Entropy-Stable Gauss Collocation Methods for Ideal Magneto-Hydrodynamics

TL;DR

This paper develops an entropy-stable Gauss collocation discontinuous Galerkin method for 3D magneto-hydrodynamics equations, improving accuracy and robustness over traditional LGL-based schemes.

Contribution

It extends entropy-stable DGSEM techniques to the non-conservative GLM-MHD system using Gauss points, enhancing accuracy and stability.

Findings

The scheme is entropy-stable and convergent on 3D curvilinear meshes.

Numerical tests show improved accuracy over LGL-based methods.

The method effectively captures MHD instabilities like Kelvin-Helmholtz.

Abstract

In this paper, we present an entropy-stable Gauss collocation discontinuous Galerkin (DG) method on 3D curvilinear meshes for the GLM-MHD equations: the single-fluid magneto-hydrodynamics (MHD) equations with a generalized Lagrange multiplier (GLM) divergence cleaning mechanism. For the continuous entropy analysis to hold and to ensure Galilean invariance in the divergence cleaning technique, the GLM-MHD system requires the use of non-conservative terms. Traditionally, entropy-stable DG discretizations have used a collocated nodal variant of the DG method, also known as the discontinuous Galerkin spectral element method (DGSEM) on Legendre-Gauss-Lobatto (LGL) points. Recently, Chan et al. ("Efficient Entropy Stable Gauss Collocation Methods". SIAM -2019) presented an entropy-stable DGSEM scheme that uses Legendre-Gauss points (instead of LGL points) for conservation laws. Our main contribution is to extend the discretization technique of Chan et al. to the non-conservative GLM-MHD system. We provide a numerical verification of the entropy behavior and convergence properties of our novel scheme on 3D curvilinear meshes. Moreover, we test the robustness and accuracy of our scheme with a magneto-hydrodynamic Kelvin-Helmholtz instability problem. The numerical experiments suggest that the entropy-stable DGSEM on Gauss points for the GLM-MHD system is more accurate than the LGL counterpart.