Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations (Communications in Computational Physics, 2018)
We design stable and high-order accurate finite volume schemes for theideal MHD equations in multi-dimensions. We obtain excellent numerical stabilitydue to some new elements in the algorithm. The schemes are based on three- andfive-wave approximate Riemann solvers of the HLL-type, with the novelty that weallow a varying normal magnetic field. This is achieved by considering the semi-conservativeGodunov-Powell form of the MHD equations. We show that it is importantto discretize the Godunov-Powell source term in the right way, and that theHLL-type solvers naturally provide a stable upwind discretization. Second-order versionsof the ENO- and WENO-type reconstructions are proposed, together with precisemodifications necessary to preserve positive pressure and density. Extending the discretesource term to second order while maintaining stability requires non-standardtechniques, which we present. The first- and second-order schemes are tested on asuite of numerical experiments demonstrating impressive numerical resolution as wellas stability, even on very fine meshes.