A Staggered Mesh Algorithm Using High Order Godunov Fluxes to Ensure Solenoidal Magnetic Fields in Magnetohydrodynamic Simulations

Abstract

The equations of magnetohydrodynamics (MHD) have been formulated as a hyperbolic system of conservation laws. In that form it becomes possible to use higher order Godunov schemes for their solution. This results in a robust and accurate solution strategy. However, the magnetic field also satisfies a constraint that requires its divergence to be zero at all times. This is a property that cannot be guaranteed in the zone centered discretizations that are favored in Godunov schemes without involving a divergence cleaning step. In this paper we present a staggered mesh strategy which directly uses the properly upwinded fluxes that are provided by a Godunov scheme. The process of directly using the upwinded fluxes relies on a duality that exists between the fluxes obtained from a higher order Godunov scheme and the electric fields in a plasma. By exploiting this duality we have been able to construct a higher order Godunov scheme that ensures that the magnetic field remains divergence-free up to the computer's round-off error. We have even presented a variant of the basic algorithm that uses multidimensional features in the flow to design an upwinded strategy that aligns itself with the predominant upwinded direction in the flow. We have devised several stringent test problems to show that the scheme works robustly and accurately in all situations. In doing so we have shown that a scheme that involves a collocation of magnetic field variables that is different from the one traditionally favored in the design of higher order Godunov schemes can nevertheless offer the same robust and accurate performance of higher order Godunov schemes provided the properly upwinded fluxes from the Godunov methodology are used in the scheme's construction.