A Finite Element Method for MHD that Preserves Energy, Cross-Helicity, Magnetic Helicity, Incompressibility, and $\operatorname{div} B = 0$

TL;DR

This paper introduces a finite element method for incompressible MHD that maintains key physical invariants like energy, helicities, and divergence constraints at discrete levels, ensuring accurate and stable simulations.

Contribution

The authors develop a structure-preserving finite element and time-stepping scheme that exactly conserves multiple physical invariants in incompressible MHD simulations.

Findings

Preserves energy, cross-helicity, and magnetic helicity exactly.

Maintains divergence-free magnetic field to machine precision.

Ensures mass and density conservation in discretized form.

Abstract

We construct a structure-preserving finite element method and time-stepping scheme for inhomogeneous, incompressible magnetohydrodynamics (MHD). The method preserves energy, cross-helicity (when the fluid density is constant), magnetic helicity, mass, total squared density, pointwise incompressibility, and the constraint $\operatorname{div} B = 0$ to machine precision, both at the spatially and temporally discrete levels.