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

T. Miyoshi, K. Kusano / Journal of Computational Physics 208 (2005) 315–344 331u g e 1 1R2 q R jv R j2 2 jB R j2¼ q Rn2 f2 þ p R f þ p Rnc 1 þ q Rn2 ðjv ? Rj 2 jv ? Rj 2 Þþ n 2 ðjB ? Rj 2 jB ? Rj 2 Þþ f 2 ðjB ? Rj 2 þjB ? Rj 2 Þþ B x ðv ?R B ?R v ? R B ? RÞ¼ q Rn21jB ?R j 2q R ng B 2 x!f 2 þ p R f þ p Rnc 1 ; ð94Þwhere v^ = (0, v, w) and B^ = (0, B y , B z ). It is noted that (94) is identical to the corresponding equation forthe HLLC solver [2] except for the correction in the first term. Since S R is expressed by k 7 of (3) and is themaximum speed in the Riemann system, the inequalities, n P c fR and g P c fR , will be satisfied. Therefore,!u 0 ¼ q Rn jB ?R j 21f 2 þ p2 q R c 2 f RB 2 R f þ p Rn6 u: ð95Þc 1xThis relation indicates that u is necessarily positive if u 0 is positive. (We point out here that Gurski [13]misled a ‘‘stronger condition’’ for positivity in view of the inequality (95).) Sinceqc 2 fjBj 2 ¼ qc 2 fB 2 xjB ? j 2 > 0 except when B^ = 0 and B 2 xP cp, the first term of (95) is necessarily positive.Note that u in the case of B^ = 0 fully corresponds to that for the HLLC solver [2], and therefore, thepositivity of u is assured in this case. Thus, since all coefficients with respect to f are positive, u 0 is positivefor any f if the discriminant of u 0 is negative:!Dðu 0 Þ¼p 2 2q R p R jB ?R j 2R1n 2 < 0:c 1 q R c 2 f RB 2 xTherefore,n 2 > ðc1Þp R2q R1!jB ?R j 2 1: ð96Þq R c 2 f RB 2 xIf B x 6¼ 0, by using the relations thatqc 2 sB 2 x ¼ B2 x jB ?j 2; c 2qc 2 fB 2 f c2 s ¼ cpB2 x;qx2(96) can be rewritten asn 2 > c 12c c2 f R: ð97ÞAlso, if B x = 0, where qc 2 f¼ cp þjB ? j 2 , the identical inequality is easily derived from (96). Therefore, from(92) and (97), in order to preserve the positivity of the pressure (90), S R must be chosen to be satisfied withthe inequality assffiffiffiffiffiffiffiffiffiffic 1S R > u R þ c fR :ð98Þ2cNote that the resultant inequality (98) is quite similar to the corresponding inequality for the HLLC solver(22) but the fast magnetosonic speed c fR must be replaced by the sound speed a R . As found form (12), (13),