On the divergence-free condition and conservation laws in numerical simulations for supersonic magnetohydrodynamic flows

An approach to maintain exactly the eight conservation laws and the divergence-free condition of magnetic fields is proposed for numerical simulations of multidimensional magnetohdyrodynamic (MHD) equations. The approach is simple and may be easily applied to both dimensionally split and unsplit Godunov schemes for supersonic MHD flows. The numerical schemes based on the approach are second-order accurate in both space and time if the original Godunov schemes are. As an example of such schemes, a scheme based on the approach and an approximate MHD Riemann solver is presented. The Riemann solver is simple and is used to approximately calculate the time-averaged flux. The correctness, accuracy, and robustness of the scheme are shown through numerical examples. A comparison in numerical solutions between the proposed scheme and a Godunov scheme without the divergence-free constraint implemented is presented.