On thermodynamic consistency of generalised Lagrange multiplier magnetohydrodynamic solvers

Abstract

This work presents a new implementation of compressible magnetohydrodynamic (MHD) models in the context of the generalised Lagrange multiplier (GLM), combined with source term techniques to retain entropy stability, necessary for thermodynamic consistency. The GLM techniques introduce a scalar field, that is evolved along the MHD quantities, in order to aid in an error control of $\nabla \cdot {\textbf {B}}$. Our implementation employs second-order HLL-type schemes in finite-volume form and an explicit time discretisation in a parallel framework. We furthermore revise and develop different GLM–MHD and source term approaches as sit-on-top solvers, that can be added to existing MHD applications. It is shown that Galilean invariance is a major factor determining the capacity of these solvers to control $\nabla \cdot {\textbf {B}}$, as achieved in GLM–MHD systems with Powell source terms. Moreover, it also influences the physical robustness of the solver, in particular its ability to maintain positive pressure during the simulation. In addition, we show that our new and easily reproducible implementation is entropy consistent.