A structure-preserving finite element method for compressible ideal and resistive MHD

TL;DR

This paper introduces a finite element method for compressible MHD that preserves key physical invariants like energy, helicity, and divergence constraints at discrete levels, applicable to both ideal and resistive cases.

Contribution

It presents a novel structure-preserving finite element scheme based on geometric variational principles for compressible MHD, ensuring conservation laws at discrete levels.

Findings

Conserves total energy and magnetic helicity in simulations.

Maintains divergence-free magnetic field to machine precision.

Cross helicity is well conserved in ideal MHD simulations.

Abstract

We construct a structure-preserving finite element method and time-stepping scheme for compressible barotropic magnetohydrodynamics (MHD) both in the ideal and resistive cases, and in the presence of viscosity. The method is deduced from the geometric variational formulation of the equations. It preserves the balance laws governing the evolution of total energy and magnetic helicity, and preserves mass and the constraint $ \operatorname{div}B = 0$ to machine precision, both at the spatially and temporally discrete levels. In particular, conservation of energy and magnetic helicity hold at the discrete levels in the ideal case. It is observed that cross helicity is well conserved in our simulation in the ideal case.