Entropy Symmetrization and High-Order Accurate Entropy Stable Numerical Schemes for Relativistic MHD Equations

Abstract

This paper presents entropy symmetrization and high-order accurate entropy stable schemes for the relativistic magnetohydrodynamic (RMHD) equations. It is shown that the conservative RMHD equations are not symmetrizable and do not admit a thermodynamic entropy pair. To address this issue, a symmetrizable RMHD system, equipped with a convex thermodynamic entropy pair, is proposed by adding a source term into the equations, providing an analogue to the non-relativistic Godunov--Powell system. Arbitrarily high-order accurate entropy stable finite difference schemes are developed on Cartesian meshes based on the symmetrizable RMHD system. The crucial ingredients of these schemes include (i) affordable explicit entropy conservative fluxes which are technically derived through carefully selected parameter variables, (ii) a special high-order discretization of the source term in the symmetrizable RMHD system, and (iii) suitable high-order dissipative operators based on essentially non-oscillatory reconstruction to ensure the entropy stability. Several numerical tests demonstrate the accuracy and robustness of the proposed entropy stable schemes.

arXiv comment32 pages; 11 figures; add some references on entropy symmetrization

References in corpus (20)

• Second Order Accurate Schemes for Magnetohydrodynamics With Divergence-Free Reconstruction
• Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier-Stokes equations
• An HLLC Solver for Relativistic Flows -- II. Magnetohydrodynamics
• The Exact Solution of the Riemann Problem in Relativistic MHD
• Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement
• Provably Positive High-Order Schemes for Ideal Magnetohydrodynamics: Analysis on General Meshes
• Ideal GLM-MHD: About the entropy consistent nine-wave magnetic field divergence diminishing ideal magnetohydrodynamics equations
• Comparison of some Entropy Conservative Numerical Fluxes for the Euler Equations
• Positivity-Preserving Analysis of Numerical Schemes for Ideal Magnetohydrodynamics
• Provably Positive Discontinuous Galerkin Methods for Multidimensional Ideal Magnetohydrodynamics
• ENO reconstruction and ENO interpolation are stable
• Relativistic Magnetohydrodynamics: Renormalized eigenvectors and full wave decomposition Riemann solver
• A multidimensional grid-adaptive relativistic magnetofluid code
• Admissible state and physical constraints preserving schemes for relativistic magnetohydrodynamic equations
• A Novel High-Order, Entropy Stable, 3D AMR MHD Solver with Guaranteed Positive Pressure
• Runge-Kutta discontinuous Galerkin methods for the special relativistic magnetohydrodynamics
• A sign preserving WENO reconstruction method
• High-order accurate entropy stable finite difference schemes for one- and two-dimensional special relativistic hydrodynamics
• Entropy-limited hydrodynamics: a novel approach to relativistic hydrodynamics
• Symmetrizations of RMHD equations and stability of relativistic current-vortex sheets

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