Well-balanced Central Scheme for the System of MHD Equations with Gravitational source term

TL;DR

This paper introduces a second-order well-balanced finite volume central scheme for MHD equations with gravity, effectively maintaining steady states and divergence-free magnetic fields without solving Riemann problems.

Contribution

It presents a novel unstaggered central scheme that is well-balanced for gravitational MHD problems and ensures divergence-free magnetic fields using constrained transport.

Findings

Successfully maintains steady states in numerical tests

Avoids solving Riemann problems at cell interfaces

Ensures divergence-free magnetic fields after each step

Abstract

A well-balanced second order finite volume central scheme for the magnetohydrodynamic (MHD) equations with gravitational source term is developed in this paper. The scheme is an unstaggered central scheme that evolves the numerical solution on a single grid and avoids solving Riemann problems at the cell interfaces using ghost staggered cells. A subtraction technique is used on the conservative variables with the support of a known steady state in order to manifest the well-balanced property of the scheme. The divergence-free constraint of the magnetic field is satisfied after applying the constrained transport method (CTM) for unstaggered central schemes at the end of each time-step by correcting the components of the magnetic field. The robustness of the proposed scheme is verified on a list of numerical test cases from the literature.