Ideal GLM-MHD: About the entropy consistent nine-wave magnetic field divergence diminishing ideal magnetohydrodynamics equations

TL;DR

This paper introduces an entropy-consistent extension of ideal MHD equations with a divergence cleaning mechanism, and develops a finite volume scheme that preserves thermodynamic entropy and magnetic divergence control, implemented in FLASH.

Contribution

The paper proposes a thermodynamically consistent extension of ideal MHD equations with divergence cleaning and a novel finite volume scheme that maintains entropy and divergence control.

Findings

The new model is thermodynamically consistent.

The finite volume scheme controls magnetic divergence error.

The scheme is implemented and compared with existing methods in FLASH.

Abstract

The paper presents two contributions in the context of the numerical simulation of magnetized fluid dynamics. First, we show how to extend the ideal magnetohydrodynamics (MHD) equations with an inbuilt magnetic field divergence cleaning mechanism in such a way that the resulting model is consistent with the second law of thermodynamics. As a byproduct of these derivations, we show that not all of the commonly used divergence cleaning extensions of the ideal MHD equations are thermodynamically consistent. Secondly, we present a numerical scheme obtained by constructing a specific finite volume discretization that is consistent with the discrete thermodynamic entropy. It includes a mechanism to control the discrete divergence error of the magnetic field by construction and is Galilean invariant. We implement the new high-order MHD solver in the adaptive mesh refinement code FLASH where we compare the divergence cleaning efficiency to the constrained transport solver available in FLASH (unsplit staggered mesh scheme).