An explicit divergence-free DG method for incompressible magnetohydrodynamics

TL;DR

This paper develops an explicit divergence-free discontinuous Galerkin method for incompressible magnetohydrodynamics, ensuring conservation, high accuracy, and stability, suitable for high-Reynolds number flows with low resistivity.

Contribution

It extends a divergence-free DG scheme to MHD, incorporating magnetic fields with conservation and stability properties, and uses hybrid-mixed Poisson solvers for time advancement.

Findings

Ensures global and local conservation of mass and magnetic flux.

Achieves high-order accuracy and energy stability.

Suitable for high-Reynolds number, low resistivity flows.

Abstract

We extend the recently introduced explicit divergence-free DG scheme for incompressible hydrodynamics [arXiv:1808.04669]. to the incompressible magnetohydrodynamics (MHD). A globally divergence-free finite element space is used for both the velocity and the magnetic field. Highlights of the scheme includes global and local conservation properties, high-order accuracy, energy-stability, pressure-robustness. When forward Euler time stepping is used, we need two symmetric positive definite (SPD) hybrid-mixed Poisson solvers (one for velocity and one for magnetic field) to advance the solution to the next time level. Since we treat both viscosity in the momentum equation and resistivity in the magnetic induction equation explicitly, the method shall be best suited for inviscid or high-Reynolds number, low resistivity flows so that the CFL constraint is not too restrictive.