A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics

A new multi-state Harten-Lax-van Leer (HLL) approximate Riemann solver for the ideal magnetohydrodynamic (MHD) equations is developed based on the assumption that the normal velocity is constant over the Riemann fan. This assumption is same as that used in the HLLC (“C” denotes Contact) approximate Riemann solver for the Euler equations. From the assumption, it is naturally derived that the Riemann fan should consist of four intermediate states for Bx ≠ 0, whereas the number of the intermediate states is reduced to two when Bx = 0. Since the intermediate states satisfied with all jump conditions in this approximate Riemann system are analytically obtained, the multi-state HLL Riemann solver can be constructed straightforwardly. It is shown that this solver can exactly resolve isolated discontinuities formed in the MHD system, and hence named as HLLD Riemann solver. (Here, “D” stands for Discontinuities.) It is also analytically proved that the HLLD Riemann solver is positively conservative like the HLLC Riemann solver. Indeed, the HLLD Riemann solver corresponds to the HLLC Riemann solver when the magnetic field vanishes. Numerical tests demonstrate that the HLLD Riemann solver is more robust and efficient than the linearized Riemann solver, and its resolution is equally good. It indicates that the HLLD solver must be useful in practical applications for the ideal MHD equations.