A simple finite difference scheme for multidimensional magnetohydrodynamical equations

An approximate MHD Riemann solver, an approach to maintain the divergence-free condition of magnetic field, and a finite difference scheme for multidimensional magnetohydrodynamical (MHD) equations are proposed in this paper. The approximate MHD Riemann solver is based on characteristic formulations. Both the conservation laws for mass, momentum, energy, and magnetic field, and the divergence-free condition of the magnetic field are exactly satisfied in the proposed scheme. The scheme does not involve any Poisson solver and is second-order accurate in both space and time. The correctness and robustness of the scheme are shown through numerical examples. The approach proposed in this paper to maintain the divergence-free condition may be applied to other dimensionally split and unsplit Godunov schemes for MHD flows. (C) 1998 Academic Press.