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

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326 T. Miyoshi, K. Kusano / Journal of Computational Physics 208 (2005) 315–344Note that the numerical operations of 0/0 seem to appear in (44)–(47) if S M = u a , S a ¼ u a c f a ,B ya ¼ B za ¼ 0, and B 2 x P cp a. In these cases, (44)–(47) can be simply replaced by v a ¼ v a; w a ¼ w a, andB y a¼ B z a¼ 0 since there is no shock across S a i.e., q a ¼ q a; u a ¼ u a; and p T a¼ p Ta. Finally, the seventhequation of (42) with (44)–(47) gives e a ase a ¼ ðS a u a Þe a p Tau a þ p T S M þ B x ðv a B a v a B a Þ : ð48ÞS a S MThe expressions (44)–(48) are the same than those employed for the intermediate states of the extendedare considered. From the jump condition of the continu-HLLC-type Riemann solver for MHD [13].Subsequently, the inner intermediate states U aity equation across an arbitrary S where S L < S < S M or S M < S < S R ,q a¼ q a ; ð49Þbecause of the relation (39).When(49) is given, the jump condition for the normal momentum across S leads top T a¼ p T a: ð50ÞThus, (40) is satisfied. Also from (49), it is appropriate that the propagation speeds of the Alfvén waves (3)in the intermediate states are evaluated byS L ¼ S Mp jB x jffiffiffiffiffi ; S R ¼ SM þ p jB ffiffiffiffiffixj : ð51Þq Lq RAlthough we should consider the other jump conditions across S awaves, the jump conditions,0q 1 0a v aq a v a S M B x B 1 0y aq 1 0a v aq a v a S M B x B 1y aq S a w aq aaB@ B Cw a S M B x B z aBy aA @ B y aS M B x v CaA ¼ q S a w aq aaB@ B Cw a S M B x B z aBy aA B y aS M B x v C@aA ;ð52ÞB z aB z aS M B x w aB z aB z aS M B x w aappear not to be solvable if S ais defined by (51).Therefore, we consider the jump conditions for the tangential components of the velocity and magneticfield across S M ,0q 1 0L v Lq L v L S M B x B 1 0y Lq 1 0R v Rq R v R S M B x B 1y Rq LS w Lq LM B@ B Cw L S M B x B z LBy LA @ B y LS M B x v CLA ¼ S q R w Rq RMB@ B Cw R S M B x B z RBy RA B y RS M B x v C@RA :ð53ÞB z LB z LS M B x w LB z RB z RS M B x w RIt is obvious that the equalitiesv L¼ v R v ;ð54Þw L¼ w R w ;B y L¼ B y R B y ;B z L¼ B z R B zð55Þð56Þð57Þare derived from (53) if B x 6¼ 0. These equalities indicate the conditions for the contact discontinuities (5).On the other hand, if B x =0,(53) also becomes unsolvable. The case of B x = 0 is discussed later. The relations(54)–(57) show that v, w, B y , and B z are approximated by three intermediate states in our assumption
(Fig. 4). Therefore, by substituting (49),(51), and (54)–(57) into the integral conservation laws over the Riemannfan,ðS R S R ÞU R þðS RS M ÞU R þðS Mit is derived thatpffiffiffiffiffivLþ ffiffiffiffiffiv ¼w ¼B yB zq LS L ÞU L þðS LS L ÞU LS R U R þ S L U L þ F R F L ¼ 0; ð58Þpq R vRþðB y RB y LÞsignðB x Þpffiffiffiffiffip ffiffiffiffiffi; ð59Þq L þ q Rpffiffiffiffiffipq L wLþ ffiffiffiffiffi q R wRþðB z RB z LÞsignðB x Þpffiffiffiffiffip ffiffiffiffiffi; ð60Þq L þ q Rpffiffiffiffiffiq L Bpy¼Rþffiffiffiffiffi q R Bpy Lþffiffiffiffiffiffiffiffiffiffi q L q Rðv Rv L ÞsignðB xÞpffiffiffiffiffip ffiffiffiffiffi; ð61Þþpffiffiffiffiffiq L¼Bz RþT. Miyoshi, K. Kusano / Journal of Computational Physics 208 (2005) 315–344 327p ffiffiffiffiffiq Lq Rp ffiffiffiffiffiffiffiffiffiffi w L ÞsignðB xÞp ; ð62Þq R Bz Lþ q L q Rðw Rpffiffiffiffiffiffiffiffiffiffiq L þ q Rwhere sign(B x ) is 1 for B x > 0 and 1 for B x < 0. Here, it is found that the jump conditions across S a (52)are satisfied by (59)–(62). Finally, the jump condition of the energy density across S awaves can be solved ase a¼ e a p ffiffiffiffiffiq a ðva B av B ÞsignðB x Þ; ð63Þwhere the minus and the plus of the right side correspond to a = L and R, respectively.In this way, we can derive the complete set of the intermediate states, U L ; U L ; U R ; and U R, and correspondingfluxes, F L ; F L ; F Rand F R, which are satisfied with all jump conditions in our approximate Riemannproblem. Therefore, the numerical fluxes of our solver are obtained by the integral of conservationlaws over the left or the right half of the Riemann fan, (S L Dt,0)· (0, Dt) or (0, S R Dt) · (0, Dt), as in (8), forinstance such thatF ¼ F L þ S L U LS L U L ¼ F Lð64Þfor S L 6 0 6 S L ,orF ¼ F L þ S L U LðS LS L ÞU LS L U L ¼ F Lð65Þfor S L 6 0 6 S M. In general, the fluxes are given by8F L if S L > 0;F Lif S L 6 0 6 S L>:F R if S R < 0:S Lt* *S LS RS RV L ∗V∗∗V R ∗V LV RxFig. 4. Schematic structure of the Riemann fan with three intermediate states.
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