Central finite volume methods for one and two-dimensional ideal magnetohydrodynamics

Abstract

Two and three-dimensional finite volume extensions of the Lax-Friedrichs (LF) and Nessyahu-Tadmor (NT) one-dimensional difference schemes were previously presented and applied to several problems for nonlinear hyperbolic systems, and in particular to typical test cases for inviscid or viscous compressible flows. These 'central' schemes by-pass the resolution of the Riemann problems, thanks to the use of the staggered Lax-Friedrichs scheme; two dual grids are used at alternate time steps. These methods are extended here to several problems in one- and multi-dimensional ideal compressible magnetohydrodynamics using a modified version of the first author's central methods with diamond shaped dual cells. In 2D the system has eight equations and solving the Riemann problem is an elaborate and time-consuming process. Central methods lead to significant computing time reductions, and the numerical experiments presented here suggest that the accuracy is quite satisfactory. For our first numerical experiments, presented here, with one and two- dimensional problems, the divB=0 constraint has been satisfied by our scheme nearly up to the accuracy of the computations. For several more elaborate numerical tests in 2 and 3D, it turned out to be necessary to apply a specific strategy to enforce this constraint. The validity of our strategy, adapted from Evans-Hawley, is clearly confirmed by the results.