Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations

We design stable and high-order accurate finite volume schemes for the ideal MHD equations in multi-dimensions.We obtain excellent numerical stability due to some new elements in the algorithm.The schemes are based on three-and five-wave approximate Riemann solvers of the HLL-type,with the novelty that we allow a varying normal magnetic field.This is achieved by considering the semiconservative Godunov-Powell form of the MHD equations.We show that it is important to discretize the Godunov-Powell source term in the right way,and that the HLL-type solvers naturally provide a stable upwind discretization.Second-order versions of the ENO-and WENO-type reconstructions are proposed,together with precise modifications necessary to preserve positive pressure and density.Extending the discrete source term to second order while maintaining stability requires non-standard techniques,which we present.The first-and second-order schemes are tested on a suite of numerical experiments demonstrating impressive numerical resolution as well as stability,even on very fine meshes.