Entropy stable finite difference schemes for Chew, Goldberger & Low anisotropic plasma flow equations

TL;DR

This paper develops entropy-stable finite difference schemes for the nonlinear, non-conservative CGL plasma flow equations, ensuring entropy stability and demonstrating their effectiveness through tests inspired by MHD problems.

Contribution

It introduces a novel approach to discretize the CGL equations with entropy stability by rewriting and symmetrizing the equations, inspired by MHD techniques.

Findings

Schemes are proven to be entropy stable in a semi-discrete sense.

Numerical tests show the schemes perform well on MHD-inspired test problems.

The approach effectively handles non-conservative terms in the CGL equations.

Abstract

In this article, we consider the Chew, Goldberger \& Low (CGL) plasma flow equations, which is a set of nonlinear, non-conservative hyperbolic PDEs modelling anisotropic plasma flows. These equations incorporate the double adiabatic approximation for the evolution of the pressure, making them very valuable for plasma physics, space physics and astrophysical applications. We first present the entropy analysis for the weak solutions. We then propose entropy-stable finite-difference schemes for the CGL equations. The key idea is to rewrite the CGL equations such that the non-conservative terms do not contribute to the entropy equations. The conservative part of the rewritten equations is very similar to the magnetohydrodynamics (MHD) equations. We then symmetrize the conservative part by following Godunov's symmetrization process for MHD. The resulting equations are then discretized by designing entropy conservative numerical flux and entropy diffusion operator based on the entropy scaled eigenvectors of the conservative part. We then prove the semi-discrete entropy stability of the schemes for CGL equations. The schemes are then tested using several test problems derived from the corresponding MHD test cases.