DISPATCH methods: An approximate, entropy-based Riemann solver for ideal magnetohydrodynamics

¹ Rosseland Centre for Solar Physics, University of Oslo, PO Box 1029, Blindern, 0315 Oslo, Norway
² Institute of Theoretical Astrophysics, University of Oslo, PO Box 1029, Blindern, 0315 Oslo, Norway

^★ Corresponding author: andrius.popovas@astro.uio.no

Received: 4 February 2025
Accepted: 29 March 2025

Abstract

With the advancement of supercomputers, we can now afford simulations with very wide scale ranges. In astrophysical applications – for example simulating solar, stellar and planetary atmospheres, interstellar medium, and so on – physical quantities such as gas pressure, density, temperature, plasma β, and Mach and Reynolds numbers can vary by orders of magnitude. This requires a robust solver, which can deal with a very wide range of conditions and maintain hydrostatic equilibrium where it is applicable. We reformulated a Godunov-type Harten–Lax–van Leer discontinuities (HLLD) approximate Riemann solver that would be suitable for maintaining hydrostatic equilibrium in atmospheric applications in a range of Mach numbers, which represent regimes where kinetic and magnetic energies dominate over thermal energy without any ad hoc corrections. We changed the solver to use entropy instead of total energy as the primary thermodynamic variable in the system of magnetohydrodynamic equations. The entropy is not conserved; it increases when kinetic and magnetic energy are converted to heat, as it should. We propose using an approximate entropy-based Riemann solver as an alternative to already widely used Riemann solver formulations where it might be beneficial. We conducted a series of standard tests with varying conditions and show that the new formulation for the Godunov-type Riemann solver works and is promising.