A finite-volume scheme for modeling compressible magnetohydrodynamic flows at low Mach numbers in stellar interiors

TL;DR

This paper introduces a new finite-volume scheme tailored for efficient and accurate simulation of low Mach number compressible magnetohydrodynamic flows in stellar interiors, addressing limitations of conventional methods.

Contribution

The authors develop a novel low-Mach MHD scheme integrated into the SLH code, combining a specialized Riemann solver, implicit-explicit time discretization, and a well-balancing method to reduce dissipation and errors.

Findings

Successfully simulates low Mach number MHD flows with reduced numerical dissipation.

Accurately models a small-scale dynamo in a star-like environment at M_son ~ 0.001.

Demonstrates effectiveness even on moderately coarse grids.

Abstract

Fully compressible magnetohydrodynamic (MHD) simulations are a fundamental tool for investigating the role of dynamo amplification in the generation of magnetic fields in deep convective layers of stars. The flows that arise in such environments are characterized by low (sonic) Mach numbers (M_son < 0.01 ). In these regimes, conventional MHD codes typically show excessive dissipation and tend to be inefficient as the Courant-Friedrichs-Lewy (CFL) constraint on the time step becomes too strict. In this work we present a new method for efficiently simulating MHD flows at low Mach numbers in a space-dependent gravitational potential while still retaining all effects of compressibility. The proposed scheme is implemented in the finite-volume Seven-League Hydro (SLH) code, and it makes use of a low-Mach version of the five-wave Harten-Lax-van Leer discontinuities (HLLD) solver to reduce numerical dissipation, an implicit-explicit time discretization technique based on Strang splitting to overcome the overly strict CFL constraint, and a well-balancing method that dramatically reduces the magnitude of spatial discretization errors in strongly stratified setups. The solenoidal constraint on the magnetic field is enforced by using a constrained transport method on a staggered grid. We carry out five verification tests, including the simulation of a small-scale dynamo in a star-like environment at M_son ~ 0.001 . We demonstrate that the proposed scheme can be used to accurately simulate compressible MHD flows in regimes of low Mach numbers and strongly stratified setups even with moderately coarse grids.