A Four-State HLL Riemann Solver for Numerical Simulation of Magneto-Hydrodynamics Based on the Least Squares Solution for the Middle Wave (Communications in Computational Physics, 2025)
By applying the least squares solution for the middle wave analysis insidethe Riemann fan, we construct a four-state Harten-Lax-van Leer (HLL) Riemann solverfor numerical simulation of magneto-hydrodynamics (MHD). First, we revisit the two-state HLL scheme and obtain the two outer intermediate states across the two out-bounding fast waves through Rankine-Hugoniot (R-H) conditions; Second, the twoinner intermediate states are calculated using a geometric interpretation of the R-Hconditions across the middle contact wave. This newly constructed four-state HLLsolver contains different wave structures from those of the HLLD Riemann solver;namely, the two Alfvén waves are replaced as the two combination waves originated from the merging of Alfvén and slow waves inside the Riemann fan. As we tested, this solver resolves the MHD discontinuities well, and has better capture ability than theHLLD solver for the slow waves, although it appears more diffusive than the latter inthe situations where the slow waves are not solely generated. Overall, the new solverhas the similar accuracy as the HLLD solver, thus it is suitable for the calculation ofnumerical fluxes for the Godunov-type numerical simulation of MHD equations wherethe slow waves are expected to be resolved.