Structure preserving numerical methods for the ideal compressible MHD system

TL;DR

This paper presents a new structure-preserving numerical method for the ideal compressible MHD system that maintains key physical invariants, is flexible with Euler solvers, and is effective for both smooth and shock-dominated problems.

Contribution

The authors develop a novel non-divergence formulation with a curl-conforming finite element discretization that preserves physical invariants and allows flexible Euler solvers, improving robustness and accuracy.

Findings

Exact preservation of involution constraints

Preservation of positivity and entropy principles

High-order accuracy and robustness in shock regimes

Abstract

We introduce a novel structure-preserving method in order to approximate the compressible ideal Magnetohydrodynamics (MHD) equations. This technique addresses the MHD equations using a non-divergence formulation, where the contributions of the magnetic field to the momentum and total mechanical energy are treated as source terms. Our approach uses the Marchuk-Strang splitting technique and involves three distinct components: a compressible Euler solver, a source-system solver, and an update procedure for the total mechanical energy. The scheme allows for significant freedom on the choice of Euler's equation solver, while the magnetic field is discretized using a curl-conforming finite element space, yielding exact preservation of the involution constraints. We prove that the method preserves invariant domain properties, including positivity of density, positivity of internal energy, and the minimum principle of the specific entropy. If the scheme used to solve Euler's equation conserves total energy, then the resulting MHD scheme can be proven to preserve total energy. Similarly, if the scheme used to solve Euler's equation is entropy-stable, then the resulting MHD scheme is entropy stable as well. In our approach, the CFL condition does not depend on magnetosonic wave-speeds, but only on the usual maximum wave speed from Euler's system. To validate the effectiveness of our method, we solve a variety of ideal MHD problems, showing that the method is capable of delivering high-order accuracy in space for smooth problems, while also offering unconditional robustness in the shock hydrodynamics regime as well.