A well-balanced semi-implicit IMEX finite volume scheme for ideal Magnetohydrodynamics at all Mach numbers

TL;DR

This paper introduces a second-order semi-implicit finite volume scheme for ideal magnetohydrodynamics that effectively handles low Mach number flows while maintaining equilibrium states and divergence-free magnetic fields.

Contribution

It develops a novel well-balanced, semi-implicit IMEX finite volume scheme combining deviation well-balancing and constrained transport for MHD at all Mach numbers.

Findings

Effective in low Mach regimes

Maintains magnetohydrostatic equilibria

Preserves divergence-free magnetic fields

Abstract

We propose a second-order accurate semi-implicit and well-balanced finite volume scheme for the equations of ideal magnetohydrodynamics (MHD) including gravitational source terms. The scheme treats all terms associated with the acoustic pressure implicitly while keeping the remaining terms part of the explicit sub-system. This semi-implicit approach makes the method particularly well suited for problems in the low Mach regime. We combine the semi-implicit scheme with the deviation well-balancing technique and prove that it maintains equilibrium solutions for the magnetohydrostatic case up to rounding errors. In order to preserve the divergence-free property of the magnetic field enforced by the solenoidal constraint, we incorporate a constrained transport method in the semi-implicit framework. Second order of accuracy is achieved by means of a standard spatial reconstruction technique with total variation diminishing (TVD) property, and by an asymptotic preserving (AP) time stepping algorithm built upon the implicit-explicit (IMEX) Runge-Kutta time integrators. Numerical tests in the low Mach regime and near magnetohydrostatic equilibria support the low Mach and well-balanced properties of the numerical method.