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A second-order direct Eulerian GRP scheme for ten-moment Gaussian closure equations with source terms

Jiangfu Wang, Huazhong Tang

TL;DR

This work develops a second-order direct Eulerian generalized Riemann problem (GRP) scheme for the 1D ten-moment Gaussian closure equations with source terms, leveraging generalized Riemann invariants and the Rankine–Hugoniot conditions to resolve left and right nonlinear waves directly in Eulerian form. The methodology includes an exact 1D RP solver and a GRP solver to obtain time derivatives at cell interfaces, with a Strang-split 2D extension. The authors derive comprehensive formulas for rarefaction, shear, and shock waves, including nonsonic, sonic, and acoustic regimes, and validate the scheme through extensive 1D and 2D numerical experiments, including novel 2D Riemann problems. The results demonstrate high-resolution, second-order accuracy, and robust handling of multiple wave types and strong discontinuities in anisotropic plasma/Gaussian closure contexts, indicating practical applicability to non-equilibrium gas/plasma dynamics.

Abstract

This paper proposes a second-order accurate direct Eulerian generalized Riemann problem (GRP) scheme for the ten-moment Gaussian closure equations with source terms. The generalized Riemann invariants associated with the rarefaction waves, the contact discontinuity and the shear waves are given, and the 1D exact Riemann solver is obtained. After that, the generalized Riemann invariants and the Rankine-Hugoniot jump conditions are directly used to resolve the left and right nonlinear waves (rarefaction wave and shock wave) of the local GRP in Eulerian formulation, and then the 1D direct Eulerian GRP scheme is derived. They are much more complicated, technical and nontrivial due to more physical variables and elementary waves. Some 1D and 2D numerical experiments are presented to check the accuracy and high resolution of the proposed GRP schemes, where the 2D direct Eulerian GRP scheme is given by using the Strang splitting method for simplicity. It should be emphasized that several examples of 2D Riemann problems are constructed for the first time.

A second-order direct Eulerian GRP scheme for ten-moment Gaussian closure equations with source terms

TL;DR

This work develops a second-order direct Eulerian generalized Riemann problem (GRP) scheme for the 1D ten-moment Gaussian closure equations with source terms, leveraging generalized Riemann invariants and the Rankine–Hugoniot conditions to resolve left and right nonlinear waves directly in Eulerian form. The methodology includes an exact 1D RP solver and a GRP solver to obtain time derivatives at cell interfaces, with a Strang-split 2D extension. The authors derive comprehensive formulas for rarefaction, shear, and shock waves, including nonsonic, sonic, and acoustic regimes, and validate the scheme through extensive 1D and 2D numerical experiments, including novel 2D Riemann problems. The results demonstrate high-resolution, second-order accuracy, and robust handling of multiple wave types and strong discontinuities in anisotropic plasma/Gaussian closure contexts, indicating practical applicability to non-equilibrium gas/plasma dynamics.

Abstract

This paper proposes a second-order accurate direct Eulerian generalized Riemann problem (GRP) scheme for the ten-moment Gaussian closure equations with source terms. The generalized Riemann invariants associated with the rarefaction waves, the contact discontinuity and the shear waves are given, and the 1D exact Riemann solver is obtained. After that, the generalized Riemann invariants and the Rankine-Hugoniot jump conditions are directly used to resolve the left and right nonlinear waves (rarefaction wave and shock wave) of the local GRP in Eulerian formulation, and then the 1D direct Eulerian GRP scheme is derived. They are much more complicated, technical and nontrivial due to more physical variables and elementary waves. Some 1D and 2D numerical experiments are presented to check the accuracy and high resolution of the proposed GRP schemes, where the 2D direct Eulerian GRP scheme is given by using the Strang splitting method for simplicity. It should be emphasized that several examples of 2D Riemann problems are constructed for the first time.
Paper Structure (31 sections, 20 theorems, 287 equations, 20 figures, 2 tables)

This paper contains 31 sections, 20 theorems, 287 equations, 20 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. 1D ten-moment equations
  4. Eigenvalues, eigenvectors and generalized Riemann invariants
  5. Numerical schemes
  6. Outline of 1D GRP scheme
  7. 2D extension of GRP scheme
  8. Exact Riemann solver
  9. Computations of $u_{1,\ast}$ and $p_{11,\ast}$
  10. Computations of $\mathbf{V}_{\text{Lfan}}$
  11. Computing $\mathbf{V}_{\ast L}$
  12. Computing $\mathbf{V}_{\ast R}$
  13. Computing $\mathbf{V}_{\ast\ast L}$ and $\mathbf{V}_{\ast\ast R}$
  14. Resolution of the GRP
  15. Nonsonic case
  16. ...and 16 more sections

Key Result

Theorem 4.1

The pressure component $p_{11,\ast}$ of the Riemann problem local RP2 is given by the root of the algebraic equation where the function $f_L$ is given by and the function $f_R$ is given by Moreover, the velocity $u_{1,\ast}$ may be given by

Figures (20)

  • Figure 1: The schematic description of a local wave configuration of the associated Riemann problem \ref{['local RP2']}.
  • Figure 2: The main steps of the exact Riemann solver in Section \ref{['exact RP']}.
  • Figure 3: The schematic description of a local wave configuration of the GRP \ref{['GRP2']} with $0\leq t\ll1$.
  • Figure 4: The main steps for the nonsonic case in Subsection \ref{['nonsonic case']}.
  • Figure 5: Example \ref{['1D_RP']}: The numerical solutions of the GRP scheme on 400 uniform cells for the 1D Riemann problem \ref{['1D_RP1']}.
  • ...and 15 more figures

Theorems & Definitions (57)

  • Theorem 4.1: Computing $p_{11,\ast}$ and $u_{1,\ast}$
  • proof
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • Lemma 5.1
  • proof
  • Lemma 5.2: Resolution of the 1-rarefaction wave
  • proof
  • Remark 5.3
  • ...and 47 more