On Godunov-type schemes for magnetohydrodynamics 1. A model system

In the light of recent analytical results on the MHD Riemann problem, Godunov-type numerical schemes for magnetohydrodynamics (MHD) are revisited. As the first step, a model system that exactly preserves the MHD hyperbolic singularities is considered. For this model, analytical results on shock waves are summarized and critical problems occurring in developing shock-capturing methods are identified. Using the results, we propose a new way to define fluxes on cell interfaces. It consists of two solvers, one on the well-posed Riemann problem and another on the evolution of Alfven waves. Numerical experiments show that the new scheme is more efficient in calculating large-time solutions. (C) 1998 Academic Press.