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

334 T. Miyoshi, K. Kusano / Journal of Computational Physics 208 (2005) 315–3441.5HLL solverHLLD solverRoe schemeρ1.61.51.4HLLD solverRoe schemeρ1.6HLLD solverRoe schemeρ1.31.21.0-0.5 0.0 0.51.11.51.0-0.22 -0.20 -0.18 -0.16 0.02 0.04 0.06 0.080.2v0.5w0.0HLL solverHLLD solverRoe scheme0.0HLL solverHLLD solverRoe scheme-0.5 0.0 0.5-0.5 0.0 0.51.5B y0.8B z1.0HLL solverHLLD solverRoe scheme0.5HLL solverHLLD solverRoe scheme-0.5 0.0 0.5-0.5 0.0 0.5Fig. 5. Results of one-dimensionalpshock tube test with the initial left states ðq; p; u; v; w; B y ; B z Þ¼ð1:08; 0:95; 1:2; 0:01; 0:5; 3:6=ffiffiffiffiffi pffiffiffiffiffip ffiffiffiffiffi pffiffiffiffiffip4p ; 2= 4p Þ, the right states ð1; 1; 0; 0; 0; 4= 4p ; 2= 4p Þ, and Bx ¼ 4= ffiffiffiffiffi4p . Numerical solutions of theHLL solver, the HLLD solver, and the Roe scheme are plotted at t = 0.2. (Top left) q, (middle left) v, (middle right) w, (bottom left) B y ,(bottom right) B z , (top middle) q around the left fast shock, (top right) q around the left slow shock are shown.the differences may not be important. The smaller resolution of the slow shocks in the HLLD solver may bebecause the slow waves in the Riemann fan are not explicitly included. On the other hand, although thesolution near the head of the rarefaction attached to the compound wave are almost equivalent, the tailof the compound rarefaction calculated by our Roe scheme seems to be very slightly undershot. This fact