and transport currents. Contrary to a wide-spread belief, the exact analytical solution unambiguously proves that there is no qualitative difference between so-called "small" (W≪1) and "large" junctions (W≫1). Another unexpected physical implication of the exact analytical solution is the existence (in the current-carrying state) of unquantized Josephson vortices carrying fractional flux and located near one of the edges of the junction. We also refine the mathematical definition of critical transport current.

Existence of solution for the problem with a concentrated source in a subdiffusive medium

NASA Astrophysics Data System (ADS)

Liu, H. Terence; Huang, Wei-Cheng

2018-01-01

Let 0 < α < 1, b, T be positive real numbers, Lau =ut-(Dt1-αu ) x x , where Dt1-αu denotes the Riemann-Liouville fractional derivative. This paper consider the problem Lau (x ,t )=δ (x -b )f (u (x ,t ))in (-∞ ,∞ )×(0 ,T ], subject to initial and boundaries condition u (x ,0 )=ϕ (x )in(-∞ ,∞ ),with ϕ (x )→as|x |→∞ u (x ,t )→0 for0 0, f″(u) > 0 for u > 0. By using Green's function, the problem is converted into an integral equation. It is shown that there exists tb such that for 0 ≤ t < tb, the integral equation has a unique nonnegative continuous solution u; if tb is finite, then u is unbounded in [0, tb). Then, u is proved to be the solution of the original problem.

Nonlinear Multidimensional Assignment Problems Efficient Conic Optimization Methods and Applications

DTIC Science & Technology

2015-06-24

WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Arizona State University School of Mathematical & Statistical Sciences 901 S...SUPPLEMENTARY NOTES 14. ABSTRACT The major goals of this project were completed: the exact solution of previously unsolved challenging combinatorial optimization... combinatorial optimization problem, the Directional Sensor Problem, was solved in two ways. First, heuristically in an engineering fashion and second, exactly

Exact solutions for species tree inference from discordant gene trees.

PubMed

Chang, Wen-Chieh; Górecki, Paweł; Eulenstein, Oliver

2013-10-01

Phylogenetic analysis has to overcome the grant challenge of inferring accurate species trees from evolutionary histories of gene families (gene trees) that are discordant with the species tree along whose branches they have evolved. Two well studied approaches to cope with this challenge are to solve either biologically informed gene tree parsimony (GTP) problems under gene duplication, gene loss, and deep coalescence, or the classic RF supertree problem that does not rely on any biological model. Despite the potential of these problems to infer credible species trees, they are NP-hard. Therefore, these problems are addressed by heuristics that typically lack any provable accuracy and precision. We describe fast dynamic programming algorithms that solve the GTP problems and the RF supertree problem exactly, and demonstrate that our algorithms can solve instances with data sets consisting of as many as 22 taxa. Extensions of our algorithms can also report the number of all optimal species trees, as well as the trees themselves. To better asses the quality of the resulting species trees that best fit the given gene trees, we also compute the worst case species trees, their numbers, and optimization score for each of the computational problems. Finally, we demonstrate the performance of our exact algorithms using empirical and simulated data sets, and analyze the quality of heuristic solutions for the studied problems by contrasting them with our exact solutions.

Exact Green's function method of solar force-free magnetic-field computations with constant alpha. I - Theory and basic test cases

NASA Technical Reports Server (NTRS)

Chiu, Y. T.; Hilton, H. H.

1977-01-01

Exact closed-form solutions to the solar force-free magnetic-field boundary-value problem are obtained for constant alpha in Cartesian geometry by a Green's function approach. The uniqueness of the physical problem is discussed. Application of the exact results to practical solar magnetic-field calculations is free of series truncation errors and is at least as economical as the approximate methods currently in use. Results of some test cases are presented.

Exact solution for spin precession in the radiationless relativistic Kepler problem

NASA Astrophysics Data System (ADS)

Mane, S. R.

2014-11-01

There is interest in circulating beams of polarized particles in all-electric storage rings to search for nonzero permanent electric dipole moments of subatomic particles. To this end, it is helpful to derive exact analytical solutions of the spin precession in idealized models, both for pedagogical reasons and to serve as benchmark tests for analysis and design of experiments. This paper derives exact solutions for the spin precession in the relativistic Kepler problem. Some counterintuitive properties of the solutions are pointed out.

Efficient numerical simulation of non-integer-order space-fractional reaction-diffusion equation via the Riemann-Liouville operator

NASA Astrophysics Data System (ADS)

Owolabi, Kolade M.

2018-03-01

In this work, we are concerned with the solution of non-integer space-fractional reaction-diffusion equations with the Riemann-Liouville space-fractional derivative in high dimensions. We approximate the Riemann-Liouville derivative with the Fourier transform method and advance the resulting system in time with any time-stepping solver. In the numerical experiments, we expect the travelling wave to arise from the given initial condition on the computational domain (-∞, ∞), which we terminate in the numerical experiments with a large but truncated value of L. It is necessary to choose L large enough to allow the waves to have enough space to distribute. Experimental results in high dimensions on the space-fractional reaction-diffusion models with applications to biological models (Fisher and Allen-Cahn equations) are considered. Simulation results reveal that fractional reaction-diffusion equations can give rise to a range of physical phenomena when compared to non-integer-order cases. As a result, most meaningful and practical situations are found to be modelled with the concept of fractional calculus.

Weyl geometry

NASA Astrophysics Data System (ADS)

Wheeler, James T.

2018-07-01

We develop the properties of Weyl geometry, beginning with a review of the conformal properties of Riemannian spacetimes. Decomposition of the Riemann curvature into trace and traceless parts allows an easy proof that the Weyl curvature tensor is the conformally invariant part of the Riemann curvature, and shows the explicit change in the Ricci and Schouten tensors required to insure conformal invariance. We include a proof of the well-known condition for the existence of a conformal transformation to a Ricci-flat spacetime. We generalize this to a derivation of the condition for the existence of a conformal transformation to a spacetime satisfying the Einstein equation with matter sources. Then, enlarging the symmetry from Poincaré to Weyl, we develop the Cartan structure equations of Weyl geometry, the form of the curvature tensor and its relationship to the Riemann curvature of the corresponding Riemannian geometry. We present a simple theory of Weyl-covariant gravity based on a curvature-linear action, and show that it is conformally equivalent to general relativity. This theory is invariant under local dilatations, but not the full conformal group.

A numerical algorithm for MHD of free surface flows at low magnetic Reynolds numbers

NASA Astrophysics Data System (ADS)

Samulyak, Roman; Du, Jian; Glimm, James; Xu, Zhiliang

2007-10-01

We have developed a numerical algorithm and computational software for the study of magnetohydrodynamics (MHD) of free surface flows at low magnetic Reynolds numbers. The governing system of equations is a coupled hyperbolic-elliptic system in moving and geometrically complex domains. The numerical algorithm employs the method of front tracking and the Riemann problem for material interfaces, second order Godunov-type hyperbolic solvers, and the embedded boundary method for the elliptic problem in complex domains. The numerical algorithm has been implemented as an MHD extension of FronTier, a hydrodynamic code with free interface support. The code is applicable for numerical simulations of free surface flows of conductive liquids or weakly ionized plasmas. The code has been validated through the comparison of numerical simulations of a liquid metal jet in a non-uniform magnetic field with experiments and theory. Simulations of the Muon Collider/Neutrino Factory target have also been discussed.

Exact Solution of a Faraday's Law Problem that Includes a Nonlinear Term and Its Implication for Perturbation Theory.

ERIC Educational Resources Information Center

Fulcher, Lewis P.

1979-01-01

Presents an exact solution to the nonlinear Faraday's law problem of a rod sliding on frictionless rails with resistance. Compares the results with perturbation calculations based on the methods of Poisson and Pincare and of Kryloff and Bogoliuboff. (Author/GA)

Raychaudhuri equation in the self-consistent Einstein-Cartan theory with spin-density

NASA Technical Reports Server (NTRS)

Fennelly, A. J.; Krisch, Jean P.; Ray, John R.; Smalley, Larry L.

1988-01-01

The physical implications of the Raychaudhuri equation for a spinning fluid in a Riemann-Cartan spacetime is developed and discussed using the self-consistent Lagrangian based formulation for the Einstein-Cartan theory. It was found that the spin-squared terms contribute to expansion (inflation) at early times and may lead to a bounce in the final collapse. The relationship between the fluid's vorticity and spin angular velocity is clarified and the effect of the interaction terms between the spin angular velocity and the spin in the Raychaudhuri equation investigated. These results should prove useful for studies of systems with an intrinsic spin angular momentum in extreme astrophysical or cosmological problems.

N-body dynamics on closed surfaces: the axioms of mechanics

PubMed Central

Dritschel, David G.; Schaefer, Rodrigo G.

2016-01-01

A major challenge for our understanding of the mathematical basis of particle dynamics is the formulation of N-body and N-vortex dynamics on Riemann surfaces. In this paper, we show how the two problems are, in fact, closely related when considering the role played by the intrinsic geometry of the surface. This enables a straightforward deduction of the dynamics of point masses, using recently derived results for point vortices on general closed differentiable surfaces M endowed with a metric g. We find, generally, that Kepler's Laws do not hold. What is more, even Newton's First Law (the law of inertia) fails on closed surfaces with variable curvature (e.g. the ellipsoid). PMID:27616915

Nonlinear runup resonance in a finite bay of variable parabolic cross-section

NASA Astrophysics Data System (ADS)

Postacioglu, Nazmi; Sinan Özeren, M.

2017-04-01

During the recent years, several interesting studies published on the non-linear runup of Tsunamis and other incident waves in channels and bays. Most of these studies use Riemann invariants to tackle the nonlinearities associate with both the incident and reflected phases. However, all of these studies assume that the waves get generated within the bay (essentially assuming bays extending into infinity), rather than conisdering a wave that has been generated in the open sea. Such waves would be reflected not just by the shoreline but also by the bay mouth. This is a setting where a resonance can be induced within the bay. We investigate this problem nonlinearly with a modal approach.

Quasi-periodic solutions of the Belov-Chaltikian lattice hierarchy

NASA Astrophysics Data System (ADS)

Geng, Xianguo; Zeng, Xin

Utilizing the characteristic polynomial of Lax matrix for the Belov-Chaltikian (BC) lattice hierarchy associated with a 3 × 3 discrete matrix spectral problem, we introduce a trigonal curve with three infinite points, from which we establish the associated Dubrovin-type equations. The essential properties of the Baker-Akhiezer function and the meromorphic function are discussed, that include their asymptotic behavior near three infinite points on the trigonal curve and the divisor of the meromorphic function. The Abel map is introduced to straighten out the continuous flow and the discrete flow in the Jacobian variety, from which the quasi-periodic solutions of the entire BC lattice hierarchy are obtained in terms of the Riemann theta function.

Canonical forms of multidimensional steady inviscid flows

NASA Technical Reports Server (NTRS)

Taasan, Shlomo

1993-01-01

Canonical forms and canonical variables for inviscid flow problems are derived. In these forms the components of the system governed by different types of operators (elliptic and hyperbolic) are separated. Both the incompressible and compressible cases are analyzed, and their similarities and differences are discussed. The canonical forms obtained are block upper triangular operator form in which the elliptic and non-elliptic parts reside in different blocks. The full nonlinear equations are treated without using any linearization process. This form enables a better analysis of the equations as well as better numerical treatment. These forms are the analog of the decomposition of the one dimensional Euler equations into characteristic directions and Riemann invariants.

Type II universal spacetimes

NASA Astrophysics Data System (ADS)

Hervik, S.; Málek, T.; Pravda, V.; Pravdová, A.

2015-12-01

We study type II universal metrics of the Lorentzian signature. These metrics simultaneously solve vacuum field equations of all theories of gravitation with the Lagrangian being a polynomial curvature invariant constructed from the metric, the Riemann tensor and its covariant derivatives of an arbitrary order. We provide examples of type II universal metrics for all composite number dimensions. On the other hand, we have no examples for prime number dimensions and we prove the non-existence of type II universal spacetimes in five dimensions. We also present type II vacuum solutions of selected classes of gravitational theories, such as Lovelock, quadratic and L({{Riemann}}) gravities.

Configuring Airspace Sectors with Approximate Dynamic Programming

NASA Technical Reports Server (NTRS)

Bloem, Michael; Gupta, Pramod

2010-01-01

In response to changing traffic and staffing conditions, supervisors dynamically configure airspace sectors by assigning them to control positions. A finite horizon airspace sector configuration problem models this supervisor decision. The problem is to select an airspace configuration at each time step while considering a workload cost, a reconfiguration cost, and a constraint on the number of control positions at each time step. Three algorithms for this problem are proposed and evaluated: a myopic heuristic, an exact dynamic programming algorithm, and a rollouts approximate dynamic programming algorithm. On problem instances from current operations with only dozens of possible configurations, an exact dynamic programming solution gives the optimal cost value. The rollouts algorithm achieves costs within 2% of optimal for these instances, on average. For larger problem instances that are representative of future operations and have thousands of possible configurations, excessive computation time prohibits the use of exact dynamic programming. On such problem instances, the rollouts algorithm reduces the cost achieved by the heuristic by more than 15% on average with an acceptable computation time.

Riemann-Hypothesis Millennium-Problem(MP) Physics Proof via CATEGORY-SEMANTICS(C-S)/F =C Aristotle SQUARE-of-OPPOSITION(SoO) DEduction-LOGIC DichotomY

NASA Astrophysics Data System (ADS)

Baez, Joao-Joan; Lapidaryus, Michelle; Siegel, Edward Carl-Ludwig

2013-03-01

Riemann-hypothesis physics-proof combines: Siegel-Antono®-Smith[AMS Joint Mtg.(2002)- Abs.973-03-126] digits on-average statistics HIll[Am. J. Math 123, 3, 887(1996)] logarithm-function's (1,0)- xed-point base =units =scale-invariance proven Newcomb [Am. J. Math. 4, 39(1881)]-Weyl[Goett. Nachr.(1914); Math. Ann.7, 313(1916)]-Benford[Proc. Am. Phil. Soc. 78, 4, 51(1938)]-law [Kac,Math. of Stat.-Reasoning(1955); Raimi, Sci. Am. 221, 109(1969)] algebraic-inversion to ONLY Bose-Einstein quantum-statistics(BEQS) with digit d = 0 gapFUL Bose-Einstein Condensation(BEC) insight that digits are quanta are bosons because bosons are and always were quanta are and always were digits, via Siegel-Baez category-semantics tabular list-format matrix truth-table analytics in Plato-Aristotle classic ''square-of-opposition'' : FUZZYICS =CATEGORYICS/Category-Semantics, with Goodkind Bose-Einstein Condensation (BEC) ABOVE ground-state with/and Rayleigh(cut-limit of ''short-cut method''1870)-Polya(1922)-''Anderson''(1958) localization [Doyle and Snell,Random-Walks and Electrical-Networks, MAA(1981)-p.99-100!!!] in Brillouin[Wave-Propagation in Periodic-Structures(1946) Dover(1922)]-Hubbard-Beeby[J.Phys.C(1967)] Siegel[J.Nonxline-Sol.40,453(1980)] generalized-disorder collective-boson negative-dispersion mode-softening universality-principle(G...P) first use of the ``square-of-opposition'' in physics since Plato and Aristote!!!

Linear stability analysis of collective neutrino oscillations without spurious modes

NASA Astrophysics Data System (ADS)

Morinaga, Taiki; Yamada, Shoichi

2018-01-01

Collective neutrino oscillations are induced by the presence of neutrinos themselves. As such, they are intrinsically nonlinear phenomena and are much more complex than linear counterparts such as the vacuum or Mikheyev-Smirnov-Wolfenstein oscillations. They obey integro-differential equations, for which it is also very challenging to obtain numerical solutions. If one focuses on the onset of collective oscillations, on the other hand, the equations can be linearized and the technique of linear analysis can be employed. Unfortunately, however, it is well known that such an analysis, when applied with discretizations of continuous angular distributions, suffers from the appearance of so-called spurious modes: unphysical eigenmodes of the discretized linear equations. In this paper, we analyze in detail the origin of these unphysical modes and present a simple solution to this annoying problem. We find that the spurious modes originate from the artificial production of pole singularities instead of a branch cut on the Riemann surface by the discretizations. The branching point singularities on the Riemann surface for the original nondiscretized equations can be recovered by approximating the angular distributions with polynomials and then performing the integrals analytically. We demonstrate for some examples that this simple prescription does remove the spurious modes. We also propose an even simpler method: a piecewise linear approximation to the angular distribution. It is shown that the same methodology is applicable to the multienergy case as well as to the dispersion relation approach that was proposed very recently.

Brain Surface Conformal Parameterization Using Riemann Surface Structure

PubMed Central

Wang, Yalin; Lui, Lok Ming; Gu, Xianfeng; Hayashi, Kiralee M.; Chan, Tony F.; Toga, Arthur W.; Thompson, Paul M.; Yau, Shing-Tung

2011-01-01

In medical imaging, parameterized 3-D surface models are useful for anatomical modeling and visualization, statistical comparisons of anatomy, and surface-based registration and signal processing. Here we introduce a parameterization method based on Riemann surface structure, which uses a special curvilinear net structure (conformal net) to partition the surface into a set of patches that can each be conformally mapped to a parallelogram. The resulting surface subdivision and the parameterizations of the components are intrinsic and stable (their solutions tend to be smooth functions and the boundary conditions of the Dirichlet problem can be enforced). Conformal parameterization also helps transform partial differential equations (PDEs) that may be defined on 3-D brain surface manifolds to modified PDEs on a two-dimensional parameter domain. Since the Jacobian matrix of a conformal parameterization is diagonal, the modified PDE on the parameter domain is readily solved. To illustrate our techniques, we computed parameterizations for several types of anatomical surfaces in 3-D magnetic resonance imaging scans of the brain, including the cerebral cortex, hippocampi, and lateral ventricles. For surfaces that are topologically homeomorphic to each other and have similar geometrical structures, we show that the parameterization results are consistent and the subdivided surfaces can be matched to each other. Finally, we present an automatic sulcal landmark location algorithm by solving PDEs on cortical surfaces. The landmark detection results are used as constraints for building conformal maps between surfaces that also match explicitly defined landmarks. PMID:17679336

The Two-Capacitor Problem Revisited: A Mechanical Harmonic Oscillator Model Approach

ERIC Educational Resources Information Center

Lee, Keeyung

2009-01-01

The well-known two-capacitor problem, in which exactly half the stored energy disappears when a charged capacitor is connected to an identical capacitor, is discussed based on the mechanical harmonic oscillator model approach. In the mechanical harmonic oscillator model, it is shown first that "exactly half" the work done by a constant applied…

An Exact Solution to the Draining Reservoir Problem of the Incompressible and Non-Viscous Liquid

ERIC Educational Resources Information Center

Hong, Seok-In

2009-01-01

The exact expressions for the drain time and the height, velocity and acceleration of the free surface are found for the draining reservoir problem of the incompressible and non-viscous liquid. Contrary to the conventional approximate results, they correctly describe the initial time dependence of the liquid velocity and acceleration. Torricelli's…

Spatial correlations and exact solution of the problem of the boson peak profile in amorphous media

NASA Astrophysics Data System (ADS)

Kirillov, Sviatoslav A.; A. Voyiatzis, George; Kolomiyets, Tatiana M.; H. Anastasiadis, Spiros

1999-11-01

Based on a model correlation function which covers spatial correlations from Gaussian to exponential, we have arrived at an exact analytic solution of the problem of the Boson peak profile in amorphous media. Probe fits made for polyisoprene and triacetin prove the working ability of the formulae obtained.

Exact Solutions to Time-dependent Mdps

NASA Technical Reports Server (NTRS)

Boyan, Justin A.; Littman, Michael L.

2000-01-01

We describe an extension of the Markov decision process model in which a continuous time dimension is included in the state space. This allows for the representation and exact solution of a wide range of problems in which transitions or rewards vary over time. We examine problems based on route planning with public transportation and telescope observation scheduling.

Exact Analytical Solutions for Elastodynamic Impact

DTIC Science & Technology

2015-11-30

corroborated by derivation of exact discrete solutions from recursive equations for the impact problems. 15. SUBJECT TERMS One-dimensional impact; Elastic...wave propagation; Laplace transform; Floor function; Discrete solutions 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT UU 18...impact Elastic wave propagation Laplace transform Floor function Discrete solutionsWe consider the one-dimensional impact problem in which a semi

Exact Electromagnetic Fields Produced by a Finite Wire with Constant Current

ERIC Educational Resources Information Center

Jimenez, J. L.; Campos, I.; Aquino, N.

2008-01-01

We solve exactly the problem of calculating the electromagnetic fields produced by a finite wire with a constant current, by using two methods: retarded potentials and Jefimenko's formalism. One result in this particular case is that the usual Biot-Savart law of magnetostatics gives the correct magnetic field of the problem. We also show…

Fully- and weakly-nonlinear biperiodic traveling waves in shallow water

NASA Astrophysics Data System (ADS)

Hirakawa, Tomoaki; Okamura, Makoto

2018-04-01

We directly calculate fully nonlinear traveling waves that are periodic in two independent horizontal directions (biperiodic) in shallow water. Based on the Riemann theta function, we also calculate exact periodic solutions to the Kadomtsev-Petviashvili (KP) equation, which can be obtained by assuming weakly-nonlinear, weakly-dispersive, weakly-two-dimensional waves. To clarify how the accuracy of the biperiodic KP solution is affected when some of the KP approximations are not satisfied, we compare the fully- and weakly-nonlinear periodic traveling waves of various wave amplitudes, wave depths, and interaction angles. As the interaction angle θ decreases, the wave frequency and the maximum wave height of the biperiodic KP solution both increase, and the central peak sharpens and grows beyond the height of the corresponding direct numerical solutions, indicating that the biperiodic KP solution cannot qualitatively model direct numerical solutions for θ ≲ 45^\\circ . To remedy the weak two-dimensionality approximation, we apply the correction of Yeh et al (2010 Eur. Phys. J. Spec. Top. 185 97-111) to the biperiodic KP solution, which substantially improves the solution accuracy and results in wave profiles that are indistinguishable from most other cases.

Tinkertoys for the E 7 theory

NASA Astrophysics Data System (ADS)

Chacaltana, Oscar; Distler, Jacques; Trimm, Anderson; Zhu, Yinan

2018-05-01

We classify the class S theories of type E 7. These are four-dimensional N=2 superconformal field theories arising from the compactification of the E 7 (2, 0) theory on a punctured Riemann surface, C. The classification is given by listing all 3-punctured spheres ("fixtures"), and connecting cylinders, which can arise in a pants-decomposition of C. We find exactly 11,000 fixtures with three regular punctures, and an additional 48 with one "irregular puncture" (in the sense used in our previous works). To organize this large number of theories, we have created a web application at https://golem.ph.utexas.edu/class-S/E7/. Among these theories, we find 10 new ones with a simple exceptional global symmetry group, as well as a new rank-2 SCFT and several new rank-3 SCFTs. As an application, we study the strong-coupling limit of the E 7 gauge theory with 3 hypermultiplets in the 56. Using our results, we also verify recent conjectures that the T 2 compactification of certain 6 d (1, 0) theories can alternatively be realized in class S as fixtures in the E 7 or E 8 theories.

Lovelock vacua with a recurrent null vector field

NASA Astrophysics Data System (ADS)

Ortaggio, Marcello

2018-02-01

Vacuum solutions of Lovelock gravity in the presence of a recurrent null vector field (a subset of Kundt spacetimes) are studied. We first discuss the general field equations, which constrain both the base space and the profile functions. While choosing a "generic" base space puts stronger constraints on the profile, in special cases there also exist solutions containing arbitrary functions (at least for certain values of the coupling constants). These and other properties (such as the p p - waves subclass and the overlap with VSI, CSI and universal spacetimes) are subsequently analyzed in more detail in lower dimensions n =5 , 6 as well as for particular choices of the base manifold. The obtained solutions describe various classes of nonexpanding gravitational waves propagating, e.g., in Nariai-like backgrounds M2×Σn -2. An Appendix contains some results about general (i.e., not necessarily Kundt) Lovelock vacua of Riemann type III/N and of Weyl and traceless-Ricci type III/N. For example, it is pointed out that for theories admitting a triply degenerate maximally symmetric vacuum, all the (reduced) field equations are satisfied identically, giving rise to large classes of exact solutions.

New definition of complexity for self-gravitating fluid distributions: The spherically symmetric, static case

NASA Astrophysics Data System (ADS)

Herrera, L.

2018-02-01

We put forward a new definition of complexity, for static and spherically symmetric self-gravitating systems, based on a quantity, hereafter referred to as complexity factor, that appears in the orthogonal splitting of the Riemann tensor, in the context of general relativity. We start by assuming that the homogeneous (in the energy density) fluid, with isotropic pressure is endowed with minimal complexity. For this kind of fluid distribution, the value of complexity factor is zero. So, the rationale behind our proposal for the definition of complexity factor stems from the fact that it measures the departure, in the value of the active gravitational mass (Tolman mass), with respect to its value for a zero complexity system. Such departure is produced by a specific combination of energy density inhomogeneity and pressure anisotropy. Thus, zero complexity factor may also be found in self-gravitating systems with inhomogeneous energy density and anisotropic pressure, provided the effects of these two factors, on the complexity factor, cancel each other. Some exact interior solutions to the Einstein equations satisfying the zero complexity criterium are found, and prospective applications of this newly defined concept, to the study of the structure and evolution of compact objects, are discussed.

Numerical treatment for solving two-dimensional space-fractional advection-dispersion equation using meshless method

NASA Astrophysics Data System (ADS)

Cheng, Rongjun; Sun, Fengxin; Wei, Qi; Wang, Jufeng

2018-02-01

Space-fractional advection-dispersion equation (SFADE) can describe particle transport in a variety of fields more accurately than the classical models of integer-order derivative. Because of nonlocal property of integro-differential operator of space-fractional derivative, it is very challenging to deal with fractional model, and few have been reported in the literature. In this paper, a numerical analysis of the two-dimensional SFADE is carried out by the element-free Galerkin (EFG) method. The trial functions for the SFADE are constructed by the moving least-square (MLS) approximation. By the Galerkin weak form, the energy functional is formulated. Employing the energy functional minimization procedure, the final algebraic equations system is obtained. The Riemann-Liouville operator is discretized by the Grünwald formula. With center difference method, EFG method and Grünwald formula, the fully discrete approximation schemes for SFADE are established. Comparing with exact results and available results by other well-known methods, the computed approximate solutions are presented in the format of tables and graphs. The presented results demonstrate the validity, efficiency and accuracy of the proposed techniques. Furthermore, the error is computed and the proposed method has reasonable convergence rates in spatial and temporal discretizations.

AN EXTENSION OF THE ATHENA++ CODE FRAMEWORK FOR GRMHD BASED ON ADVANCED RIEMANN SOLVERS AND STAGGERED-MESH CONSTRAINED TRANSPORT

DOE Office of Scientific and Technical Information (OSTI.GOV)

White, Christopher J.; Stone, James M.; Gammie, Charles F.

2016-08-01

We present a new general relativistic magnetohydrodynamics (GRMHD) code integrated into the Athena++ framework. Improving upon the techniques used in most GRMHD codes, ours allows the use of advanced, less diffusive Riemann solvers, in particular HLLC and HLLD. We also employ a staggered-mesh constrained transport algorithm suited for curvilinear coordinate systems in order to maintain the divergence-free constraint of the magnetic field. Our code is designed to work with arbitrary stationary spacetimes in one, two, or three dimensions, and we demonstrate its reliability through a number of tests. We also report on its promising performance and scalability.

Riemann surfaces of complex classical trajectories and tunnelling splitting in one-dimensional systems

NASA Astrophysics Data System (ADS)

Harada, Hiromitsu; Mouchet, Amaury; Shudo, Akira

2017-10-01

The topology of complex classical paths is investigated to discuss quantum tunnelling splittings in one-dimensional systems. Here the Hamiltonian is assumed to be given as polynomial functions, so the fundamental group for the Riemann surface provides complete information on the topology of complex paths, which allows us to enumerate all the possible candidates contributing to the semiclassical sum formula for tunnelling splittings. This naturally leads to action relations among classically disjoined regions, revealing entirely non-local nature in the quantization condition. The importance of the proper treatment of Stokes phenomena is also discussed in Hamiltonians in the normal form.

Some applications of the multi-dimensional fractional order for the Riemann-Liouville derivative

NASA Astrophysics Data System (ADS)

Ahmood, Wasan Ajeel; Kiliçman, Adem

2017-01-01

In this paper, the aim of this work is to study theorem for the one-dimensional space-time fractional deriative, generalize some function for the one-dimensional fractional by table represents the fractional Laplace transforms of some elementary functions to be valid for the multi-dimensional fractional Laplace transform and give the definition of the multi-dimensional fractional Laplace transform. This study includes that, dedicate the one-dimensional fractional Laplace transform for functions of only one independent variable and develop of the one-dimensional fractional Laplace transform to multi-dimensional fractional Laplace transform based on the modified Riemann-Liouville derivative.

Analysis of the single-vehicle cyclic inventory routing problem

NASA Astrophysics Data System (ADS)

Aghezzaf, El-Houssaine; Zhong, Yiqing; Raa, Birger; Mateo, Manel

2012-11-01

The single-vehicle cyclic inventory routing problem (SV-CIRP) consists of a repetitive distribution of a product from a single depot to a selected subset of customers. For each customer, selected for replenishments, the supplier collects a corresponding fixed reward. The objective is to determine the subset of customers to replenish, the quantity of the product to be delivered to each and to design the vehicle route so that the resulting profit (difference between the total reward and the total logistical cost) is maximised while preventing stockouts at each of the selected customers. This problem appears often as a sub-problem in many logistical problems. In this article, the SV-CIRP is formulated as a mixed-integer program with a nonlinear objective function. After a thorough analysis of the structure of the problem and its features, an exact algorithm for its solution is proposed. This exact algorithm requires only solutions of linear mixed-integer programs. Values of a savings-based heuristic for this problem are compared to the optimal values obtained for a set of some test problems. In general, the gap may get as large as 25%, which justifies the effort to continue exploring and developing exact and approximation algorithms for the SV-CIRP.

Lefschetz thimbles in fermionic effective models with repulsive vector-field

NASA Astrophysics Data System (ADS)

Mori, Yuto; Kashiwa, Kouji; Ohnishi, Akira

2018-06-01

We discuss two problems in complexified auxiliary fields in fermionic effective models, the auxiliary sign problem associated with the repulsive vector-field and the choice of the cut for the scalar field appearing from the logarithmic function. In the fermionic effective models with attractive scalar and repulsive vector-type interaction, the auxiliary scalar and vector fields appear in the path integral after the bosonization of fermion bilinears. When we make the path integral well-defined by the Wick rotation of the vector field, the oscillating Boltzmann weight appears in the partition function. This "auxiliary" sign problem can be solved by using the Lefschetz-thimble path-integral method, where the integration path is constructed in the complex plane. Another serious obstacle in the numerical construction of Lefschetz thimbles is caused by singular points and cuts induced by multivalued functions of the complexified scalar field in the momentum integration. We propose a new prescription which fixes gradient flow trajectories on the same Riemann sheet in the flow evolution by performing the momentum integration in the complex domain.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Burke, J.V.

The published work on exact penalization is indeed vast. Recently this work has indicated an intimate relationship between exact penalization, Lagrange multipliers, and problem stability or calmness. In the present work we chronicle this development within a simple idealized problem framework, wherein we unify, extend, and refine much of the known theory. In particular, most of the foundations for constrained optimization are developed with the aid of exact penalization techniques. Our approach is highly geometric and is based upon the elementary subdifferential theory for distance functions. It is assumed that the reader is familiar with the theory of convex setsmore » and functions. 54 refs.« less

Topological String Theory and Enumerative Geometry

DOE Office of Scientific and Technical Information (OSTI.GOV)

Song, Y. S

In this thesis we investigate several problems which have their roots in both topological string theory and enumerative geometry. In the former case, underlying theories are topological field theories, whereas the latter case is concerned with intersection theories on moduli spaces. A permeating theme in this thesis is to examine the close interplay between these two complementary fields of study. The main problems addressed are as follows: In considering the Hurwitz enumeration problem of branched covers of compact connected Riemann surfaces, we completely solve the problem in the case of simple Hurwitz numbers. In addition, utilizing the connection between Hurwitzmore » numbers and Hodge integrals, we derive a generating function for the latter on the moduli space {bar M}{sub g,2} of 2-pointed, genus-g Deligne-Mumford stable curves. We also investigate Givental's recent conjecture regarding semisimple Frobenius structures and Gromov-Witten invariants, both of which are closely related to topological field theories; we consider the case of a complex projective line P{sup 1} as a specific example and verify his conjecture at low genera. In the last chapter, we demonstrate that certain topological open string amplitudes can be computed via relative stable morphisms in the algebraic category.« less

Quantum simulation of the integer factorization problem: Bell states in a Penning trap

NASA Astrophysics Data System (ADS)

Rosales, Jose Luis; Martin, Vicente

2018-03-01

The arithmetic problem of factoring an integer N can be translated into the physics of a quantum device, a result that supports Pólya's and Hilbert's conjecture to demonstrate Riemann's hypothesis. The energies of this system, being univocally related to the factors of N , are the eigenvalues of a bounded Hamiltonian. Here we solve the quantum conditions and show that the histogram of the discrete energies, provided by the spectrum of the system, should be interpreted in number theory as the relative probability for a prime to be a factor candidate of N . This is equivalent to a quantum sieve that is shown to require only o (ln√{N}) 3 energy measurements to solve the problem, recovering Shor's complexity result. Hence the outcome can be seen as a probability map that a pair of primes solve the given factorization problem. Furthermore, we show that a possible embodiment of this quantum simulator corresponds to two entangled particles in a Penning trap. The possibility to build the simulator experimentally is studied in detail. The results show that factoring numbers, many orders of magnitude larger than those computed with experimentally available quantum computers, is achievable using typical parameters in Penning traps.

Relativistic shock waves and Mach cones in viscous gluon matter

NASA Astrophysics Data System (ADS)

Bouras, Ioannis; Molnár, Etele; Niemi, Harri; Xu, Zhe; El, Andrej; Fochler, Oliver; Lauciello, Francesco; Greiner, Carsten; Rischke, Dirk H.

2010-06-01

To investigate the formation and the propagation of relativistic shock waves in viscous gluon matter we solve the relativistic Riemann problem using a microscopic parton cascade. We demonstrate the transition from ideal to viscous shock waves by varying the shear viscosity to entropy density ratio η/s. Furthermore we compare our results with those obtained by solving the relativistic causal dissipative fluid equations of Israel and Stewart (IS), in order to show the validity of the IS hydrodynamics. Employing the parton cascade we also investigate the formation of Mach shocks induced by a high-energy gluon traversing viscous gluon matter. For η/s = 0.08 a Mach cone structure is observed, whereas the signal smears out for η/s >= 0.32.

Relaxation approximations to second-order traffic flow models by high-resolution schemes

DOE Office of Scientific and Technical Information (OSTI.GOV)

Nikolos, I.K.; Delis, A.I.; Papageorgiou, M.

2015-03-10

A relaxation-type approximation of second-order non-equilibrium traffic models, written in conservation or balance law form, is considered. Using the relaxation approximation, the nonlinear equations are transformed to a semi-linear diagonilizable problem with linear characteristic variables and stiff source terms with the attractive feature that neither Riemann solvers nor characteristic decompositions are in need. In particular, it is only necessary to provide the flux and source term functions and an estimate of the characteristic speeds. To discretize the resulting relaxation system, high-resolution reconstructions in space are considered. Emphasis is given on a fifth-order WENO scheme and its performance. The computations reportedmore » demonstrate the simplicity and versatility of relaxation schemes as numerical solvers.« less

GENASIS: General Astrophysical Simulation System. I. Refinable Mesh and Nonrelativistic Hydrodynamics

NASA Astrophysics Data System (ADS)

Cardall, Christian Y.; Budiardja, Reuben D.; Endeve, Eirik; Mezzacappa, Anthony

2014-02-01

GenASiS (General Astrophysical Simulation System) is a new code being developed initially and primarily, though by no means exclusively, for the simulation of core-collapse supernovae on the world's leading capability supercomputers. This paper—the first in a series—demonstrates a centrally refined coordinate patch suitable for gravitational collapse and documents methods for compressible nonrelativistic hydrodynamics. We benchmark the hydrodynamics capabilities of GenASiS against many standard test problems; the results illustrate the basic competence of our implementation, demonstrate the strengths and limitations of the HLLC relative to the HLL Riemann solver in a number of interesting cases, and provide preliminary indications of the code's ability to scale and to function with cell-by-cell fixed-mesh refinement.

Newly-Developed 3D GRMHD Code and its Application to Jet Formation

NASA Technical Reports Server (NTRS)

Mizuno, Y.; Nishikawa, K.-I.; Koide, S.; Hardee, P.; Fishman, G. J.

2006-01-01

We have developed a new three-dimensional general relativistic magnetohydrodynamic code by using a conservative, high-resolution shock-capturing scheme. The numerical fluxes are calculated using the HLL approximate Riemann solver scheme. The flux-interpolated constrained transport scheme is used to maintain a divergence-free magnetic field. We have performed various 1-dimensional test problems in both special and general relativity by using several reconstruction methods and found that the new 3D GRMHD code shows substantial improvements over our previous model. The . preliminary results show the jet formations from a geometrically thin accretion disk near a non-rotating and a rotating black hole. We will discuss the jet properties depended on the rotation of a black hole and the magnetic field strength.

Evolution of Advection Upstream Splitting Method Schemes

NASA Technical Reports Server (NTRS)

Liou, Meng-Sing

2010-01-01

This paper focuses on the evolution of advection upstream splitting method(AUSM) schemes. The main ingredients that have led to the development of modern computational fluid dynamics (CFD) methods have been reviewed, thus the ideas behind AUSM. First and foremost is the concept of upwinding. Second, the use of Riemann problem in constructing the numerical flux in the finite-volume setting. Third, the necessity of including all physical processes, as characterised by the linear (convection) and nonlinear (acoustic) fields. Fourth, the realisation of separating the flux into convection and pressure fluxes. The rest of this review briefly outlines the technical evolution of AUSM and more details can be found in the cited references. Keywords: Computational fluid dynamics methods, hyperbolic systems, advection upstream splitting method, conservation laws, upwinding, CFD

A Hermite WENO reconstruction for fourth order temporal accurate schemes based on the GRP solver for hyperbolic conservation laws

NASA Astrophysics Data System (ADS)

Du, Zhifang; Li, Jiequan

2018-02-01

This paper develops a new fifth order accurate Hermite WENO (HWENO) reconstruction method for hyperbolic conservation schemes in the framework of the two-stage fourth order accurate temporal discretization in Li and Du (2016) [13]. Instead of computing the first moment of the solution additionally in the conventional HWENO or DG approach, we can directly take the interface values, which are already available in the numerical flux construction using the generalized Riemann problem (GRP) solver, to approximate the first moment. The resulting scheme is fourth order temporal accurate by only invoking the HWENO reconstruction twice so that it becomes more compact. Numerical experiments show that such compactness makes significant impact on the resolution of nonlinear waves.

Structural factoring approach for analyzing stochastic networks

NASA Technical Reports Server (NTRS)

Hayhurst, Kelly J.; Shier, Douglas R.

1991-01-01

The problem of finding the distribution of the shortest path length through a stochastic network is investigated. A general algorithm for determining the exact distribution of the shortest path length is developed based on the concept of conditional factoring, in which a directed, stochastic network is decomposed into an equivalent set of smaller, generally less complex subnetworks. Several network constructs are identified and exploited to reduce significantly the computational effort required to solve a network problem relative to complete enumeration. This algorithm can be applied to two important classes of stochastic path problems: determining the critical path distribution for acyclic networks and the exact two-terminal reliability for probabilistic networks. Computational experience with the algorithm was encouraging and allowed the exact solution of networks that have been previously analyzed only by approximation techniques.

Exact solutions for the collaborative pickup and delivery problem.

PubMed

Gansterer, Margaretha; Hartl, Richard F; Salzmann, Philipp E H

2018-01-01

In this study we investigate the decision problem of a central authority in pickup and delivery carrier collaborations. Customer requests are to be redistributed among participants, such that the total cost is minimized. We formulate the problem as multi-depot traveling salesman problem with pickups and deliveries. We apply three well-established exact solution approaches and compare their performance in terms of computational time. To avoid unrealistic solutions with unevenly distributed workload, we extend the problem by introducing minimum workload constraints. Our computational results show that, while for the original problem Benders decomposition is the method of choice, for the newly formulated problem this method is clearly dominated by the proposed column generation approach. The obtained results can be used as benchmarks for decentralized mechanisms in collaborative pickup and delivery problems.

Ice cream and orbifold Riemann-Roch

NASA Astrophysics Data System (ADS)

Buckley, Anita; Reid, Miles; Zhou, Shengtian

2013-06-01

We give an orbifold Riemann-Roch formula in closed form for the Hilbert series of a quasismooth polarized n-fold (X,D), under the assumption that X is projectively Gorenstein with only isolated orbifold points. Our formula is a sum of parts each of which is integral and Gorenstein symmetric of the same canonical weight; the orbifold parts are called ice cream functions. This form of the Hilbert series is particularly useful for computer algebra, and we illustrate it on examples of {K3} surfaces and Calabi-Yau 3-folds. These results apply also with higher dimensional orbifold strata (see [1] and [2]), although the precise statements are considerably trickier. We expect to return to this in future publications.

A universal counting of black hole microstates in AdS4

NASA Astrophysics Data System (ADS)

Azzurli, Francesco; Bobev, Nikolay; Crichigno, P. Marcos; Min, Vincent S.; Zaffaroni, Alberto

2018-02-01

Many three-dimensional N=2 SCFTs admit a universal partial topological twist when placed on hyperbolic Riemann surfaces. We exploit this fact to derive a universal formula which relates the planar limit of the topologically twisted index of these SCFTs and their three-sphere partition function. We then utilize this to account for the entropy of a large class of supersymmetric asymptotically AdS4 magnetically charged black holes in M-theory and massive type IIA string theory. In this context we also discuss novel AdS2 solutions of eleven-dimensional supergravity which describe the near horizon region of large new families of supersymmetric black holes arising from M2-branes wrapping Riemann surfaces.

Mass-deformed ABJM and black holes in AdS4

NASA Astrophysics Data System (ADS)

Bobev, Nikolay; Min, Vincent S.; Pilch, Krzysztof

2018-03-01

We find a class of new supersymmetric dyonic black holes in four-dimensional maximal gauged supergravity which are asymptotic to the SU(3) × U(1) invariant AdS4 Warner vacuum. These black holes can be embedded in eleven-dimensional supergravity where they describe the backreaction of M2-branes wrapped on a Riemann surface. The holographic dual description of these supergravity backgrounds is given by a partial topological twist on a Riemann surface of a three-dimensional N=2 SCFT that is obtained by a mass-deformation of the ABJM theory. We compute explicitly the topologically twisted index of this SCFT and show that it accounts for the entropy of the black holes.

A superparticle on the super Riemann surface

NASA Astrophysics Data System (ADS)

Matsumoto, Shuji; Uehara, Shozo; Yasui, Yukinori

1990-02-01

The free motion of a nonrelativistic superparticle on the super Riemann surface (SRS) of genus h≥2 is investigated. Geodesics or classical paths are given explicitly on the super Poincaré upper half-plane SH, a universal covering space of the SRS, and the paths with some suitable initial conditions yield periodic orbits on the SRS. The periodic orbits are unstable and the system is chaotic. Quantum mechanics is solved on the universal covering space SH and the heat kernel is given on the SRS. This leads to a superanalog of the Selberg trace formula. The Selberg super zeta function is introduced whose zero points and poles determine the energy spectrum on the SRS.

High-speed water impacts of flat plates in different ditching configuration through a Riemann-ALE SPH model

NASA Astrophysics Data System (ADS)

Marrone, S.; Colagrossi, A.; Chiron, L.; De Leffe, M.; Le Touzé, D.

2018-02-01

The violent water entry of flat plates is investigated using a Riemann-arbitrary Eulerian-Lagrangian (ALE) smoothed particle hydrodynamics (SPH) model. The test conditions are of interest for problems related to aircraft and helicopter emergency landing in water. Three main parameters are considered: the horizontal velocity, the approach angle (i.e., vertical to horizontal velocity ratio) and the pitch angle, α. Regarding the latter, small angles are considered in this study. As described in the theoretical work by Zhao and Faltinsen (1993), for small α a very thin, high-speed jet of water is formed, and the time-spatial gradients of the pressure field are extremely high. These test conditions are very challenging for numerical solvers. In the present study an enhanced SPH model is firstly tested on a purely vertical impact with deadrise angle α = 4°. An in-depth validation against analytical solutions and experimental results is carried out, highlighting the several critical aspects of the numerical modelling of this kind of flow, especially when pressure peaks are to be captured. A discussion on the main difficulties when comparing to model scale experiments is also provided. Then, the more realistic case of a plate with both horizontal and vertical velocity components is discussed and compared to ditching experiments recently carried out at CNR-INSEAN. In the latter case both 2-D and 3-D simulations are considered and the importance of 3-D effects on the pressure peak is discussed for α = 4° and α = 10°.

A second-order shock-adaptive Godunov scheme based on the generalized Lagrangian formulation

NASA Astrophysics Data System (ADS)

Lepage, Claude

Application of the Godunov scheme to the Euler equations of gas dynamics, based on the Eulerian formulation of flow, smears discontinuities (especially sliplines) over several computational cells, while the accuracy in the smooth flow regions is of the order of a function of the cell width. Based on the generalized Lagrangian formulation (GLF), the Godunov scheme yields far superior results. By the use of coordinate streamlines in the GLF, the slipline (itself a streamline) is resolved crisply. Infinite shock resolution is achieved through the splitting of shock cells, while the accuracy in the smooth flow regions is improved using a nonconservative formulation of the governing equations coupled to a second order extension of the Godunov scheme. Furthermore, GLF requires no grid generation for boundary value problems and the simple structure of the solution to the Riemann problem in the GLF is exploited in the numerical implementation of the shock adaptive scheme. Numerical experiments reveal high efficiency and unprecedented resolution of shock and slipline discontinuities.

A class of high resolution explicit and implicit shock-capturing methods

NASA Technical Reports Server (NTRS)

Yee, H. C.

1989-01-01

An attempt is made to give a unified and generalized formulation of a class of high resolution, explicit and implicit shock capturing methods, and to illustrate their versatility in various steady and unsteady complex shock wave computations. Included is a systematic review of the basic design principle of the various related numerical methods. Special emphasis is on the construction of the basis nonlinear, spatially second and third order schemes for nonlinear scalar hyperbolic conservation laws and the methods of extending these nonlinear scalar schemes to nonlinear systems via the approximate Riemann solvers and the flux vector splitting approaches. Generalization of these methods to efficiently include equilibrium real gases and large systems of nonequilibrium flows are discussed. Some issues concerning the applicability of these methods that were designed for homogeneous hyperbolic conservation laws to problems containing stiff source terms and shock waves are also included. The performance of some of these schemes is illustrated by numerical examples for 1-, 2- and 3-dimensional gas dynamics problems.

An exact algorithm for optimal MAE stack filter design.

PubMed

Dellamonica, Domingos; Silva, Paulo J S; Humes, Carlos; Hirata, Nina S T; Barrera, Junior

2007-02-01

We propose a new algorithm for optimal MAE stack filter design. It is based on three main ingredients. First, we show that the dual of the integer programming formulation of the filter design problem is a minimum cost network flow problem. Next, we present a decomposition principle that can be used to break this dual problem into smaller subproblems. Finally, we propose a specialization of the network Simplex algorithm based on column generation to solve these smaller subproblems. Using our method, we were able to efficiently solve instances of the filter problem with window size up to 25 pixels. To the best of our knowledge, this is the largest dimension for which this problem was ever solved exactly.

ADHD and math - The differential effect on calculation and estimation.

PubMed

Ganor-Stern, Dana; Steinhorn, Ofir

2018-05-31

Adults with ADHD were compared to controls when solving multiplication problems exactly and when estimating the results of multidigit multiplication problems relative to reference numbers. The ADHD participants were slower than controls in the exact calculation and in the estimation tasks, but not less accurate. The ADHD participants were similar to controls in showing enhanced accuracy and speed for smaller problem sizes, for trials in which the reference numbers were smaller (vs. larger) than the exact answers and for reference numbers that were far (vs. close) from the exact answer. The two groups similarly used the approximated calculation and the sense of magnitude strategies. They differed however in strategy execution, mainly of the approximated calculation strategy, which requires working memory resources. The increase in reaction time associated with using the approximated calculation strategy was larger for the ADHD compared to the control participants. Thus, ADHD seems to selectively impair calculation processes in estimation tasks that rely on working memory, but it does not hamper estimation skills that are based on sense of magnitude. The educational implications of these findings are discussed. Copyright © 2018. Published by Elsevier B.V.

Riemann solvers and Alfven waves in black hole magnetospheres

NASA Astrophysics Data System (ADS)

Punsly, Brian; Balsara, Dinshaw; Kim, Jinho; Garain, Sudip

2016-09-01

In the magnetosphere of a rotating black hole, an inner Alfven critical surface (IACS) must be crossed by inflowing plasma. Inside the IACS, Alfven waves are inward directed toward the black hole. The majority of the proper volume of the active region of spacetime (the ergosphere) is inside of the IACS. The charge and the totally transverse momentum flux (the momentum flux transverse to both the wave normal and the unperturbed magnetic field) are both determined exclusively by the Alfven polarization. Thus, it is important for numerical simulations of black hole magnetospheres to minimize the dissipation of Alfven waves. Elements of the dissipated wave emerge in adjacent cells regardless of the IACS, there is no mechanism to prevent Alfvenic information from crossing outward. Thus, numerical dissipation can affect how simulated magnetospheres attain the substantial Goldreich-Julian charge density associated with the rotating magnetic field. In order to help minimize dissipation of Alfven waves in relativistic numerical simulations we have formulated a one-dimensional Riemann solver, called HLLI, which incorporates the Alfven discontinuity and the contact discontinuity. We have also formulated a multidimensional Riemann solver, called MuSIC, that enables low dissipation propagation of Alfven waves in multiple dimensions. The importance of higher order schemes in lowering the numerical dissipation of Alfven waves is also catalogued.

Revealing the Physics of Galactic Winds Through Massively-Parallel Hydrodynamics Simulations

NASA Astrophysics Data System (ADS)

Schneider, Evan Elizabeth

This thesis documents the hydrodynamics code Cholla and a numerical study of multiphase galactic winds. Cholla is a massively-parallel, GPU-based code designed for astrophysical simulations that is freely available to the astrophysics community. A static-mesh Eulerian code, Cholla is ideally suited to carrying out massive simulations (> 20483 cells) that require very high resolution. The code incorporates state-of-the-art hydrodynamics algorithms including third-order spatial reconstruction, exact and linearized Riemann solvers, and unsplit integration algorithms that account for transverse fluxes on multidimensional grids. Operator-split radiative cooling and a dual-energy formalism for high mach number flows are also included. An extensive test suite demonstrates Cholla's superior ability to model shocks and discontinuities, while the GPU-native design makes the code extremely computationally efficient - speeds of 5-10 million cell updates per GPU-second are typical on current hardware for 3D simulations with all of the aforementioned physics. The latter half of this work comprises a comprehensive study of the mixing between a hot, supernova-driven wind and cooler clouds representative of those observed in multiphase galactic winds. Both adiabatic and radiatively-cooling clouds are investigated. The analytic theory of cloud-crushing is applied to the problem, and adiabatic turbulent clouds are found to be mixed with the hot wind on similar timescales as the classic spherical case (4-5 t cc) with an appropriate rescaling of the cloud-crushing time. Radiatively cooling clouds survive considerably longer, and the differences in evolution between turbulent and spherical clouds cannot be reconciled with a simple rescaling. The rapid incorporation of low-density material into the hot wind implies efficient mass-loading of hot phases of galactic winds. At the same time, the extreme compression of high-density cloud material leads to long-lived but slow-moving clumps that are unlikely to escape the galaxy.

Volumetric formulation for a class of kinetic models with energy conservation.

PubMed

Sbragaglia, M; Sugiyama, K

2010-10-01

We analyze a volumetric formulation of lattice Boltzmann for compressible thermal fluid flows. The velocity set is chosen with the desired accuracy, based on the Gauss-Hermite quadrature procedure, and tested against controlled problems in bounded and unbounded fluids. The method allows the simulation of thermohydrodyamical problems without the need to preserve the exact space-filling nature of the velocity set, but still ensuring the exact conservation laws for density, momentum, and energy. Issues related to boundary condition problems and improvements based on grid refinement are also investigated.

Finite element analysis of wrinkling membranes

NASA Technical Reports Server (NTRS)

Miller, R. K.; Hedgepeth, J. M.; Weingarten, V. I.; Das, P.; Kahyai, S.

1984-01-01

The development of a nonlinear numerical algorithm for the analysis of stresses and displacements in partly wrinkled flat membranes, and its implementation on the SAP VII finite-element code are described. A comparison of numerical results with exact solutions of two benchmark problems reveals excellent agreement, with good convergence of the required iterative procedure. An exact solution of a problem involving axisymmetric deformations of a partly wrinkled shallow curved membrane is also reported.

A Riemann solver for single-phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pelanti, Marica, E-mail: Marica.Pelanti@ens.f; Bouchut, Francois, E-mail: francois.bouchut@univ-mlv.f; Mangeney, Anne, E-mail: mangeney@ipgp.jussieu.f

2011-02-01

We present a Riemann solver derived by a relaxation technique for classical single-phase shallow flow equations and for a two-phase shallow flow model describing a mixture of solid granular material and fluid. Our primary interest is the numerical approximation of this two-phase solid/fluid model, whose complexity poses numerical difficulties that cannot be efficiently addressed by existing solvers. In particular, we are concerned with ensuring a robust treatment of dry bed states. The relaxation system used by the proposed solver is formulated by introducing auxiliary variables that replace the momenta in the spatial gradients of the original model systems. The resultingmore » relaxation solver is related to Roe solver in that its Riemann solution for the flow height and relaxation variables is formally computed as Roe's Riemann solution. The relaxation solver has the advantage of a certain degree of freedom in the specification of the wave structure through the choice of the relaxation parameters. This flexibility can be exploited to handle robustly vacuum states, which is a well known difficulty of standard Roe's method, while maintaining Roe's low diffusivity. For the single-phase model positivity of flow height is rigorously preserved. For the two-phase model positivity of volume fractions in general is not ensured, and a suitable restriction on the CFL number might be needed. Nonetheless, numerical experiments suggest that the proposed two-phase flow solver efficiently models wet/dry fronts and vacuum formation for a large range of flow conditions. As a corollary of our study, we show that for single-phase shallow flow equations the relaxation solver is formally equivalent to the VFRoe solver with conservative variables of Gallouet and Masella [T. Gallouet, J.-M. Masella, Un schema de Godunov approche C.R. Acad. Sci. Paris, Serie I, 323 (1996) 77-84]. The relaxation interpretation allows establishing positivity conditions for this VFRoe method.« less

Efficient algorithms for a class of partitioning problems

NASA Technical Reports Server (NTRS)

Iqbal, M. Ashraf; Bokhari, Shahid H.

1990-01-01

The problem of optimally partitioning the modules of chain- or tree-like tasks over chain-structured or host-satellite multiple computer systems is addressed. This important class of problems includes many signal processing and industrial control applications. Prior research has resulted in a succession of faster exact and approximate algorithms for these problems. Polynomial exact and approximate algorithms are described for this class that are better than any of the previously reported algorithms. The approach is based on a preprocessing step that condenses the given chain or tree structured task into a monotonic chain or tree. The partitioning of this monotonic take can then be carried out using fast search techniques.

Numerical method for solving the nonlinear four-point boundary value problems

NASA Astrophysics Data System (ADS)

Lin, Yingzhen; Lin, Jinnan

2010-12-01

In this paper, a new reproducing kernel space is constructed skillfully in order to solve a class of nonlinear four-point boundary value problems. The exact solution of the linear problem can be expressed in the form of series and the approximate solution of the nonlinear problem is given by the iterative formula. Compared with known investigations, the advantages of our method are that the representation of exact solution is obtained in a new reproducing kernel Hilbert space and accuracy of numerical computation is higher. Meanwhile we present the convergent theorem, complexity analysis and error estimation. The performance of the new method is illustrated with several numerical examples.

Solving standard traveling salesman problem and multiple traveling salesman problem by using branch-and-bound

NASA Astrophysics Data System (ADS)

Saad, Shakila; Wan Jaafar, Wan Nurhadani; Jamil, Siti Jasmida

2013-04-01

The standard Traveling Salesman Problem (TSP) is the classical Traveling Salesman Problem (TSP) while Multiple Traveling Salesman Problem (MTSP) is an extension of TSP when more than one salesman is involved. The objective of MTSP is to find the least costly route that the traveling salesman problem can take if he wishes to visit exactly once each of a list of n cities and then return back to the home city. There are a few methods that can be used to solve MTSP. The objective of this research is to implement an exact method called Branch-and-Bound (B&B) algorithm. Briefly, the idea of B&B algorithm is to start with the associated Assignment Problem (AP). A branching strategy will be applied to the TSP and MTSP which is Breadth-first-Search (BFS). 11 nodes of cities are implemented for both problem and the solutions to the problem are presented.

Alternate solution to generalized Bernoulli equations via an integrating factor: an exact differential equation approach

NASA Astrophysics Data System (ADS)

Tisdell, C. C.

2017-08-01

Solution methods to exact differential equations via integrating factors have a rich history dating back to Euler (1740) and the ideas enjoy applications to thermodynamics and electromagnetism. Recently, Azevedo and Valentino presented an analysis of the generalized Bernoulli equation, constructing a general solution by linearizing the problem through a substitution. The purpose of this note is to present an alternative approach using 'exact methods', illustrating that a substitution and linearization of the problem is unnecessary. The ideas may be seen as forming a complimentary and arguably simpler approach to Azevedo and Valentino that have the potential to be assimilated and adapted to pedagogical needs of those learning and teaching exact differential equations in schools, colleges, universities and polytechnics. We illustrate how to apply the ideas through an analysis of the Gompertz equation, which is of interest in biomathematical models of tumour growth.

Non-oscillatory central differencing for hyperbolic conservation laws

NASA Technical Reports Server (NTRS)

Nessyahu, Haim; Tadmor, Eitan

1988-01-01

Many of the recently developed high resolution schemes for hyperbolic conservation laws are based on upwind differencing. The building block for these schemes is the averaging of an appropriate Godunov solver; its time consuming part involves the field-by-field decomposition which is required in order to identify the direction of the wind. Instead, the use of the more robust Lax-Friedrichs (LxF) solver is proposed. The main advantage is simplicity: no Riemann problems are solved and hence field-by-field decompositions are avoided. The main disadvantage is the excessive numerical viscosity typical to the LxF solver. This is compensated for by using high-resolution MUSCL-type interpolants. Numerical experiments show that the quality of results obtained by such convenient central differencing is comparable with those of the upwind schemes.

Application of the trigonal curve to the Blaszak-Marciniak lattice hierarchy

NASA Astrophysics Data System (ADS)

Geng, Xianguo; Zeng, Xin

2017-01-01

We develop a method for constructing algebro-geometric solutions of the Blaszak-Marciniak ( BM) lattice hierarchy based on the theory of trigonal curves. We first derive the BM lattice hierarchy associated with a discrete (3×3)- matrix spectral problem using Lenard recurrence relations. Using the characteristic polynomial of the Lax matrix for the BM lattice hierarchy, we introduce a trigonal curve with two infinite points, which we use to establish the associated Dubrovin-type equations. We then study the asymptotic properties of the algebraic function carrying the data of the divisor and the Baker-Akhiezer function near the two infinite points on the trigonal curve. We finally obtain algebro-geometric solutions of the entire BM lattice hierarchy in terms of the Riemann theta function.

Comments regarding two upwind methods for solving two-dimensional external flows using unstructured grids

NASA Technical Reports Server (NTRS)

Kleb, W. L.

1994-01-01

Steady flow over the leading portion of a multicomponent airfoil section is studied using computational fluid dynamics (CFD) employing an unstructured grid. To simplify the problem, only the inviscid terms are retained from the Reynolds-averaged Navier-Stokes equations - leaving the Euler equations. The algorithm is derived using the finite-volume approach, incorporating explicit time-marching of the unsteady Euler equations to a time-asymptotic, steady-state solution. The inviscid fluxes are obtained through either of two approximate Riemann solvers: Roe's flux difference splitting or van Leer's flux vector splitting. Results are presented which contrast the solutions given by the two flux functions as a function of Mach number and grid resolution. Additional information is presented concerning code verification techniques, flow recirculation regions, convergence histories, and computational resources.

Performance comparison of a new hybrid conjugate gradient method under exact and inexact line searches

NASA Astrophysics Data System (ADS)

Ghani, N. H. A.; Mohamed, N. S.; Zull, N.; Shoid, S.; Rivaie, M.; Mamat, M.

2017-09-01

Conjugate gradient (CG) method is one of iterative techniques prominently used in solving unconstrained optimization problems due to its simplicity, low memory storage, and good convergence analysis. This paper presents a new hybrid conjugate gradient method, named NRM1 method. The method is analyzed under the exact and inexact line searches in given conditions. Theoretically, proofs show that the NRM1 method satisfies the sufficient descent condition with both line searches. The computational result indicates that NRM1 method is capable in solving the standard unconstrained optimization problems used. On the other hand, the NRM1 method performs better under inexact line search compared with exact line search.

Exact solution for the optimal neuronal layout problem.

PubMed

Chklovskii, Dmitri B

2004-10-01

Evolution perfected brain design by maximizing its functionality while minimizing costs associated with building and maintaining it. Assumption that brain functionality is specified by neuronal connectivity, implemented by costly biological wiring, leads to the following optimal design problem. For a given neuronal connectivity, find a spatial layout of neurons that minimizes the wiring cost. Unfortunately, this problem is difficult to solve because the number of possible layouts is often astronomically large. We argue that the wiring cost may scale as wire length squared, reducing the optimal layout problem to a constrained minimization of a quadratic form. For biologically plausible constraints, this problem has exact analytical solutions, which give reasonable approximations to actual layouts in the brain. These solutions make the inverse problem of inferring neuronal connectivity from neuronal layout more tractable.

Fermions tunneling from a general static Riemann black hole

NASA Astrophysics Data System (ADS)

Chen, Ge-Rui; Huang, Yong-Chang

2015-05-01

In this paper we investigate the tunneling of fermions from a general static Riemann black hole by following Kerner and Mann (Class Quantum Gravit 25:095014, 2008a; Phys Lett B 665:277-283, 2008b) methods. By applying the WKB approximation and the Hamilton-Jacobi ansatz to the Dirac equation, we obtain the standard Hawking temperature. Furthermore, Kerner and Mann (Class Quantum Gravit 25:095014, 2008a; Phys Lett B 665:277-283, 2008b) only calculated the tunneling spectrum of the Dirac particles with spin-up, and we extend the methods to investigate the tunneling of Dirac particles with arbitrary spin directions and also obtain the expected Hawking temperature. Our result provides further evidence for the universality of black hole radiation.

On the Behavior of Eisenstein Series Through Elliptic Degeneration

NASA Astrophysics Data System (ADS)

Garbin, D.; Pippich, A.-M. V.

2009-12-01

Let Γ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane {mathbb{H}}, and let {M = Γbackslash mathbb{H}} be the associated finite volume hyperbolic Riemann surface. If γ is a primitive parabolic, hyperbolic, resp. elliptic element of Γ, there is an associated parabolic, hyperbolic, resp. elliptic Eisenstein series. In this article, we study the limiting behavior of these Eisenstein series on an elliptically degenerating family of finite volume hyperbolic Riemann surfaces. In particular, we prove the following result. The elliptic Eisenstein series associated to a degenerating elliptic element converges up to a factor to the parabolic Eisenstein series associated to the parabolic element which fixes the newly developed cusp on the limit surface.

An approximate Riemann solver for hypervelocity flows

NASA Technical Reports Server (NTRS)

Jacobs, Peter A.

1991-01-01

We describe an approximate Riemann solver for the computation of hypervelocity flows in which there are strong shocks and viscous interactions. The scheme has three stages, the first of which computes the intermediate states assuming isentropic waves. A second stage, based on the strong shock relations, may then be invoked if the pressure jump across either wave is large. The third stage interpolates the interface state from the two initial states and the intermediate states. The solver is used as part of a finite-volume code and is demonstrated on two test cases. The first is a high Mach number flow over a sphere while the second is a flow over a slender cone with an adiabatic boundary layer. In both cases the solver performs well.

Generalized Toda theory from six dimensions and the conifold

NASA Astrophysics Data System (ADS)

van Leuven, Sam; Oling, Gerben

2017-12-01

Recently, a physical derivation of the Alday-Gaiotto-Tachikawa correspondence has been put forward. A crucial role is played by the complex Chern-Simons theory arising in the 3d-3d correspondence, whose boundary modes lead to Toda theory on a Riemann surface. We explore several features of this derivation and subsequently argue that it can be extended to a generalization of the AGT correspondence. The latter involves codimension two defects in six dimensions that wrap the Riemann surface. We use a purely geometrical description of these defects and find that the generalized AGT setup can be modeled in a pole region using generalized conifolds. Furthermore, we argue that the ordinary conifold clarifies several features of the derivation of the original AGT correspondence.

Opera: reconstructing optimal genomic scaffolds with high-throughput paired-end sequences.

PubMed

Gao, Song; Sung, Wing-Kin; Nagarajan, Niranjan

2011-11-01

Scaffolding, the problem of ordering and orienting contigs, typically using paired-end reads, is a crucial step in the assembly of high-quality draft genomes. Even as sequencing technologies and mate-pair protocols have improved significantly, scaffolding programs still rely on heuristics, with no guarantees on the quality of the solution. In this work, we explored the feasibility of an exact solution for scaffolding and present a first tractable solution for this problem (Opera). We also describe a graph contraction procedure that allows the solution to scale to large scaffolding problems and demonstrate this by scaffolding several large real and synthetic datasets. In comparisons with existing scaffolders, Opera simultaneously produced longer and more accurate scaffolds demonstrating the utility of an exact approach. Opera also incorporates an exact quadratic programming formulation to precisely compute gap sizes (Availability: http://sourceforge.net/projects/operasf/ ).

Opera: Reconstructing Optimal Genomic Scaffolds with High-Throughput Paired-End Sequences

PubMed Central

Gao, Song; Sung, Wing-Kin

2011-01-01

Abstract Scaffolding, the problem of ordering and orienting contigs, typically using paired-end reads, is a crucial step in the assembly of high-quality draft genomes. Even as sequencing technologies and mate-pair protocols have improved significantly, scaffolding programs still rely on heuristics, with no guarantees on the quality of the solution. In this work, we explored the feasibility of an exact solution for scaffolding and present a first tractable solution for this problem (Opera). We also describe a graph contraction procedure that allows the solution to scale to large scaffolding problems and demonstrate this by scaffolding several large real and synthetic datasets. In comparisons with existing scaffolders, Opera simultaneously produced longer and more accurate scaffolds demonstrating the utility of an exact approach. Opera also incorporates an exact quadratic programming formulation to precisely compute gap sizes (Availability: http://sourceforge.net/projects/operasf/). PMID:21929371

The Poisson-Boltzmann theory for the two-plates problem: some exact results.

PubMed

Xing, Xiang-Jun

2011-12-01

The general solution to the nonlinear Poisson-Boltzmann equation for two parallel charged plates, either inside a symmetric electrolyte, or inside a 2q:-q asymmetric electrolyte, is found in terms of Weierstrass elliptic functions. From this we derive some exact asymptotic results for the interaction between charged plates, as well as the exact form of the renormalized surface charge density.

A model for solving the prescribed burn planning problem.

PubMed

Rachmawati, Ramya; Ozlen, Melih; Reinke, Karin J; Hearne, John W

2015-01-01

The increasing frequency of destructive wildfires, with a consequent loss of life and property, has led to fire and land management agencies initiating extensive fuel management programs. This involves long-term planning of fuel reduction activities such as prescribed burning or mechanical clearing. In this paper, we propose a mixed integer programming (MIP) model that determines when and where fuel reduction activities should take place. The model takes into account multiple vegetation types in the landscape, their tolerance to frequency of fire events, and keeps track of the age of each vegetation class in each treatment unit. The objective is to minimise fuel load over the planning horizon. The complexity of scheduling fuel reduction activities has led to the introduction of sophisticated mathematical optimisation methods. While these approaches can provide optimum solutions, they can be computationally expensive, particularly for fuel management planning which extends across the landscape and spans long term planning horizons. This raises the question of how much better do exact modelling approaches compare to simpler heuristic approaches in their solutions. To answer this question, the proposed model is run using an exact MIP (using commercial MIP solver) and two heuristic approaches that decompose the problem into multiple single-period sub problems. The Knapsack Problem (KP), which is the first heuristic approach, solves the single period problems, using an exact MIP approach. The second heuristic approach solves the single period sub problem using a greedy heuristic approach. The three methods are compared in term of model tractability, computational time and the objective values. The model was tested using randomised data from 711 treatment units in the Barwon-Otway district of Victoria, Australia. Solutions for the exact MIP could be obtained for up to a 15-year planning only using a standard implementation of CPLEX. Both heuristic approaches can solve significantly larger problems, involving 100-year or even longer planning horizons. Furthermore there are no substantial differences in the solutions produced by the three approaches. It is concluded that for practical purposes a heuristic method is to be preferred to the exact MIP approach.

Exact and explicit optimal solutions for trajectory planning and control of single-link flexible-joint manipulators

NASA Technical Reports Server (NTRS)

Chen, Guanrong

1991-01-01

An optimal trajectory planning problem for a single-link, flexible joint manipulator is studied. A global feedback-linearization is first applied to formulate the nonlinear inequality-constrained optimization problem in a suitable way. Then, an exact and explicit structural formula for the optimal solution of the problem is derived and the solution is shown to be unique. It turns out that the optimal trajectory planning and control can be done off-line, so that the proposed method is applicable to both theoretical analysis and real time tele-robotics control engineering.

Numerical Solution of the Electron Transport Equation in the Upper Atmosphere

DOE Office of Scientific and Technical Information (OSTI.GOV)

Woods, Mark Christopher; Holmes, Mark; Sailor, William C

A new approach for solving the electron transport equation in the upper atmosphere is derived. The problem is a very stiff boundary value problem, and to obtain an accurate numerical solution, matrix factorizations are used to decouple the fast and slow modes. A stable finite difference method is applied to each mode. This solver is applied to a simplifieed problem for which an exact solution exists using various versions of the boundary conditions that might arise in a natural auroral display. The numerical and exact solutions are found to agree with each other to at least two significant digits.

Lax-Wendroff and TVD finite volume methods for unidimensional thermomechanical numerical simulations of impacts on elastic-plastic solids

NASA Astrophysics Data System (ADS)

Heuzé, Thomas

2017-10-01

We present in this work two finite volume methods for the simulation of unidimensional impact problems, both for bars and plane waves, on elastic-plastic solid media within the small strain framework. First, an extension of Lax-Wendroff to elastic-plastic constitutive models with linear and nonlinear hardenings is presented. Second, a high order TVD method based on flux-difference splitting [1] and Superbee flux limiter [2] is coupled with an approximate elastic-plastic Riemann solver for nonlinear hardenings, and follows that of Fogarty [3] for linear ones. Thermomechanical coupling is accounted for through dissipation heating and thermal softening, and adiabatic conditions are assumed. This paper essentially focuses on one-dimensional problems since analytical solutions exist or can easily be developed. Accordingly, these two numerical methods are compared to analytical solutions and to the explicit finite element method on test cases involving discontinuous and continuous solutions. This allows to study in more details their respective performance during the loading, unloading and reloading stages. Particular emphasis is also paid to the accuracy of the computed plastic strains, some differences being found according to the numerical method used. Lax-Wendoff two-dimensional discretization of a one-dimensional problem is also appended at the end to demonstrate the extensibility of such numerical scheme to multidimensional problems.

Determining linear vibration frequencies of a ferromagnetic shell

NASA Astrophysics Data System (ADS)

Bagdoev, A. G.; Vardanyan, A. V.; Vardanyan, S. V.; Kukudzhanov, V. N.

2007-10-01

The problems of determining the roots of dispersion equations for free bending vibrations of thin magnetoelastic plates and shells are of both theoretical and practical interest, in particular, in studying vibrations of metallic structures used in controlled thermonuclear reactors. These problems were solved on the basis of the Kirchhoff hypothesis in [1-5]. In [6], an exact spatial approach to determining the vibration frequencies of thin plates was suggested, and it was shown that it completely agrees with the solution obtained according to the Kirchhoff hypothesis. In [7-9], this exact approach was used to solve the problem on vibrations of thin magnetoelastic plates, and it was shown by cumbersome calculations that the solutions obtained according to the exact theory and the Kirchhoff hypothesis differ substantially except in a single case. In [10], the equations of the dynamic theory of elasticity in the axisymmetric problem are given. In [11], the equations for the vibration frequencies of thin ferromagnetic plates with arbitrary conductivity were obtained in the exact statement. In [12], the Kirchhoff hypothesis was used to obtain dispersion relations for a magnetoelastic thin shell. In [5, 13-16], the relations for the Maxwell tensor and the ponderomotive force for magnetics were presented. In [17], the dispersion relations for thin ferromagnetic plates in the transverse field in the spatial statement were studied analytically and numerically. In the present paper, on the basis of the exact approach, we study free bending vibrations of a thin ferromagnetic cylindrical shell. We obtain the exact dispersion equation in the form of a sixth-order determinant, which can be solved numerically in the case of a magnetoelastic thin shell. The numerical results are presented in tables and compared with the results obtained by the Kirchhoff hypothesis. We show a large number of differences in the results, even for the least frequency.

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.

What Is Geometry?

ERIC Educational Resources Information Center

Chern, Shiing-Shen

1990-01-01

Discussed are the major historical developments of geometry. Euclid, Descartes, Klein's Erlanger Program, Gaus and Riemann, globalization, topology, Elie Cartan, and an application to molecular biology are included as topics. (KR)

Fast and Exact Continuous Collision Detection with Bernstein Sign Classification

PubMed Central

Tang, Min; Tong, Ruofeng; Wang, Zhendong; Manocha, Dinesh

2014-01-01

We present fast algorithms to perform accurate CCD queries between triangulated models. Our formulation uses properties of the Bernstein basis and Bézier curves and reduces the problem to evaluating signs of polynomials. We present a geometrically exact CCD algorithm based on the exact geometric computation paradigm to perform reliable Boolean collision queries. Our algorithm is more than an order of magnitude faster than prior exact algorithms. We evaluate its performance for cloth and FEM simulations on CPUs and GPUs, and highlight the benefits. PMID:25568589

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

NASA Technical Reports Server (NTRS)

Atkins, Harold L.; Pampell, Alyssa

2011-01-01

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

Graphing as a Problem-Solving Strategy.

ERIC Educational Resources Information Center

Cohen, Donald

1984-01-01

The focus is on how line graphs can be used to approximate solutions to rate problems and to suggest equations that offer exact algebraic solutions to the problem. Four problems requiring progressively greater graphing sophistication are presented plus four exercises. (MNS)

Using exact solutions to develop an implicit scheme for the baroclinic primitive equations

NASA Technical Reports Server (NTRS)

Marchesin, D.

1984-01-01

The exact solutions presently obtained by means of a novel method for nonlinear initial value problems are used in the development of numerical schemes for the computer solution of these problems. The method is applied to a new, fully implicit scheme on a vertical slice of the isentropic baroclinic equations. It was not possible to find a global scale phenomenon that could be simulated by the baroclinic primitive equations on a vertical slice.

Ion and Electron Energization in Guide Field Reconnection Outflows with Kinetic Riemann Simulations and Parallel Shock Simulations

NASA Astrophysics Data System (ADS)

Zhang, Q.; Drake, J. F.; Swisdak, M.

2017-12-01

How ions and electrons are energized in magnetic reconnection outflows is an essential topic throughout the heliosphere. Here we carry out guide field PIC Riemann simulations to explore the ion and electron energization mechanisms far downstream of the x-line. Riemann simulations, with their simple magnetic geometry, facilitate the study of the reconnection outflow far downstream of the x-line in much more detail than is possible with conventional reconnection simulations. We find that the ions get accelerated at rotational discontinuities, counter stream, and give rise to two slow shocks. We demonstrate that the energization mechanism at the slow shocks is essentially the same as that of parallel electrostatic shocks. Also, the electron confining electric potential at the slow shocks is driven by the counterstreaming beams, which tend to break the quasi-neutrality. Based on this picture, we build a kinetic model to self consistently predict the downstream ion and electron temperatures. Additional explorations using parallel shock simulations also imply that in a very low beta(0.001 0.01 for a modest guide field) regime, electron energization will be insignificant compared to the ion energization. Our model and the parallel shock simulations might be used as simple tools to understand and estimate the energization of ions and electrons and the energy partition far downstream of the x-line.

Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders. Part I: The one-dimensional case

NASA Astrophysics Data System (ADS)

Vilar, François; Shu, Chi-Wang; Maire, Pierre-Henri

2016-05-01

One of the main issues in the field of numerical schemes is to ally robustness with accuracy. Considering gas dynamics, numerical approximations may generate negative density or pressure, which may lead to nonlinear instability and crash of the code. This phenomenon is even more critical using a Lagrangian formalism, the grid moving and being deformed during the calculation. Furthermore, most of the problems studied in this framework contain very intense rarefaction and shock waves. In this paper, the admissibility of numerical solutions obtained by high-order finite-volume-scheme-based methods, such as the discontinuous Galerkin (DG) method, the essentially non-oscillatory (ENO) and the weighted ENO (WENO) finite volume schemes, is addressed in the one-dimensional Lagrangian gas dynamics framework. After briefly recalling how to derive Lagrangian forms of the 1D gas dynamics system of equations, a discussion on positivity-preserving approximate Riemann solvers, ensuring first-order finite volume schemes to be positive, is then given. This study is conducted for both ideal gas and non-ideal gas equations of state (EOS), such as the Jones-Wilkins-Lee (JWL) EOS or the Mie-Grüneisen (MG) EOS, and relies on two different techniques: either a particular definition of the local approximation of the acoustic impedances arising from the approximate Riemann solver, or an additional time step constraint relative to the cell volume variation. Then, making use of the work presented in [89,90,22], this positivity study is extended to high-orders of accuracy, where new time step constraints are obtained, and proper limitation is required. Through this new procedure, scheme robustness is highly improved and hence new problems can be tackled. Numerical results are provided to demonstrate the effectiveness of these methods. This paper is the first part of a series of two. The whole analysis presented here is extended to the two-dimensional case in [85], and proves to fit a wide range of numerical schemes in the literature, such as those presented in [19,64,15,82,84].

Fluid displacement between two parallel plates: a non-empirical model displaying change of type from hyperbolic to elliptic equations

NASA Astrophysics Data System (ADS)

Shariati, M.; Talon, L.; Martin, J.; Rakotomalala, N.; Salin, D.; Yortsos, Y. C.

2004-11-01

We consider miscible displacement between parallel plates in the absence of diffusion, with a concentration-dependent viscosity. By selecting a piecewise viscosity function, this can also be considered as ‘three-fluid’ flow in the same geometry. Assuming symmetry across the gap and based on the lubrication (‘equilibrium’) approximation, a description in terms of two quasi-linear hyperbolic equations is obtained. We find that the system is hyperbolic and can be solved analytically, when the mobility profile is monotonic, or when the mobility of the middle phase is smaller than its neighbours. When the mobility of the middle phase is larger, a change of type is displayed, an elliptic region developing in the composition space. Numerical solutions of Riemann problems of the hyperbolic system spanning the elliptic region, with small diffusion added, show good agreement with the analytical outside, but an unstable behaviour inside the elliptic region. In these problems, the elliptic region arises precisely at the displacement front. Crossing the elliptic region requires the solution of essentially an eigenvalue problem of the full higher-dimensional model, obtained here using lattice BGK simulations. The hyperbolic-to-elliptic change-of-type reflects the failing of the lubrication approximation, underlying the quasi-linear hyperbolic formalism, to describe the problem uniformly. The obtained solution is analogous to non-classical shocks recently suggested in problems with change of type.

Patch-based Adaptive Mesh Refinement for Multimaterial Hydrodynamics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lomov, I; Pember, R; Greenough, J

2005-10-18

We present a patch-based direct Eulerian adaptive mesh refinement (AMR) algorithm for modeling real equation-of-state, multimaterial compressible flow with strength. Our approach to AMR uses a hierarchical, structured grid approach first developed by (Berger and Oliger 1984), (Berger and Oliger 1984). The grid structure is dynamic in time and is composed of nested uniform rectangular grids of varying resolution. The integration scheme on the grid hierarchy is a recursive procedure in which the coarse grids are advanced, then the fine grids are advanced multiple steps to reach the same time, and finally the coarse and fine grids are synchronized tomore » remove conservation errors during the separate advances. The methodology presented here is based on a single grid algorithm developed for multimaterial gas dynamics by (Colella et al. 1993), refined by(Greenough et al. 1995), and extended to the solution of solid mechanics problems with significant strength by (Lomov and Rubin 2003). The single grid algorithm uses a second-order Godunov scheme with an approximate single fluid Riemann solver and a volume-of-fluid treatment of material interfaces. The method also uses a non-conservative treatment of the deformation tensor and an acoustic approximation for shear waves in the Riemann solver. This departure from a strict application of the higher-order Godunov methodology to the equation of solid mechanics is justified due to the fact that highly nonlinear behavior of shear stresses is rare. This algorithm is implemented in two codes, Geodyn and Raptor, the latter of which is a coupled rad-hydro code. The present discussion will be solely concerned with hydrodynamics modeling. Results from a number of simulations for flows with and without strength will be presented.« less

Exact analytic solution for the spin-up maneuver of an axially symmetric spacecraft

NASA Astrophysics Data System (ADS)

Ventura, Jacopo; Romano, Marcello

2014-11-01

The problem of spinning-up an axially symmetric spacecraft subjected to an external torque constant in magnitude and parallel to the symmetry axis is considered. The existing exact analytic solution for an axially symmetric body is applied for the first time to this problem. The proposed solution is valid for any initial conditions of attitude and angular velocity and for any length of time and rotation amplitude. Furthermore, the proposed solution can be numerically evaluated up to any desired level of accuracy. Numerical experiments and comparison with an existing approximated solution and with the integration of the equations of motion are reported in the paper. Finally, a new approximated solution obtained from the exact one is introduced in this paper.

An exact stiffness theory for unidirectional xFRP composites

NASA Astrophysics Data System (ADS)

Klasztorny, M.; Konderla, P.; Piekarski, R.

2009-01-01

UD xFRP composites, i.e., isotropic plastics reinforced with long transversely isotropic fibres packed unidirectionally according to the hexagonal scheme are considered. The constituent materials are geometrically and physically linear. The previous formulations of the exact stiffness theory of such composites are revised, and the theory is developed further based on selected boundary-value problems of elasticity theory. The numerical examples presented are focussed on testing the theory with account of previous variants of this theory and experimental values of the effective elastic constants. The authors have pointed out that the exact stiffness theory of UD xFRP composites, with the modifications proposed in our study, will be useful in the engineering practice and in solving the current problems of the mechanics of composite materials.

Exact axisymmetric solutions of the Maxwell equations in a nonlinear nondispersive medium.

PubMed

Petrov, E Yu; Kudrin, A V

2010-05-14

The features of propagation of intense waves are of great interest for theory and experiment in electrodynamics and acoustics. The behavior of nonlinear waves in a bounded volume is of special importance and, at the same time, is an extremely complicated problem. It seems almost impossible to find a rigorous solution to such a problem even for any model of nonlinearity. We obtain the first exact solution of this type. We present a new method for deriving exact solutions of the Maxwell equations in a nonlinear medium without dispersion and give examples of the obtained solutions that describe propagation of cylindrical electromagnetic waves in a nonlinear nondispersive medium and free electromagnetic oscillations in a cylindrical cavity resonator filled with such a medium.

The Davey-Stewartson Equation on the Half-Plane

NASA Astrophysics Data System (ADS)

Fokas, A. S.

2009-08-01

The Davey-Stewartson (DS) equation is a nonlinear integrable evolution equation in two spatial dimensions. It provides a multidimensional generalisation of the celebrated nonlinear Schrödinger (NLS) equation and it appears in several physical situations. The implementation of the Inverse Scattering Transform (IST) to the solution of the initial-value problem of the NLS was presented in 1972, whereas the analogous problem for the DS equation was solved in 1983. These results are based on the formulation and solution of certain classical problems in complex analysis, namely of a Riemann Hilbert problem (RH) and of either a d-bar or a non-local RH problem respectively. A method for solving the mathematically more complicated but physically more relevant case of boundary-value problems for evolution equations in one spatial dimension, like the NLS, was finally presented in 1997, after interjecting several novel ideas to the panoply of the IST methodology. Here, this method is further extended so that it can be applied to evolution equations in two spatial dimensions, like the DS equation. This novel extension involves several new steps, including the formulation of a d-bar problem for a sectionally non-analytic function, i.e. for a function which has different non-analytic representations in different domains of the complex plane. This, in addition to the computation of a d-bar derivative, also requires the computation of the relevant jumps across the different domains. This latter step has certain similarities (but is more complicated) with the corresponding step for those initial-value problems in two dimensions which can be solved via a non-local RH problem, like KPI.

On a New Trigonometric Identity

ERIC Educational Resources Information Center

Chen, Hongwei

2002-01-01

A new trigonometric identity derived from factorizations and partial fractions is given. This identity is used to evaluate the Poisson integral via Riemann sum and to establish some trigonometric summation identities.

Riemann sum method for non-line-of-sight ultraviolet communication in noncoplanar geometry

NASA Astrophysics Data System (ADS)

Song, Peng; Zhou, Xianli; Song, Fei; Zhao, Taifei; Li, Yunhong

2017-12-01

The non-line-of-sight ultraviolet (UV) communication relies on the scattering common volume, however, it is difficult to carry out the triple integral operation of the scattering common volume. Based on UV single-scattering propagation theory and the spherical coordinate, we propose to use the Riemann sum method (RSM) to analyze the link path loss (PL) of UV communication system in noncoplanar geometries, and carried out related simulations. In addition, an outdoor testbed using UV light-emitting diode was set up to provide support for the validity of the RSM. When the elevation angles of the transmitter or the receiver are small, using RSM, the channel PL and temporal response of UV communication systems can be effectively and efficiently calculated. It is useful in UV embedded system design.

The range and valence of a real Smirnov function

NASA Astrophysics Data System (ADS)

Ferguson, Timothy; Ross, William T.

2018-02-01

We give a complete description of the possible ranges of real Smirnov functions (quotients of two bounded analytic functions on the open unit disk where the denominator is outer and such that the radial boundary values are real almost everywhere on the unit circle). Our techniques use the theory of unbounded symmetric Toeplitz operators, some general theory of unbounded symmetric operators, classical Hardy spaces, and an application of the uniformization theorem. In addition, we completely characterize the possible valences for these real Smirnov functions when the valence is finite. To do so we construct Riemann surfaces we call disk trees by welding together copies of the unit disk and its complement in the Riemann sphere. We also make use of certain trees we call valence trees that mirror the structure of disk trees.

Solving the Cauchy-Riemann equations on parallel computers

NASA Technical Reports Server (NTRS)

Fatoohi, Raad A.; Grosch, Chester E.

1987-01-01

Discussed is the implementation of a single algorithm on three parallel-vector computers. The algorithm is a relaxation scheme for the solution of the Cauchy-Riemann equations; a set of coupled first order partial differential equations. The computers were chosen so as to encompass a variety of architectures. They are: the MPP, and SIMD machine with 16K bit serial processors; FLEX/32, an MIMD machine with 20 processors; and CRAY/2, an MIMD machine with four vector processors. The machine architectures are briefly described. The implementation of the algorithm is discussed in relation to these architectures and measures of the performance on each machine are given. Simple performance models are used to describe the performance. These models highlight the bottlenecks and limiting factors for this algorithm on these architectures. Conclusions are presented.

The median problems on linear multichromosomal genomes: graph representation and fast exact solutions.

PubMed

Xu, Andrew Wei

2010-09-01

In genome rearrangement, given a set of genomes G and a distance measure d, the median problem asks for another genome q that minimizes the total distance [Formula: see text]. This is a key problem in genome rearrangement based phylogenetic analysis. Although this problem is known to be NP-hard, we have shown in a previous article, on circular genomes and under the DCJ distance measure, that a family of patterns in the given genomes--represented by adequate subgraphs--allow us to rapidly find exact solutions to the median problem in a decomposition approach. In this article, we extend this result to the case of linear multichromosomal genomes, in order to solve more interesting problems on eukaryotic nuclear genomes. A multi-way capping problem in the linear multichromosomal case imposes an extra computational challenge on top of the difficulty in the circular case, and this difficulty has been underestimated in our previous study and is addressed in this article. We represent the median problem by the capped multiple breakpoint graph, extend the adequate subgraphs into the capped adequate subgraphs, and prove optimality-preserving decomposition theorems, which give us the tools to solve the median problem and the multi-way capping optimization problem together. We also develop an exact algorithm ASMedian-linear, which iteratively detects instances of (capped) adequate subgraphs and decomposes problems into subproblems. Tested on simulated data, ASMedian-linear can rapidly solve most problems with up to several thousand genes, and it also can provide optimal or near-optimal solutions to the median problem under the reversal/HP distance measures. ASMedian-linear is available at http://sites.google.com/site/andrewweixu .

Electromagnetic Radiation Reaction in General Relativity.

NASA Astrophysics Data System (ADS)

O'Donnell, Nuala

Available from UMI in association with The British Library. This thesis examines the electromagnetic radiation reaction felt by a charged body falling freely in an external gravitational field in general relativity. The original objective was to find a new derivation of the radiation reaction force F^{i} of DeWitt and DeWitt^1 which was calculated for the special case of a point charge falling in slow motion in a weak, static gravitational field: F ^{i} = {2over 3}e^2R^{i}_{0j0 }v^{j}. This may be thought of as a local expression since it involves the particle's velocity v^{j } and the local Riemann curvature tensor R ^{i}_{0j0}. Its derivation involves integrals over the whole history of the particle, covering distances of approximately the length scale on which R^{i}_{0j0 } changes. This is different from calculations of the Abraham-Lorentz force of flat space-time involving integrals over distances only a few times the size of the charge. This work was motivated by the wish to find a "local" derivation of the local reaction force. Using Schutz's^2 local initial value method to solve the problem of a charged, rigid, spherically symmetric body moving in an external gravitational field of arbitrary metric. Calculations are done in a Riemann normal coordinate system ^3 and are only valid in a normal neighbourhood of the origin, where geodesics have not begun to cross one another. We solve Maxwell's equations for the self -force by making a slow-motion approximation and keeping terms to first order only in the Riemann tensor and velocity. It is surprising that we find no local radiation reaction. Consider two particles in a static spacetime with the same initial conditions at t = 0. Particle A is that of DeWitt and DeWitt; it feels a reaction force F^{i} = {2over 3}e^2R^{i }_{0j0}v^{j}. Particle B is accelerated from rest to the same small velocity; it feels no reaction force. The two particles therefore follow different trajectories. We conclude that there is a certain amount of history dependence in curved spacetime which is absent in flat spacetime where the Abraham-Lorentz reaction force acts equally on both particles. ftn ^1C. M. DeWitt and B. S. Brehme, Falling Charges, Phys., 1, 3 (1964). ^2B. F. Schutz, Statistical Formulation of Gravitational Radiation Reaction, Phys. Rev. D., 22, 249 (1980). ^3See for example A. Z. Petrov, Einstein Spaces, p.33, Pergamon Press (1969).

Entanglement bases and general structures of orthogonal complete bases

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zhong Zaizhe

2004-10-01

In quantum mechanics and quantum information, to establish the orthogonal bases is a useful means. The existence of unextendible product bases impels us to study the 'entanglement bases' problems. In this paper, the concepts of entanglement bases and exact-entanglement bases are defined, and a theorem about exact-entanglement bases is given. We discuss the general structures of the orthogonal complete bases. Two examples of applications are given. At last, we discuss the problem of transformation of the general structure forms.

Efficient exact-exchange time-dependent density-functional theory methods and their relation to time-dependent Hartree-Fock.

PubMed

Hesselmann, Andreas; Görling, Andreas

2011-01-21

A recently introduced time-dependent exact-exchange (TDEXX) method, i.e., a response method based on time-dependent density-functional theory that treats the frequency-dependent exchange kernel exactly, is reformulated. In the reformulated version of the TDEXX method electronic excitation energies can be calculated by solving a linear generalized eigenvalue problem while in the original version of the TDEXX method a laborious frequency iteration is required in the calculation of each excitation energy. The lowest eigenvalues of the new TDEXX eigenvalue equation corresponding to the lowest excitation energies can be efficiently obtained by, e.g., a version of the Davidson algorithm appropriate for generalized eigenvalue problems. Alternatively, with the help of a series expansion of the new TDEXX eigenvalue equation, standard eigensolvers for large regular eigenvalue problems, e.g., the standard Davidson algorithm, can be used to efficiently calculate the lowest excitation energies. With the help of the series expansion as well, the relation between the TDEXX method and time-dependent Hartree-Fock is analyzed. Several ways to take into account correlation in addition to the exact treatment of exchange in the TDEXX method are discussed, e.g., a scaling of the Kohn-Sham eigenvalues, the inclusion of (semi)local approximate correlation potentials, or hybrids of the exact-exchange kernel with kernels within the adiabatic local density approximation. The lowest lying excitations of the molecules ethylene, acetaldehyde, and pyridine are considered as examples.

Entanglement dynamics in a non-Markovian environment: An exactly solvable model

NASA Astrophysics Data System (ADS)

Wilson, Justin H.; Fregoso, Benjamin M.; Galitski, Victor M.

2012-05-01

We study the non-Markovian effects on the dynamics of entanglement in an exactly solvable model that involves two independent oscillators, each coupled to its own stochastic noise source. First, we develop Lie algebraic and functional integral methods to find an exact solution to the single-oscillator problem which includes an analytic expression for the density matrix and the complete statistics, i.e., the probability distribution functions for observables. For long bath time correlations, we see nonmonotonic evolution of the uncertainties in observables. Further, we extend this exact solution to the two-particle problem and find the dynamics of entanglement in a subspace. We find the phenomena of “sudden death” and “rebirth” of entanglement. Interestingly, all memory effects enter via the functional form of the energy and hence the time of death and rebirth is controlled by the amount of noisy energy added into each oscillator. If this energy increases above (decreases below) a threshold, we obtain sudden death (rebirth) of entanglement.

Exact-Output Tracking Theory for Systems with Parameter Jumps

NASA Technical Reports Server (NTRS)

Devasia, Santosh; Paden, Brad; Rossi, Carlo

1996-01-01

In this paper we consider the exact output tracking problem for systems with parameter jumps. Necessary and sufficient conditions are derived for the elimination of switching-introduced output transient. Previous works have studied this problem by developing a regulator that maintains exact tracking through parameter jumps (switches). Such techniques are, however, only applicable to minimum-phase systems. In contrast, our approach is applicable to nonminimum-phase systems and obtains bounded but possibly non-causal solutions. If the reference trajectories are generated by an exo-system, then we develop an exact-tracking controller in a feedback form. As in standard regulator theory, we obtain a linear map from the states of the exo-system to the desired system state which is defined via a matrix differential equation. The constant solution of this differential equation provides asymptotic tracking, and coincides with the feedback law used in standard regulator theory. The obtained results are applied to a simple flexible manipulator with jumps in the pay-load mass.

Exact-Output Tracking Theory for Systems with Parameter Jumps

NASA Technical Reports Server (NTRS)

Devasia, Santosh; Paden, Brad; Rossi, Carlo

1997-01-01

We consider the exact output tracking problem for systems with parameter jumps. Necessary and sufficient conditions are derived for the elimination of switching-introduced output transient. Previous works have studied this problem by developing a regulator that maintains exact tracking through parameter jumps (switches). Such techniques are, however, only applicable to minimum-phase systems. In contrast, our approach is applicable to non-minimum-phase systems and it obtains bounded but possibly non-causal solutions. If the reference trajectories are generated by an exosystem, then we develop an exact-tracking controller in a feed-back form. As in standard regulator theory, we obtain a linear map from the states of the exosystem to the desired system state which is defined via a matrix differential equation. The constant solution of this differential equation provides asymptotic tracking, and coincides with the feedback law used in standard regulator theory. The obtained results are applied to a simple flexible manipulator with jumps in the pay-load mass.

Eshelby problem of polygonal inclusions in anisotropic piezoelectric full- and half-planes

NASA Astrophysics Data System (ADS)

Pan, E.

2004-03-01

This paper presents an exact closed-form solution for the Eshelby problem of polygonal inclusion in anisotropic piezoelectric full- and half-planes. Based on the equivalent body-force concept of eigenstrain, the induced elastic and piezoelectric fields are first expressed in terms of line integral on the boundary of the inclusion with the integrand being the Green's function. Using the recently derived exact closed-form line-source Green's function, the line integral is then carried out analytically, with the final expression involving only elementary functions. The exact closed-form solution is applied to a square-shaped quantum wire within semiconductor GaAs full- and half-planes, with results clearly showing the importance of material orientation and piezoelectric coupling. While the elastic and piezoelectric fields within the square-shaped quantum wire could serve as benchmarks to other numerical methods, the exact closed-form solution should be useful to the analysis of nanoscale quantum-wire structures where large strain and electric fields could be induced by the misfit strain.

A new exact method for line radiative transfer

NASA Astrophysics Data System (ADS)

Elitzur, Moshe; Asensio Ramos, Andrés

2006-01-01

We present a new method, the coupled escape probability (CEP), for exact calculation of line emission from multi-level systems, solving only algebraic equations for the level populations. The CEP formulation of the classical two-level problem is a set of linear equations, and we uncover an exact analytic expression for the emission from two-level optically thick sources that holds as long as they are in the `effectively thin' regime. In a comparative study of a number of standard problems, the CEP method outperformed the leading line transfer methods by substantial margins. The algebraic equations employed by our new method are already incorporated in numerous codes based on the escape probability approximation. All that is required for an exact solution with these existing codes is to augment the expression for the escape probability with simple zone-coupling terms. As an application, we find that standard escape probability calculations generally produce the correct cooling emission by the CII 158-μm line but not by the 3P lines of OI.

The exact thermal rotational spectrum of a two-dimensional rigid rotor obtained using Gaussian wave packet dynamics

NASA Technical Reports Server (NTRS)

Reimers, J. R.; Heller, E. J.

1985-01-01

The exact thermal rotational spectrum of a two-dimensional rigid rotor is obtained using Gaussian wave packet dynamics. The spectrum is obtained by propagating, without approximation, infinite sets of Gaussian wave packets. These sets are constructed so that collectively they have the correct periodicity, and indeed, are coherent states appropriate to this problem. Also, simple, almost classical, approximations to full wave packet dynamics are shown to give results which are either exact or very nearly exact. Advantages of the use of Gaussian wave packet dynamics over conventional linear response theory are discussed.

Evolution of initial discontinuities in the Riemann problem for the Kaup-Boussinesq equation with positive dispersion

NASA Astrophysics Data System (ADS)

Congy, T.; Ivanov, S. K.; Kamchatnov, A. M.; Pavloff, N.

2017-08-01

We consider the space-time evolution of initial discontinuities of depth and flow velocity for an integrable version of the shallow water Boussinesq system introduced by Kaup. We focus on a specific version of this "Kaup-Boussinesq model" for which a flat water surface is modulationally stable, we speak below of "positive dispersion" model. This model also appears as an approximation to the equations governing the dynamics of polarisation waves in two-component Bose-Einstein condensates. We describe its periodic solutions and the corresponding Whitham modulation equations. The self-similar, one-phase wave structures are composed of different building blocks, which are studied in detail. This makes it possible to establish a classification of all the possible wave configurations evolving from initial discontinuities. The analytic results are confirmed by numerical simulations.

Evolution of initial discontinuities in the Riemann problem for the Kaup-Boussinesq equation with positive dispersion.

PubMed

Congy, T; Ivanov, S K; Kamchatnov, A M; Pavloff, N

2017-08-01

We consider the space-time evolution of initial discontinuities of depth and flow velocity for an integrable version of the shallow water Boussinesq system introduced by Kaup. We focus on a specific version of this "Kaup-Boussinesq model" for which a flat water surface is modulationally stable, we speak below of "positive dispersion" model. This model also appears as an approximation to the equations governing the dynamics of polarisation waves in two-component Bose-Einstein condensates. We describe its periodic solutions and the corresponding Whitham modulation equations. The self-similar, one-phase wave structures are composed of different building blocks, which are studied in detail. This makes it possible to establish a classification of all the possible wave configurations evolving from initial discontinuities. The analytic results are confirmed by numerical simulations.

EOSlib, Version 3

DOE Office of Scientific and Technical Information (OSTI.GOV)

Woods, Nathan; Menikoff, Ralph

2017-02-03

Equilibrium thermodynamics underpins many of the technologies used throughout theoretical physics, yet verification of the various theoretical models in the open literature remains challenging. EOSlib provides a single, consistent, verifiable implementation of these models, in a single, easy-to-use software package. It consists of three parts: a software library implementing various published equation-of-state (EOS) models; a database of fitting parameters for various materials for these models; and a number of useful utility functions for simplifying thermodynamic calculations such as computing Hugoniot curves or Riemann problem solutions. Ready availability of this library will enable reliable code-to- code testing of equation-of-state implementations, asmore » well as a starting point for more rigorous verification work. EOSlib also provides a single, consistent API for its analytic and tabular EOS models, which simplifies the process of comparing models for a particular application.« less

Studies of Plasma Instabilities using Unstructured Discontinuous Galerkin Method with the Two-Fluid Plasma Model

NASA Astrophysics Data System (ADS)

Song, Yang; Srinivasan, Bhuvana

2017-10-01

The discontinuous Galerkin (DG) method has the advantage of resolving shocks and sharp gradients that occur in neutral fluids and plasmas. An unstructured DG code has been developed in this work to study plasma instabilities using the two-fluid plasma model. Unstructured meshes are known to produce small and randomized grid errors compared to traditional structured meshes. Computational tests for Rayleigh-Taylor instabilities in radially-converging flows are performed using the MHD model. Choice of grid geometry is not obvious for simulations of instabilities in these circular configurations. Comparisons of the effects for different grids are made. A 2D magnetic nozzle simulation using the two-fluid plasma model is also performed. A vacuum boundary condition technique is applied to accurately solve the Riemann problem on the edge of the plume.

Treatment of pairing correlations based on the equations of motion for zero-coupled pair operators

NASA Astrophysics Data System (ADS)

Andreozzi, F.; Covello, A.; Gargano, A.; Ye, Liu Jian; Porrino, A.

1985-07-01

The pairing problem is treated by means of the equations of motion for zero-coupled pair operators. Exact equations for the seniority-v states of N particles are derived. These equations can be solved by a step-by-step procedure which consists of progressively adding pairs of particles to a core. The theory can be applied at several levels of approximation depending on the number of core states which are taken into account. Some numerical applications to the treatment of v=0, v=1, and v=2 states in the Ni isotopes are performed. The accuracy of various approximations is tested by comparison with exact results. For the seniority-one and seniority-two problems it turns out that the results obtained from the first-order theory are very accurate, while those of higher order calculations are practically exact. Concerning the seniority-zero problem, a fifth-order calculation reproduces quite well the three lowest states.

An updated Lagrangian discontinuous Galerkin hydrodynamic method for gas dynamics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Wu, Tong; Shashkov, Mikhail Jurievich; Morgan, Nathaniel Ray

Here, we present a new Lagrangian discontinuous Galerkin (DG) hydrodynamic method for gas dynamics. The new method evolves conserved unknowns in the current configuration, which obviates the Jacobi matrix that maps the element in a reference coordinate system or the initial coordinate system to the current configuration. The density, momentum, and total energy (ρ, ρu, E) are approximated with conservative higher-order Taylor expansions over the element and are limited toward a piecewise constant field near discontinuities using a limiter. Two new limiting methods are presented for enforcing the bounds on the primitive variables of density, velocity, and specific internal energymore » (ρ, u, e). The nodal velocity, and the corresponding forces, are calculated by solving an approximate Riemann problem at the element nodes. An explicit second-order method is used to temporally advance the solution. This new Lagrangian DG hydrodynamic method conserves mass, momentum, and total energy. 1D Cartesian coordinates test problem results are presented to demonstrate the accuracy and convergence order of the new DG method with the new limiters.« less

A Lagrangian discontinuous Galerkin hydrodynamic method

DOE Office of Scientific and Technical Information (OSTI.GOV)

Liu, Xiaodong; Morgan, Nathaniel Ray; Burton, Donald E.

Here, we present a new Lagrangian discontinuous Galerkin (DG) hydrodynamic method for solving the two-dimensional gas dynamic equations on unstructured hybrid meshes. The physical conservation laws for the momentum and total energy are discretized using a DG method based on linear Taylor expansions. Three different approaches are investigated for calculating the density variation over the element. The first approach evolves a Taylor expansion of the specific volume field. The second approach follows certain finite element methods and uses the strong mass conservation to calculate the density field at a location inside the element or on the element surface. The thirdmore » approach evolves a Taylor expansion of the density field. The nodal velocity, and the corresponding forces, are explicitly calculated by solving a multidirectional approximate Riemann problem. An effective limiting strategy is presented that ensures monotonicity of the primitive variables. This new Lagrangian DG hydrodynamic method conserves mass, momentum, and total energy. Results from a suite of test problems are presented to demonstrate the robustness and expected second-order accuracy of this new method.« less

Experience in using a numerical scheme with artificial viscosity at solving the Riemann problem for a multi-fluid model of multiphase flow

NASA Astrophysics Data System (ADS)

Bulovich, S. V.; Smirnov, E. M.

2018-05-01

The paper covers application of the artificial viscosity technique to numerical simulation of unsteady one-dimensional multiphase compressible flows on the base of the multi-fluid approach. The system of the governing equations is written under assumption of the pressure equilibrium between the "fluids" (phases). No interfacial exchange is taken into account. A model for evaluation of the artificial viscosity coefficient that (i) assumes identity of this coefficient for all interpenetrating phases and (ii) uses the multiphase-mixture Wood equation for evaluation of a scale speed of sound has been suggested. Performance of the artificial viscosity technique has been evaluated via numerical solution of a model problem of pressure discontinuity breakdown in a three-fluid medium. It has been shown that a relatively simple numerical scheme, explicit and first-order, combined with the suggested artificial viscosity model, predicts a physically correct behavior of the moving shock and expansion waves, and a subsequent refinement of the computational grid results in a monotonic approaching to an asymptotic time-dependent solution, without non-physical oscillations.

Coupled double-distribution-function lattice Boltzmann method for the compressible Navier-Stokes equations.

PubMed

Li, Q; He, Y L; Wang, Y; Tao, W Q

2007-11-01

A coupled double-distribution-function lattice Boltzmann method is developed for the compressible Navier-Stokes equations. Different from existing thermal lattice Boltzmann methods, this method can recover the compressible Navier-Stokes equations with a flexible specific-heat ratio and Prandtl number. In the method, a density distribution function based on a multispeed lattice is used to recover the compressible continuity and momentum equations, while the compressible energy equation is recovered by an energy distribution function. The energy distribution function is then coupled to the density distribution function via the thermal equation of state. In order to obtain an adjustable specific-heat ratio, a constant related to the specific-heat ratio is introduced into the equilibrium energy distribution function. Two different coupled double-distribution-function lattice Boltzmann models are also proposed in the paper. Numerical simulations are performed for the Riemann problem, the double-Mach-reflection problem, and the Couette flow with a range of specific-heat ratios and Prandtl numbers. The numerical results are found to be in excellent agreement with analytical and/or other solutions.

A Lagrangian discontinuous Galerkin hydrodynamic method

DOE PAGES

Liu, Xiaodong; Morgan, Nathaniel Ray; Burton, Donald E.

2017-12-11

Here, we present a new Lagrangian discontinuous Galerkin (DG) hydrodynamic method for solving the two-dimensional gas dynamic equations on unstructured hybrid meshes. The physical conservation laws for the momentum and total energy are discretized using a DG method based on linear Taylor expansions. Three different approaches are investigated for calculating the density variation over the element. The first approach evolves a Taylor expansion of the specific volume field. The second approach follows certain finite element methods and uses the strong mass conservation to calculate the density field at a location inside the element or on the element surface. The thirdmore » approach evolves a Taylor expansion of the density field. The nodal velocity, and the corresponding forces, are explicitly calculated by solving a multidirectional approximate Riemann problem. An effective limiting strategy is presented that ensures monotonicity of the primitive variables. This new Lagrangian DG hydrodynamic method conserves mass, momentum, and total energy. Results from a suite of test problems are presented to demonstrate the robustness and expected second-order accuracy of this new method.« less

Whitham modulation theory for (2  +  1)-dimensional equations of Kadomtsev–Petviashvili type

NASA Astrophysics Data System (ADS)

Ablowitz, Mark J.; Biondini, Gino; Rumanov, Igor

2018-05-01

Whitham modulation theory for certain two-dimensional evolution equations of Kadomtsev–Petviashvili (KP) type is presented. Three specific examples are considered in detail: the KP equation, the two-dimensional Benjamin–Ono (2DBO) equation and a modified KP (m2KP) equation. A unified derivation is also provided. In the case of the m2KP equation, the corresponding Whitham modulation system exhibits features different from the other two. The approach presented here does not require integrability of the original evolution equation. Indeed, while the KP equation is known to be a completely integrable equation, the 2DBO equation and the m2KP equation are not known to be integrable. In each of the cases considered, the Whitham modulation system obtained consists of five first-order quasilinear partial differential equations. The Riemann problem (i.e. the analogue of the Gurevich–Pitaevskii problem) for the one-dimensional reduction of the m2KP equation is studied. For the m2KP equation, the system of modulation equations is used to analyze the linear stability of traveling wave solutions.

Transient fields produced by a cylindrical electron beam flowing through a plasma

NASA Astrophysics Data System (ADS)

Firpo, Marie-Christine

2012-10-01

Fast ignition schemes (FIS) for inertial confinement fusion should involve in their final stage the interaction of an ignition beam composed of MeV electrons laser generated at the critical density surface with a dense plasma target. In this study, the out-of-equilibrium situation in which an initially sharp-edged cylindrical electron beam, that could e.g. model electrons flowing within a wire [1], is injected into a plasma is considered. A detailed computation of the subsequently produced magnetic field is presented [2]. The control parameter of the problem is shown to be the ratio of the beam radius to the electron skin depth. Two alternative ways to address analytically the problem are considered: one uses the usual Laplace transform approach, the other one involves Riemann's method in which causality conditions manifest through some integrals of triple products of Bessel functions.[4pt] [1] J.S. Green et al., Surface heating of wire plasmas using laser-irradiated cone geometries, Nature Physics 3, 853--856 (2007).[0pt] [2] M.-C. Firpo, http://hal.archives-ouvertes.fr/hal-00695629, to be published (2012).

An updated Lagrangian discontinuous Galerkin hydrodynamic method for gas dynamics

DOE PAGES

Wu, Tong; Shashkov, Mikhail Jurievich; Morgan, Nathaniel Ray; ...

2018-04-09

Here, we present a new Lagrangian discontinuous Galerkin (DG) hydrodynamic method for gas dynamics. The new method evolves conserved unknowns in the current configuration, which obviates the Jacobi matrix that maps the element in a reference coordinate system or the initial coordinate system to the current configuration. The density, momentum, and total energy (ρ, ρu, E) are approximated with conservative higher-order Taylor expansions over the element and are limited toward a piecewise constant field near discontinuities using a limiter. Two new limiting methods are presented for enforcing the bounds on the primitive variables of density, velocity, and specific internal energymore » (ρ, u, e). The nodal velocity, and the corresponding forces, are calculated by solving an approximate Riemann problem at the element nodes. An explicit second-order method is used to temporally advance the solution. This new Lagrangian DG hydrodynamic method conserves mass, momentum, and total energy. 1D Cartesian coordinates test problem results are presented to demonstrate the accuracy and convergence order of the new DG method with the new limiters.« less

Exact solutions for the source-excited cylindrical electromagnetic waves in a nonlinear nondispersive medium.

PubMed

Es'kin, V A; Kudrin, A V; Petrov, E Yu

2011-06-01

The behavior of electromagnetic fields in nonlinear media has been a topical problem since the discovery of materials with a nonlinearity of electromagnetic properties. The problem of finding exact solutions for the source-excited nonlinear waves in curvilinear coordinates has been regarded as unsolvable for a long time. In this work, we present the first solution of this type for a cylindrically symmetric field excited by a pulsed current filament in a nondispersive medium that is simultaneously inhomogeneous and nonlinear. Assuming that the medium has a power-law permittivity profile in the linear regime and lacks a center of inversion, we derive an exact solution for the electromagnetic field excited by a current filament in such a medium and discuss the properties of this solution.

JANUS: a bit-wise reversible integrator for N-body dynamics

NASA Astrophysics Data System (ADS)

Rein, Hanno; Tamayo, Daniel

2018-01-01

Hamiltonian systems such as the gravitational N-body problem have time-reversal symmetry. However, all numerical N-body integration schemes, including symplectic ones, respect this property only approximately. In this paper, we present the new N-body integrator JANUS , for which we achieve exact time-reversal symmetry by combining integer and floating point arithmetic. JANUS is explicit, formally symplectic and satisfies Liouville's theorem exactly. Its order is even and can be adjusted between two and ten. We discuss the implementation of JANUS and present tests of its accuracy and speed by performing and analysing long-term integrations of the Solar system. We show that JANUS is fast and accurate enough to tackle a broad class of dynamical problems. We also discuss the practical and philosophical implications of running exactly time-reversible simulations.

Fahrenheit to Celsius: An Exploration in College Algebra.

ERIC Educational Resources Information Center

Fay, Temple H.; Hardie, Keith A.

2003-01-01

Suggests that the classical exact formula for the conversion of degrees Celsius and degrees Fahrenheit is not user-friendly. Offers an approximate linear transformation that is easier to remember and use. Investigates both the exact conversion and the approximate conversion and provides interesting and relevant problems for small group…

A computational exploration of the McCoy-Tracy-Wu solutions of the third Painlevé equation

NASA Astrophysics Data System (ADS)

Fasondini, Marco; Fornberg, Bengt; Weideman, J. A. C.

2018-01-01

The method recently developed by the authors for the computation of the multivalued Painlevé transcendents on their Riemann surfaces (Fasondini et al., 2017) is used to explore families of solutions to the third Painlevé equation that were identified by McCoy et al. (1977) and which contain a pole-free sector. Limiting cases, in which the solutions are singular functions of the parameters, are also investigated and it is shown that a particular set of limiting solutions is expressible in terms of special functions. Solutions that are single-valued, logarithmically (infinitely) branched and algebraically branched, with any number of distinct sheets, are encountered. The algebraically branched solutions have multiple pole-free sectors on their Riemann surfaces that are accounted for by using asymptotic formulae and Bäcklund transformations.

Riemann curvature of a boosted spacetime geometry

NASA Astrophysics Data System (ADS)

Battista, Emmanuele; Esposito, Giampiero; Scudellaro, Paolo; Tramontano, Francesco

2016-10-01

The ultrarelativistic boosting procedure had been applied in the literature to map the metric of Schwarzschild-de Sitter spacetime into a metric describing de Sitter spacetime plus a shock-wave singularity located on a null hypersurface. This paper evaluates the Riemann curvature tensor of the boosted Schwarzschild-de Sitter metric by means of numerical calculations, which make it possible to reach the ultrarelativistic regime gradually by letting the boost velocity approach the speed of light. Thus, for the first time in the literature, the singular limit of curvature, through Dirac’s δ distribution and its derivatives, is numerically evaluated for this class of spacetimes. Moreover, the analysis of the Kretschmann invariant and the geodesic equation shows that the spacetime possesses a “scalar curvature singularity” within a 3-sphere and it is possible to define what we here call “boosted horizon”, a sort of elastic wall where all particles are surprisingly pushed away, as numerical analysis demonstrates. This seems to suggest that such “boosted geometries” are ruled by a sort of “antigravity effect” since all geodesics seem to refuse to enter the “boosted horizon” and are “reflected” by it, even though their initial conditions are aimed at driving the particles toward the “boosted horizon” itself. Eventually, the equivalence with the coordinate shift method is invoked in order to demonstrate that all δ2 terms appearing in the Riemann curvature tensor give vanishing contribution in distributional sense.

The Riesz-Radon-Fréchet problem of characterization of integrals

NASA Astrophysics Data System (ADS)

Zakharov, Valerii K.; Mikhalev, Aleksandr V.; Rodionov, Timofey V.

2010-11-01

This paper is a survey of results on characterizing integrals as linear functionals. It starts from the familiar result of F. Riesz (1909) on integral representation of bounded linear functionals by Riemann-Stieltjes integrals on a closed interval, and is directly connected with Radon's famous theorem (1913) on integral representation of bounded linear functionals by Lebesgue integrals on a compact subset of {R}^n. After the works of Radon, Fréchet, and Hausdorff, the problem of characterizing integrals as linear functionals took the particular form of the problem of extending Radon's theorem from {R}^n to more general topological spaces with Radon measures. This problem turned out to be difficult, and its solution has a long and rich history. Therefore, it is natural to call it the Riesz-Radon-Fréchet problem of characterization of integrals. Important stages of its solution are associated with such eminent mathematicians as Banach (1937-1938), Saks (1937-1938), Kakutani (1941), Halmos (1950), Hewitt (1952), Edwards (1953), Prokhorov (1956), Bourbaki (1969), and others. Essential ideas and technical tools were developed by A.D. Alexandrov (1940-1943), Stone (1948-1949), Fremlin (1974), and others. Most of this paper is devoted to the contemporary stage of the solution of the problem, connected with papers of König (1995-2008), Zakharov and Mikhalev (1997-2009), and others. The general solution of the problem is presented in the form of a parametric theorem on characterization of integrals which directly implies the characterization theorems of the indicated authors. Bibliography: 60 titles.

Solving large-scale fixed cost integer linear programming models for grid-based location problems with heuristic techniques

NASA Astrophysics Data System (ADS)

Noor-E-Alam, Md.; Doucette, John

2015-08-01

Grid-based location problems (GBLPs) can be used to solve location problems in business, engineering, resource exploitation, and even in the field of medical sciences. To solve these decision problems, an integer linear programming (ILP) model is designed and developed to provide the optimal solution for GBLPs considering fixed cost criteria. Preliminary results show that the ILP model is efficient in solving small to moderate-sized problems. However, this ILP model becomes intractable in solving large-scale instances. Therefore, a decomposition heuristic is proposed to solve these large-scale GBLPs, which demonstrates significant reduction of solution runtimes. To benchmark the proposed heuristic, results are compared with the exact solution via ILP. The experimental results show that the proposed method significantly outperforms the exact method in runtime with minimal (and in most cases, no) loss of optimality.

Geometrically derived difference formulae for the numerical integration of trajectory problems

NASA Technical Reports Server (NTRS)

Mcleod, R. J. Y.; Sanz-Serna, J. M.

1982-01-01

An initial value problem for the autonomous system of ordinary differential equations dy/dt = f(y), where y is a vector, is considered. In a number of practical applications the interest lies in obtaining the curve traced by the solution y. These applications include the computation of trajectories in mechanical problems. The term 'trajectory problem' is employed to refer to these cases. Lambert and McLeod (1979) have introduced a method involving local rotation of the axes in the y-plane for the two-dimensional case. The present investigation continues the study of difference schemes specifically derived for trajectory problems. A simple geometrical way of constructing such methods is presented, and the local accuracy of the schemes is investigated. A circularly exact, fixed-step predictor-corrector algorithm is defined, and a variable-step version of a circularly exact algorithm is presented.

Exact finite element method analysis of viscoelastic tapered structures to transient loads

NASA Technical Reports Server (NTRS)

Spyrakos, Constantine Chris

1987-01-01

A general method is presented for determining the dynamic torsional/axial response of linear structures composed of either tapered bars or shafts to transient excitations. The method consists of formulating and solving the dynamic problem in the Laplace transform domain by the finite element method and obtaining the response by a numerical inversion of the transformed solution. The derivation of the torsional and axial stiffness matrices is based on the exact solution of the transformed governing equation of motion, and it consequently leads to the exact solution of the problem. The solution permits treatment of the most practical cases of linear tapered bars and shafts, and employs modeling of structures with only one element per member which reduces the number of degrees of freedom involved. The effects of external viscous or internal viscoelastic damping are also taken into account.

Continuum-Kinetic Models and Numerical Methods for Multiphase Applications

NASA Astrophysics Data System (ADS)

Nault, Isaac Michael

This thesis presents a continuum-kinetic approach for modeling general problems in multiphase solid mechanics. In this context, a continuum model refers to any model, typically on the macro-scale, in which continuous state variables are used to capture the most important physics: conservation of mass, momentum, and energy. A kinetic model refers to any model, typically on the meso-scale, which captures the statistical motion and evolution of microscopic entitites. Multiphase phenomena usually involve non-negligible micro or meso-scopic effects at the interfaces between phases. The approach developed in the thesis attempts to combine the computational performance benefits of a continuum model with the physical accuracy of a kinetic model when applied to a multiphase problem. The approach is applied to modeling a single particle impact in Cold Spray, an engineering process that intimately involves the interaction of crystal grains with high-magnitude elastic waves. Such a situation could be classified a multiphase application due to the discrete nature of grains on the spatial scale of the problem. For this application, a hyper elasto-plastic model is solved by a finite volume method with approximate Riemann solver. The results of this model are compared for two types of plastic closure: a phenomenological macro-scale constitutive law, and a physics-based meso-scale Crystal Plasticity model.

Two-dimensional CFD modeling of wave rotor flow dynamics

NASA Technical Reports Server (NTRS)

Welch, Gerard E.; Chima, Rodrick V.

1994-01-01

A two-dimensional Navier-Stokes solver developed for detailed study of wave rotor flow dynamics is described. The CFD model is helping characterize important loss mechanisms within the wave rotor. The wave rotor stationary ports and the moving rotor passages are resolved on multiple computational grid blocks. The finite-volume form of the thin-layer Navier-Stokes equations with laminar viscosity are integrated in time using a four-stage Runge-Kutta scheme. Roe's approximate Riemann solution scheme or the computationally less expensive advection upstream splitting method (AUSM) flux-splitting scheme is used to effect upwind-differencing of the inviscid flux terms, using cell interface primitive variables set by MUSCL-type interpolation. The diffusion terms are central-differenced. The solver is validated using a steady shock/laminar boundary layer interaction problem and an unsteady, inviscid wave rotor passage gradual opening problem. A model inlet port/passage charging problem is simulated and key features of the unsteady wave rotor flow field are identified. Lastly, the medium pressure inlet port and high pressure outlet port portion of the NASA Lewis Research Center experimental divider cycle is simulated and computed results are compared with experimental measurements. The model accurately predicts the wave timing within the rotor passages and the distribution of flow variables in the stationary inlet port region.

Two-dimensional CFD modeling of wave rotor flow dynamics

NASA Technical Reports Server (NTRS)

Welch, Gerard E.; Chima, Rodrick V.

1993-01-01

A two-dimensional Navier-Stokes solver developed for detailed study of wave rotor flow dynamics is described. The CFD model is helping characterize important loss mechanisms within the wave rotor. The wave rotor stationary ports and the moving rotor passages are resolved on multiple computational grid blocks. The finite-volume form of the thin-layer Navier-Stokes equations with laminar viscosity are integrated in time using a four-stage Runge-Kutta scheme. The Roe approximate Riemann solution scheme or the computationally less expensive Advection Upstream Splitting Method (AUSM) flux-splitting scheme are used to effect upwind-differencing of the inviscid flux terms, using cell interface primitive variables set by MUSCL-type interpolation. The diffusion terms are central-differenced. The solver is validated using a steady shock/laminar boundary layer interaction problem and an unsteady, inviscid wave rotor passage gradual opening problem. A model inlet port/passage charging problem is simulated and key features of the unsteady wave rotor flow field are identified. Lastly, the medium pressure inlet port and high pressure outlet port portion of the NASA Lewis Research Center experimental divider cycle is simulated and computed results are compared with experimental measurements. The model accurately predicts the wave timing within the rotor passage and the distribution of flow variables in the stationary inlet port region.

Jet Simulation in a Diesel Engine

NASA Astrophysics Data System (ADS)

Xu, Zhiliang

2005-03-01

We present a numerical study of the jet breakup and spray formation in a diesel engine by the Front Tracking method. The mechanisms of jet breakup and spray formation of a high speed diesel jet injected through a circular nozzle are the key to design a fuel efficient, nonpolluting diesel engine. We conduct the simulations for the jet breakup within a 2D axis-symmetric geometry. Our goal is to model the spray at a micro-physical level, with the creation of individual droplets. The problem is multiscale. The droplets are a few microns in size. The nozzle is about 0.2 mm in diameter and 1 mm in length. To resolve various physical patterns such as vortex, shock waves, vacuum and track droplets and spray, the Burger-Colella adaptive mesh refinement technique is used. To simulate the spray formation, we model mixed vapor-liquid region through a heterogeneous model with dynamic vapor bubble insertion. The formation of the cavitation is represented by the dynamic creation of vapor bubbles. On the liquid/vapor interface, a phase transition problem is solved numerically. The phase transition is governed by the compressible Euler equations with heat diffusion. Our solution is a new description for the Riemann problem associated with a phase transition in a fully compressible fluid.

Generalized Riemann hypothesis and stochastic time series

NASA Astrophysics Data System (ADS)

Mussardo, Giuseppe; LeClair, André

2018-06-01

Using the Dirichlet theorem on the equidistribution of residue classes modulo q and the Lemke Oliver–Soundararajan conjecture on the distribution of pairs of residues on consecutive primes, we show that the domain of convergence of the infinite product of Dirichlet L-functions of non-principal characters can be extended from down to , without encountering any zeros before reaching this critical line. The possibility of doing so can be traced back to a universal diffusive random walk behavior of a series C N over the primes which underlies the convergence of the infinite product of the Dirichlet functions. The series C N presents several aspects in common with stochastic time series and its control requires to address a problem similar to the single Brownian trajectory problem in statistical mechanics. In the case of the Dirichlet functions of non principal characters, we show that this problem can be solved in terms of a self-averaging procedure based on an ensemble of block variables computed on extended intervals of primes. Those intervals, called inertial intervals, ensure the ergodicity and stationarity of the time series underlying the quantity C N . The infinity of primes also ensures the absence of rare events which would have been responsible for a different scaling behavior than the universal law of the random walks.

Improved method for implicit Monte Carlo

DOE Office of Scientific and Technical Information (OSTI.GOV)

Brown, F. B.; Martin, W. R.

2001-01-01

The Implicit Monte Carlo (IMC) method has been used for over 30 years to analyze radiative transfer problems, such as those encountered in stellar atmospheres or inertial confinement fusion. Reference [2] provided an exact error analysis of IMC for 0-D problems and demonstrated that IMC can exhibit substantial errors when timesteps are large. These temporal errors are inherent in the method and are in addition to spatial discretization errors and approximations that address nonlinearities (due to variation of physical constants). In Reference [3], IMC and four other methods were analyzed in detail and compared on both theoretical grounds and themore » accuracy of numerical tests. As discussed in, two alternative schemes for solving the radiative transfer equations, the Carter-Forest (C-F) method and the Ahrens-Larsen (A-L) method, do not exhibit the errors found in IMC; for 0-D, both of these methods are exact for all time, while for 3-D, A-L is exact for all time and C-F is exact within a timestep. These methods can yield substantially superior results to IMC.« less

Exact zeros of entanglement for arbitrary rank-two mixtures derived from a geometric view of the zero polytope

NASA Astrophysics Data System (ADS)

Osterloh, Andreas

2016-12-01

Here I present a method for how intersections of a certain density matrix of rank 2 with the zero polytope can be calculated exactly. This is a purely geometrical procedure which thereby is applicable to obtaining the zeros of SL- and SU-invariant entanglement measures of arbitrary polynomial degree. I explain this method in detail for a recently unsolved problem. In particular, I show how a three-dimensional view, namely, in terms of the Bloch-sphere analogy, solves this problem immediately. To this end, I determine the zero polytope of the three-tangle, which is an exact result up to computer accuracy, and calculate upper bounds to its convex roof which are below the linearized upper bound. The zeros of the three-tangle (in this case) induced by the zero polytope (zero simplex) are exact values. I apply this procedure to a superposition of the four-qubit Greenberger-Horne-Zeilinger and W state. It can, however, be applied to every case one has under consideration, including an arbitrary polynomial convex-roof measure of entanglement and for arbitrary local dimension.

Exact and Heuristic Algorithms for Runway Scheduling

NASA Technical Reports Server (NTRS)

Malik, Waqar A.; Jung, Yoon C.

2016-01-01

This paper explores the Single Runway Scheduling (SRS) problem with arrivals, departures, and crossing aircraft on the airport surface. Constraints for wake vortex separations, departure area navigation separations and departure time window restrictions are explicitly considered. The main objective of this research is to develop exact and heuristic based algorithms that can be used in real-time decision support tools for Air Traffic Control Tower (ATCT) controllers. The paper provides a multi-objective dynamic programming (DP) based algorithm that finds the exact solution to the SRS problem, but may prove unusable for application in real-time environment due to large computation times for moderate sized problems. We next propose a second algorithm that uses heuristics to restrict the search space for the DP based algorithm. A third algorithm based on a combination of insertion and local search (ILS) heuristics is then presented. Simulation conducted for the east side of Dallas/Fort Worth International Airport allows comparison of the three proposed algorithms and indicates that the ILS algorithm performs favorably in its ability to find efficient solutions and its computation times.

Hierarchic models for laminated plates. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Actis, Ricardo Luis

1991-01-01

Structural plates and shells are three-dimensional bodies, one dimension of which happens to be much smaller than the other two. Thus, the quality of a plate or shell model must be judged on the basis of how well its exact solution approximates the corresponding three-dimensional problem. Of course, the exact solution depends not only on the choice of the model but also on the topology, material properties, loading and constraints. The desired degree of approximation depends on the analyst's goals in performing the analysis. For these reasons models have to be chosen adaptively. Hierarchic sequences of models make adaptive selection of the model which is best suited for the purposes of a particular analysis possible. The principles governing the formulation of hierarchic models for laminated plates are presented. The essential features of the hierarchic models described models are: (1) the exact solutions corresponding to the hierarchic sequence of models converge to the exact solution of the corresponding problem of elasticity for a fixed laminate thickness; and (2) the exact solution of each model converges to the same limit as the exact solution of the corresponding problem of elasticity with respect to the laminate thickness approaching zero. The formulation is based on one parameter (beta) which characterizes the hierarchic sequence of models, and a set of constants whose influence was assessed by a numerical sensitivity study. The recommended selection of these constants results in the number of fields increasing by three for each increment in the power of beta. Numerical examples analyzed with the proposed sequence of models are included and good correlation with the reference solutions was found. Results were obtained for laminated strips (plates in cylindrical bending) and for square and rectangular plates with uniform loading and with homogeneous boundary conditions. Cross-ply and angle-ply laminates were evaluated and the results compared with those of MSC/PROBE. Hierarchic models make the computation of any engineering data possible to an arbitrary level of precision within the framework of the theory of elasticity.

Exact solution for a two-phase Stefan problem with variable latent heat and a convective boundary condition at the fixed face

NASA Astrophysics Data System (ADS)

Bollati, Julieta; Tarzia, Domingo A.

2018-04-01

Recently, in Tarzia (Thermal Sci 21A:1-11, 2017) for the classical two-phase Lamé-Clapeyron-Stefan problem an equivalence between the temperature and convective boundary conditions at the fixed face under a certain restriction was obtained. Motivated by this article we study the two-phase Stefan problem for a semi-infinite material with a latent heat defined as a power function of the position and a convective boundary condition at the fixed face. An exact solution is constructed using Kummer functions in case that an inequality for the convective transfer coefficient is satisfied generalizing recent works for the corresponding one-phase free boundary problem. We also consider the limit to our problem when that coefficient goes to infinity obtaining a new free boundary problem, which has been recently studied in Zhou et al. (J Eng Math 2017. https://doi.org/10.1007/s10665-017-9921-y).

Improving Transportation Services for the University of the Thai Chamber of Commerce: A Case Study on Solving the Mixed-Fleet Vehicle Routing Problem with Split Deliveries

NASA Astrophysics Data System (ADS)

Suthikarnnarunai, N.; Olinick, E.

2009-01-01

We present a case study on the application of techniques for solving the Vehicle Routing Problem (VRP) to improve the transportation service provided by the University of The Thai Chamber of Commerce to its staff. The problem is modeled as VRP with time windows, split deliveries, and a mixed fleet. An exact algorithm and a heuristic solution procedure are developed to solve the problem and implemented in the AMPL modeling language and CPLEX Integer Programming solver. Empirical results indicate that the heuristic can find relatively good solutions in a small fraction of the time required by the exact method. We also perform sensitivity analysis and find that a savings in outsourcing cost can be achieved with a small increase in vehicle capacity.

Iso-geometric analysis for neutron diffusion problems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hall, S. K.; Eaton, M. D.; Williams, M. M. R.

Iso-geometric analysis can be viewed as a generalisation of the finite element method. It permits the exact representation of a wider range of geometries including conic sections. This is possible due to the use of concepts employed in computer-aided design. The underlying mathematical representations from computer-aided design are used to capture both the geometry and approximate the solution. In this paper the neutron diffusion equation is solved using iso-geometric analysis. The practical advantages are highlighted by looking at the problem of a circular fuel pin in a square moderator. For this problem the finite element method requires the geometry tomore » be approximated. This leads to errors in the shape and size of the interface between the fuel and the moderator. In contrast to this iso-geometric analysis allows the interface to be represented exactly. It is found that, due to a cancellation of errors, the finite element method converges more quickly than iso-geometric analysis for this problem. A fuel pin in a vacuum was then considered as this problem is highly sensitive to the leakage across the interface. In this case iso-geometric analysis greatly outperforms the finite element method. Due to the improvement in the representation of the geometry iso-geometric analysis can outperform traditional finite element methods. It is proposed that the use of iso-geometric analysis on neutron transport problems will allow deterministic solutions to be obtained for exact geometries. Something that is only currently possible with Monte Carlo techniques. (authors)« less

Shock Dynamics for particle-laden thin film

NASA Astrophysics Data System (ADS)

Wang, Li; Bertozzi, Andrea

2013-11-01

We study the shock dynamics for a recently proposed system of conservation laws (Murisic et al. [J. Fluid Mech. 2013]) describing gravity-driven thin film flow of a suspension of particles down an incline. When the particle concentration is above a critical value, singular shock solutions can occur. We analyze the Hugoniot topology associated with the Riemann problem for this system, describing in detail how the transition from a double shock to a singular shock happen. We also derive the singular shock speed based on a key observation that the particles pilling up at the maximum packing fraction near the contact line. These results are further applied to constant volume case to generate a rarefaction-singular shock solution. The particle/fluid front are shown to move linearly to the leading order with time to the one-third power as predicted by the Huppert solution for clear fluid.

The problem of the Grand Unification Theory

NASA Astrophysics Data System (ADS)

Treder, H.-J.

The evolution and fundamental questions of physical theories unifying the gravitational, electromagnetic, and quantum-mechanical interactions are explored, taking Pauli's aphorism as a motto: 'Let no man join what God has cast asunder.' The contributions of Faraday and Riemann, Lorentz, Einstein, and others are discussed, and the criterion of Pauli is applied to Grand Unification Theories (GUT) in general and to those seeking to link gravitation and electromagnetism in particular. Formal mathematical symmetry principles must be shown to have real physical relevance by predicting measurable phenomena not explainable without a GUT; these phenomena must be macroscopic because gravitational effects are to weak to be measured on the microscopic level. It is shown that empirical and theoretical studies of 'gravomagnetism', 'gravoelectricity', or possible links between gravoelectrity and the cosmic baryon assymmetry eventually lead back to basic questions which appear philosophical or purely mathematical but actually challenge physics to seek verifiable answers.

Domain decomposition methods in aerodynamics

NASA Technical Reports Server (NTRS)

Venkatakrishnan, V.; Saltz, Joel

1990-01-01

Compressible Euler equations are solved for two-dimensional problems by a preconditioned conjugate gradient-like technique. An approximate Riemann solver is used to compute the numerical fluxes to second order accuracy in space. Two ways to achieve parallelism are tested, one which makes use of parallelism inherent in triangular solves and the other which employs domain decomposition techniques. The vectorization/parallelism in triangular solves is realized by the use of a recording technique called wavefront ordering. This process involves the interpretation of the triangular matrix as a directed graph and the analysis of the data dependencies. It is noted that the factorization can also be done in parallel with the wave front ordering. The performances of two ways of partitioning the domain, strips and slabs, are compared. Results on Cray YMP are reported for an inviscid transonic test case. The performances of linear algebra kernels are also reported.

A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations

NASA Astrophysics Data System (ADS)

Bhrawy, A. H.; Zaky, M. A.

2015-01-01

In this paper, we propose and analyze an efficient operational formulation of spectral tau method for multi-term time-space fractional differential equation with Dirichlet boundary conditions. The shifted Jacobi operational matrices of Riemann-Liouville fractional integral, left-sided and right-sided Caputo fractional derivatives are presented. By using these operational matrices, we propose a shifted Jacobi tau method for both temporal and spatial discretizations, which allows us to present an efficient spectral method for solving such problem. Furthermore, the error is estimated and the proposed method has reasonable convergence rates in spatial and temporal discretizations. In addition, some known spectral tau approximations can be derived as special cases from our algorithm if we suitably choose the corresponding special cases of Jacobi parameters θ and ϑ. Finally, in order to demonstrate its accuracy, we compare our method with those reported in the literature.

Einsteinian cubic gravity

NASA Astrophysics Data System (ADS)

Bueno, Pablo; Cano, Pablo A.

2016-11-01

We drastically simplify the problem of linearizing a general higher-order theory of gravity. We reduce it to the evaluation of its Lagrangian on a particular Riemann tensor depending on two parameters, and the computation of two derivatives with respect to one of those parameters. We use our method to construct a D -dimensional cubic theory of gravity which satisfies the following properties: (1) it shares the spectrum of Einstein gravity, i.e., it only propagates a transverse and massless graviton on a maximally symmetric background; (2) it is defined in the same way in general dimensions; (3) it is neither trivial nor topological in four dimensions. Up to cubic order in curvature, the only previously known theories satisfying the first two requirements are the Lovelock ones. We show that, up to cubic order, there exists only one additional theory satisfying requirements (1) and (2). Interestingly, this theory is, along with Einstein gravity, the only one which also satisfies (3).

Study of the Ernst metric

DOE Office of Scientific and Technical Information (OSTI.GOV)

Esteban, E.P.

In this thesis some properties of the Ernst metric are studied. This metric could provide a model for a Schwarzschild black hole immersed in a magnetic field. In chapter I, some standard propertiess of the Ernst's metric such as the affine connections, the Riemann, the Ricci, and the Weyl conformal tensor are calculated. In chapter II, the geodesics described by test particles in the Ernst space-time are studied. As an application a formula for the perihelion shift is derived. In the last chapter a null tetrad analysis of the Ernst metric is carried out and the resulting formalism applied tomore » the study of three problems. First, the algebraic classification of the Ernst metric is determined to be of type I in the Petrov scheme. Secondly, an explicit formula for the Gaussian curvature for the event horizon is derived. Finally, the form of the electromagnetic field is evaluated.« less

Observation of Self-Cavitating Envelope Dispersive Shock Waves in Yttrium Iron Garnet Thin Films

NASA Astrophysics Data System (ADS)

Janantha, P. A. Praveen; Sprenger, Patrick; Hoefer, Mark A.; Wu, Mingzhong

2017-07-01

The formation and properties of envelope dispersive shock wave (DSW) excitations from repulsive nonlinear waves in a magnetic film are studied. Experiments involve the excitation of a spin wave step pulse in a low-loss magnetic Y3Fe5O12 thin film strip, in which the spin wave amplitude increases rapidly, realizing the canonical Riemann problem of shock theory. Under certain conditions, the envelope of the spin wave pulse evolves into a DSW that consists of an expanding train of nonlinear oscillations with amplitudes increasing from front to back, terminated by a black soliton. The onset of DSW self-cavitation, indicated by a point of zero power and a concomitant 180° phase jump, is observed for sufficiently large steps, indicative of the bidirectional dispersive hydrodynamic nature of the DSW. The experimental observations are interpreted with theory and simulations of the nonlinear Schrödinger equation.

Number-Theory in Nuclear-Physics in Number-Theory: Non-Primality Factorization As Fission VS. Primality As Fusion; Composites' Islands of INstability: Feshbach-Resonances?

NASA Astrophysics Data System (ADS)

Siegel, Edward

2011-10-01

Numbers: primality/indivisibility/non-factorization versus compositeness/divisibility /factor-ization, often in tandem but not always, provocatively close analogy to nuclear-physics: (2 + 1)=(fusion)=3; (3+1)=(fission)=4[=2 × 2]; (4+1)=(fusion)=5; (5 +1)=(fission)=6[=2 × 3]; (6 + 1)=(fusion)=7; (7+1)=(fission)=8[= 2 × 4 = 2 × 2 × 2]; (8 + 1) =(non: fission nor fusion)= 9[=3 × 3]; then ONLY composites' Islands of fusion-INstability: 8, 9, 10; then 14, 15, 16,... Could inter-digit Feshbach-resonances exist??? Applications to: quantum-information/computing non-Shore factorization, millennium-problem Riemann-hypotheses proof as Goodkin BEC intersection with graph-theory ``short-cut'' method: Rayleigh(1870)-Polya(1922)-``Anderson'' (1958)-localization, Goldbach-conjecture, financial auditing/accounting as quantum-statistical-physics;... abound!!!

Number-Theory in Nuclear-Physics in Number-Theory: Non-Primality Factorization As Fission VS. Primality As Fusion; Composites' Islands of INstability: Feshbach-Resonances?

NASA Astrophysics Data System (ADS)

Siegel, Edward

2011-04-01

Numbers: primality/indivisibility/non-factorization versus compositeness/divisibility /factor-ization, often in tandem but not always, provocatively close analogy to nuclear-physics: (2 + 1)=(fusion)=3; (3+1)=(fission)=4[=2 x 2]; (4+1)=(fusion)=5; (5+1)=(fission)=6[=2 x 3]; (6 + 1)=(fusion)=7; (7+1)=(fission)=8[= 2 x 4 = 2 x 2 x 2]; (8 + 1) =(non: fission nor fusion)= 9[=3 x 3]; then ONLY composites' Islands of fusion-INstability: 8, 9, 10; then 14, 15, 16,... Could inter-digit Feshbach-resonances exist??? Applications to: quantum-information and computing non-Shore factorization, millennium-problem Riemann-hypotheses physics-proof as numbers/digits Goodkin Bose-Einstein Condensation intersection with graph-theory ``short-cut'' method: Rayleigh(1870)-Polya(1922)-``Anderson'' (1958)-localization, Goldbach-conjecture, financial auditing/accounting as quantum-statistical-physics;... abound!!!

Mesoscopic Fluctuations for the Thinned Circular Unitary Ensemble

NASA Astrophysics Data System (ADS)

Berggren, Tomas; Duits, Maurice

2017-09-01

In this paper we study the asymptotic behavior of mesoscopic fluctuations for the thinned Circular Unitary Ensemble. The effect of thinning is that the eigenvalues start to decorrelate. The decorrelation is stronger on the larger scales than on the smaller scales. We investigate this behavior by studying mesoscopic linear statistics. There are two regimes depending on the scale parameter and the thinning parameter. In one regime we obtain a CLT of a classical type and in the other regime we retrieve the CLT for CUE. The two regimes are separated by a critical line. On the critical line the limiting fluctuations are no longer Gaussian, but described by infinitely divisible laws. We argue that this transition phenomenon is universal by showing that the same transition and their laws appear for fluctuations of the thinned sine process in a growing box. The proofs are based on a Riemann-Hilbert problem for integrable operators.

Generalization of Einstein's gravitational field equations

NASA Astrophysics Data System (ADS)

Moulin, Frédéric

2017-12-01

The Riemann tensor is the cornerstone of general relativity, but as is well known it does not appear explicitly in Einstein's equation of gravitation. This suggests that the latter may not be the most general equation. We propose here for the first time, following a rigorous mathematical treatment based on the variational principle, that there exists a generalized 4-index gravitational field equation containing the Riemann curvature tensor linearly, and thus the Weyl tensor as well. We show that this equation, written in n dimensions, contains the energy-momentum tensor for matter and that of the gravitational field itself. This new 4-index equation remains completely within the framework of general relativity and emerges as a natural generalization of the familiar 2-index Einstein equation. Due to the presence of the Weyl tensor, we show that this equation contains much more information, which fully justifies the use of a fourth-order theory.

Number Theoretic Background

NASA Astrophysics Data System (ADS)

Rudnick, Z.

Contents: 1. Introduction 2. Divisibility 2.1. Basics on Divisibility 2.2. The Greatest Common Divisor 2.3. The Euclidean Algorithm 2.4. The Diophantine Equation ax+by=c 3. Prime Numbers 3.1. The Fundamental Theorem of Arithmetic 3.2. There Are Infinitely Many Primes 3.3. The Density of Primes 3.4. Primes in Arithmetic Progressions 4. Continued Fractions 5. Modular Arithmetic 5.1. Congruences 5.2. Modular Inverses 5.3. The Chinese Remainder Theorem 5.4. The Structure of the Multiplicative Group (Z/NZ)^* 5.5. Primitive Roots 6. Quadratic Congruences 6.1. Euler's Criterion 6.2. The Legendre Symbol and Quadratic Reciprocity 7. Pell's Equation 7.1. The Group Law 7.2. Integer Solutions 7.3. Finding the Fundamental Solution 8. The Riemann Zeta Function 8.1 Analytic Continuation and Functinal Equation of ζ(s) 8.2 Connecting the Primes and the Zeros of ζ(s) 8.3 The Riemann Hypothesis References

Extension of the root-locus method to a certain class of fractional-order systems.

PubMed

Merrikh-Bayat, Farshad; Afshar, Mahdi; Karimi-Ghartemani, Masoud

2009-01-01

In this paper, the well-known root-locus method is developed for the special subset of linear time-invariant systems commonly known as fractional-order systems. Transfer functions of these systems are rational functions with polynomials of rational powers of the Laplace variable s. Such systems are defined on a Riemann surface because of their multi-valued nature. A set of rules for plotting the root loci on the first Riemann sheet is presented. The important features of the classical root-locus method such as asymptotes, roots condition on the real axis and breakaway points are extended to the fractional case. It is also shown that the proposed method can assess the closed-loop stability of fractional-order systems in the presence of a varying gain in the loop. Moreover, the effect of perturbation on the root loci is discussed. Three illustrative examples are presented to confirm the effectiveness of the proposed algorithm.

Holomorphic curves in surfaces of general type.

PubMed Central

Lu, S S; Yau, S T

1990-01-01

This note answers some questions on holomorphic curves and their distribution in an algebraic surface of positive index. More specifically, we exploit the existence of natural negatively curved "pseudo-Finsler" metrics on a surface S of general type whose Chern numbers satisfy c(2)1>2c2 to show that a holomorphic map of a Riemann surface to S whose image is not in any rational or elliptic curve must satisfy a distance decreasing property with respect to these metrics. We show as a consequence that such a map extends over isolated punctures. So assuming that the Riemann surface is obtained from a compact one of genus q by removing a finite number of points, then the map is actually algebraic and defines a compact holomorphic curve in S. Furthermore, the degree of the curve with respect to a fixed polarization is shown to be bounded above by a multiple of q - 1 irrespective of the map. PMID:11607050