A multi-state HLL approximate Riemann solver for ideal ...

338 T. Miyoshi, K. Kusano / Journal of Computational Physics 208 (2005) 315–3441.5ρ1.5ρ1.0HLL solverHLLD solverRoe scheme1.0HLL solverHLLD solverRoe scheme-0.5 0.0 0.50.15 0.20 0.251.0v1.0HLL solverHLLD solverRoe schemeB y0.50.50.0HLL solverHLLD solverRoe scheme0.0-0.5 0.0 0.5-0.5 0.0 0.5Fig. 9. Results of one-dimensional shock tube test with the initial left states (q, p, u, v, w, B y , B z ) = (1.368, 1.769, 0.269, 1, 0, 0, 0), theright states (1, 1, 0, 0, 0, 1, 0), and B x = 1. Numerical solutions of the HLL solver, the HLLD solver, and the Roe scheme are plotted att = 0.2. (Top left) q, (bottom left) v, (bottom right) B y , (top right) q around the slow switch-off shock are shown.In the final shock tube test, we consider super-fast expansions which may be rather extreme situations.The initial states are given by (q, p, u, v, w, B y , B z ) = (1, 0.45, u 0 , 0, 0, 0.5, 0) for x 0, with B x = 0. The fast magnetosonic Mach number M f of the expansionwave is given by u 0 since the fast magnetosonic speed c f is 1 at the initial states. This type of problemfor the Euler equations was shown not to be linearizable for certain Mach numbers [9]. This will also betrue for the MHD equations. Indeed, as shown in Fig. 11, although the physically realistic solutions,non-negative density and pressure, can be obtained in the problem with M f = 3 for all solvers, the Roescheme even with the entropy correction fails in the problem with M f = 3.1. On the other hand, theHLL and the HLLD solvers preserve the positivity without any extra numerical dissipation as expectedanalytically.6.2. Applicability to multi-dimensionsIt is known that the extension of the one-dimensional upwind-type MHD solver to multi-dimensions isnot straightforward because the solenoidal condition of the magnetic field is broken numerically. In orderto remove the numerical divergence errors, several approaches have been applied to the upwind-type solverand compared with one another (e.g., [8,30]). Although the so-called constrained transport (CT) method by