Monolithic parabolic regularization of the MHD equations and entropy principles

TL;DR

This paper introduces a monolithic parabolic regularization for ideal MHD equations that maintains entropy principles and positivity, and demonstrates its effectiveness through high-order accurate numerical methods.

Contribution

It develops a regularization compatible with entropy principles and positivity, and validates its performance with high-order numerical schemes for smooth and discontinuous problems.

Findings

Regularization preserves all generalized entropies.

Method maintains positivity of density and internal energy.

High accuracy for smooth problems and effective shock capturing.

Abstract

We show at the PDE level that the monolithic parabolic regularization of the equations of ideal magnetohydrodynamics (MHD) is compatible with all the generalized entropies, fulfills the minimum entropy principle, and preserves the positivity of density and internal energy. We then numerically investigate this regularization for the MHD equations using continuous finite elements in space and explicit strong stability preserving Runge-Kuta methods in time. The artificial viscosity coefficient of the regularization term is constructed to be proportional to the entropy residual of MHD. It is shown that the method has a high order of accuracy for smooth problems and captures strong shocks and discontinuities accurately for non-smooth problems.