Hydrostatic equilibrium preservation in MHD numerical simulation with stratified atmospheres: Explicit Godunov-type schemes with MUSCL reconstruction

Abstract

Context. Many astrophysical processes involving plasma flows are produced in the context of a gravitationally stratified atmosphere in hydrostatic equilibrium, in which strong gradients can exist with gas properties that vary in small regions by several orders of magnitude. The standard Godunov-type schemes with polynomial reconstruction used to numerically solve these problems fail to preserve the hydrostatic equilibrium owing to the appearance of spurious fluxes generated by the numerical unbalance between gravitational forces and pressure gradients. Aims. The aim of this work is to present local hydrostatic reconstruction techniques that can be implemented in existing codes with Godunov-type methods to obtain well-balanced schemes that numerically satisfy the hydrostatic equilibrium for various conditions. Methods. The proposed numerical scheme is based on the Godunov method with second order MUSCL-type reconstruction, as is extensively used in astrophysical applications. The difference between the scheme and the standard formulations is only given by calculating the pressure and density Riemann states on each intercell face and by computing the gravitational source term on each cell. Results. The local hydrostatic reconstruction scheme is implemented in the FLASH code to verify the well-balanced property for hydrostatic equilibrium with constant or linearly variable temperature and constant or variable gravity. In addition, the behavior of the scheme for hydrostatic equilibrium with arbitrary temperature distributions is also analyzed together with the ability to propagate low-amplitude waves and to capture shock waves. Conclusions. The scheme is demonstrated to be robust and relatively simple to implement in existing codes. This approach produces good results in hydrostatic equilibrium preservation, satisfying the well-balanced property for the preset conditions and strongly reducing the spurious fluxes for extreme configurations.