Exploring various flux vector splittings for the magnetohydrodynamic system

Abstract

In this paper we explore flux vector splittings for the MHD system of equations. Our approach follows the strategy that was initially put forward in Toro and Vázquez-Cendón (2012) [55] . We split the flux vector into an advected sub-system and a pressure sub-system. The eigenvalues and eigenvectors of the split sub-systems are then studied for physical suitability. Not all flux vector splittings for MHD yield physically meaningful results. We find one that is completely useless, another that is only marginally useful and one that should work well in all regimes where the MHD equations are used. Unfortunately, this successful flux vector splitting turns out to be different from the Zha–Bilgen flux vector splitting. The eigenvalues and eigenvectors of this favorable FVS are explored in great detail in this paper. The pressure sub-system holds the key to finding a successful flux vector splitting. The eigenstructure of the successful flux vector splitting for MHD is thoroughly explored and orthonormalized left and right eigenvectors are explicitly catalogued. We present a novel approach to the solution of the Riemann problem formed by the pressure sub-system for the MHD equations. Once the pressure sub-system is solved, the advection sub-system follows naturally. Our method also works very well for the Euler system. Our FVS successfully captures isolated, stationary contact discontinuities in MHD. However, we explain why any FVS for MHD is not adept at capturing isolated, stationary Alfvenic discontinuities. Several stringent one-dimensional Riemann problems are presented to show that the method works successfully and can effectively capture the full panoply of wave structures that arise in MHD. This includes compound waves and switch-on and switch-off shocks that arise because of the non-convex nature of the MHD system.