An HLLC-Type Approximate Riemann Solver for Ideal Magnetohydrodynamics

This paper presents a solver based on the HLLC (Harten--Lax--van Leer contact wave) approximate nonlinear Riemann solver for gas dynamics for the ideal magnetohydrodynamics (MHD) equations written in conservation form. It is shown how this solver also can be considered a modification of Linde's "adequate" solver. This approximation method is intended to resolve slow, Alfvén, and contact waves better than the original HLL (Harten--Lax--van Leer) solver. Compared to exact nonlinear solvers and Roe's solver, this new solver is computationally inexpensive. In addition, the method will exactly resolve isolated contacts and fast shocks. The method also preserves positive density and pressure with two caveats: first, the numerical signal velocities (the eigenvalues of the Roe average matrix) do not underestimate the physical signal velocities, and second, in a very few cases it may be required to change the wavespeeds of the Riemann fan for the underlying HLL method to guarantee positive pressures. These conditions are less restrictive on the definitions of the wavespeeds than the conditions needed to make the HLLC method positively conservative for gas dynamics. While the method is intended for a three-dimensional MHD problem, the simulation results concentrate on one-dimensional test cases.