A multiwave approximate Riemann solver for ideal MHD based on relaxation II – Numerical implementation with 3 and 5 waves
François Bouchut, Christian Klingenberg and Knut Waagan
Abstract: In the first part of this work we introduced an approximate Riemann solver for one-dimensional ideal MHD derived from a relaxation system. We gave sufficient conditions for the solver to satisfy discrete entropy inequalities, and to preserve positivity of density and internal energy. In this paper we consider the practical implementation, and derive explicit wave speed estimates satisfying the stability conditions of Part 1. We present a 3-wave solver that well resolves fast waves and material contacts, and a 5-wave solver that accurately resolves the cases when two eigenvalues coincide. A full 7-wave solver, which is highly accurate on all types of waves, will be described in a follow-up paper. We test the solvers on one-dimensional shock tube data and smooth shear waves.