A globally divergence-free entropy stable nodal DG method for conservative ideal MHD equations

TL;DR

This paper introduces a high-order, entropy stable nodal DG method for ideal MHD equations that maintains a divergence-free magnetic field and effectively handles shocks, verified through numerical experiments.

Contribution

It presents a novel high-order DG scheme that ensures a globally divergence-free magnetic field and entropy stability for ideal MHD equations.

Findings

Method achieves high accuracy in numerical tests.

Effectively suppresses unphysical oscillations near shocks.

Verifies the divergence-free property and stability through experiments.

Abstract

We propose an arbitrarily high-order globally divergence-free entropy stable nodal discontinuous Galerkin (DG) method to directly solve the conservative form of the ideal MHD equations using appropriate quadrature rules. The method ensures a globally divergence-free magnetic field by updating it at interfaces with a constraint-preserving formulation [5] and employing a novel least-squares reconstruction technique. Leveraging this property, the semi-discrete nodal DG scheme is proven to be entropy stable. To handle the problems with strong shocks, we introduce a novel limiting strategy that suppresses unphysical oscillations while preserving the globally divergence-free property. Numerical experiments verify the accuracy and efficacy of our method.