A Central, Diamond-Staggered Dual Cell, Finite Volume Method for 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 successfully applied to several problems for nonlinear hyperbolic systems, and in particular to typical test cases for both inviscid and viscous compressible flows. These ’’central” schemes by-pass the resolution, at the cell interfaces, of the Riemann problems, thanks to the use of the staggered Lax-Friedrichs scheme which serves as the base scheme on which high order finite volume methods can be constructed using van Leer’s MUSCL-type limited reconstruction principle. For this purpose, 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 oblique (diamond shaped) dual cells. In two dimensions the system has eight equations and solving the corresponding Riemann problem is an elaborate and time-consuming process. Central methods lead to significant computing time reductions, and the numerical experiments presented here suggest the accuracy is quite satisfactory.