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

T. Miyoshi, K. Kusano / Journal of Computational Physics 208 (2005) 315–344 317another essential problem for multi-dimensional MHD concerned with the violation of the divergenceconstraint (see, e.g., [8,16]), discussions of positivity here are restricted to one-dimensional MHD. OurHLL based solver for MHD can exactly resolve not only isolated contact discontinuities but also all otherisolated discontinuities formed in the ideal MHD system, and therefore, the solver is named as HLLD.The organization of the paper is as follows: After introduction, the MHD equations and their nature arepresented in Section 2. We briefly review the single-state and two-state HLL approximate Riemann solversin Sections 3 and 4. In Section 5.1, new multi-state approximate Riemann solver for MHD, called HLLDsolver, is proposed. The important properties of the HLLD solver such as an exact resolution of discontinuitiesand a positivity preserving property are argued in Sections 5.2 and 5.3. In Section 6, several numericaltests are performed, and those results are discussed. Finally, concluding remarks are given in Section 7.2. Governing equationsGeneral one-dimensional hyperbolic conservation laws can be written asoUot þ oFox ¼ 0;ð1Þwhere all eigenvalues of the Jacobian A = oF/oU are real and distinct, and the set of corresponding eigenvectorsis complete. In one-dimensional ideal MHD equations, the conservative variable vector U and theflux function F are given by0 1 01qquququ 2 þ p T B 2 xqvqvu B x B yU ¼qw; F ¼qwu B x B z; ð2ÞB yB y u B x vB C BC@ B z A @ B z u B x w Aeðe þ p T Þu B x ðv BÞwhere v =(u, v, w), B =(B x , B y , B z ), p T denotes the total pressure, and other notations are standard. Here,due to the divergence free condition of the magnetic field, B x is given as a constant in one dimension. Thepressure p and the total pressure p T are given byp ¼ðc1Þ e1 12 qjvj2 2 jBj2and p T ¼ p þ 1 2 jBj2 ;respectively.It is well known that these equations have seven eigenvalues which correspond to two Alfvén waves, fourmagneto-acoustic (two fast and two slow) waves, and one entropy wave:k 2;6 ¼ u c a ; k 1;7 ¼ u c f ; k 3;5 ¼ u c s ; k 4 ¼ u; ð3Þwhere8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 92c a ¼ jB >< cp þjBj 2 cp þjBj 2 4cpBxj2 x>=p ffiffiffi ; c f;s ¼q2q>:>;1=2:

318 T. Miyoshi, K. Kusano / Journal of Computational Physics 208 (2005) 315–344It is obvious that the inequalitiesk 1 6 k 2 6 k 3 6 k 4 6 k 5 6 k 6 6 k 7are satisfied, and thus, some eigenvalues may coincide depending on the direction and the strength of themagnetic field. Therefore, the MHD equations are not strictly hyperbolic, and, as a consequence, the completeset of the eigenvectors is not obtained straightforwardly [1,4,5]. In addition, as first pointed out byBrio and Wu [1], the flux function F is not convex in the vector U in the MHD system.Since the MHD equations possess non-convexity as well as non-strict hyperbolicity, a solution of theRiemann problem may be composed not only of ordinary shock and rarefaction waves but also other wavesas compound waves and overcompressible shocks [4,17]. Although it is not so easy to solve the Riemannproblem for such MHD system in general, the discontinuous solutions formed by the regular waves can bediscussed easily using the Rankine–Hugoniot relations, S[U] =[F], where S denotes the speed of the discontinuitiesand [Æ] indicates the jump of the states ahead of and behind the discontinuities, U 1 and U 2 . Whenwe consider solutions for which u 1 , u 2 6¼ S, i.e., there is net particle transport across the discontinuities, andB x 6¼ 0, three types of shock solutions are obtained. In the cases of the compressible solutions, that is,q 1 < q 2 and u 1 > u 2 , the fast and slow shocks are realized, in which both shocks conserve the signs of thetangential components of the magnetic field. While the magnetic field is strengthened behind the fast shock,it is weakened behind the slow shock. On the other hand, the incompressible condition for which q 1 = q 2and u 1 = u 2 gives the following relations:h i½qŠ ¼½pŠ ¼ B 2 y þ p B2 z¼ 0; ffiffiffipq ½vŠ ¼½By Š; ffiffiffi q ½wŠ ¼½Bz Š;ð4Þacross the discontinuity which moves with the Alfvén wave speed k 2 or k 6 . This solution is called the Alfvénshock. Since thermodynamical quantities, q and p, are not changed across the discontinuity, it is also calledthe rotational discontinuity. Also, the discontinuity moving with the entropy wave speed k 4 must be satisfiedwith½B y Š¼½B z Š¼½vŠ ¼½wŠ ¼½pŠ ¼0for the case of B x 6¼ 0, or"p þ B2 y þ #B2 z¼ 02ð5Þð6Þfor B x = 0. The former discontinuity is called the contact discontinuity. At the contact discontinuity, the tangentialcomponents of the velocity and magnetic field must be continuous. The latter relation, on the otherhand, indicates that the tangential velocity and the tangential magnetic field may have a jump across thediscontinuity. Thus, that is called the tangential discontinuity.3. HLL Riemann solverLet us consider general hyperbolic conservation laws (1), where U and F are not specified here. The integralform of the conservation laws for a rectangle (x 1 , x 2 ) · (t 1 , t 2 ) is given byZ x2x 1Uðx; t 2 ÞdxZ x2x 1Uðx; t 1 Þdx þZ t2t 1FUðx ð 2 ; tÞÞdtZ t2t 1FUðx ð 1 ; tÞÞdt ¼ 0: ð7ÞHarten et al. [15] showed that the Godunov-type scheme for (1) can be written in conservative form:

U nþ1i¼ U n iT. Miyoshi, K. Kusano / Journal of Computational Physics 208 (2005) 315–344 319Dt Dx FRð0; Un i ; Un iþ1 Þ FRð0; U n i 1 ; Un i Þ ;where n and i indicate a time step and a cell number, respectively, and Rðx=t; U n i ; Un iþ1Þ is the approximatesolution of the Riemann problem around the interface x i+1/2 . In this form, the appropriate numerical fluxesare obtained by applying the integral conservation laws (7) over the rectangle (x i , x i+1/2 ) · (t n , t n+1 )asZ1xiþ1=2 F iþ1=2 ¼ F i R x x iþ1=2; U n iDt x iDt; Un iþ1dx þ x iþ1=2 x iU n iDt;ð8Þwhere F iþ1=2 ¼ FðRð0; U n i ; Un iþ1 ÞÞ; F i ¼ FðU n iÞ,andDt = tn+1 t n . We note that the exact solution of the Riemannproblem R exact produces the fluxes of the original Godunov scheme. The numerical fluxes F i+1/2 obtainedby the other integral conservation laws over (x i+1/2 , x i+1 ) · (t n , t n+1 ) must coincide with (8) due to theconsistency with the integral form of conservation laws over (x i , x i+1 ) · (t n , t n+1 ).Particularly, Harten et al. [15] proposed one of the simplest Godunov-type scheme, the so-called HLLapproximate Riemann solver. The HLL Riemann solver is constructed by assuming an average intermediatestate between the fastest and slowest waves. Consider a ‘‘subsonic’’ solution of the single-state approximateRiemann problem at the interface between the left and right states, U L and U R , where the minimumsignal speed S L and the maximum signal speed S R are negative and positive, respectively (Fig. 1). By applyingthe integral conservation laws (7) over the Riemann fan, (DtS L , DtS R ) · (0, Dt), the intermediate state isgiven byU ¼ S RU R S L U L F R þ F L: ð9ÞS R S LAfter that, as denoted by (8), the integral over (DtS L ,0)· (0, Dt) gives the HLL fluxes,F ¼ S RF L S L F R þ S R S L ðU R U L Þ: ð10ÞS R S LIf both signal speeds are of the same sign, the fluxes must be evaluated only from the upstream side. Therefore,in general, the HLL fluxes become8>< F L if S L > 0;F HLL ¼ F if S L 6 0 6 S R ;ð11Þ>:F R if S R < 0:Practically, (11) can be unified with (10) if the signal speeds are replaced by S L = min(S L , 0) andS R = max(S R , 0).In order to complete the HLL Riemann solver, S R and S L must be estimated appropriately. Correctlyspeaking, the upper and lower bounds of the signal speed in the system cannot be obtained without informationof the exact Riemann solution [2]. Particularly, the difficulty for the MHD equations may beS = x/t LtU∗S = x/t RU LU RxFig. 1. Schematic structure of the Riemann fan with one intermediate state.

322 T. Miyoshi, K. Kusano / Journal of Computational Physics 208 (2005) 315–344estimation of S L and S R , the algorithm (13) is adopted. Also, the middle wave S M is evaluated from theHLL average like the HLLC. In MHD, the total pressure must be continuous across the middle wave insteadof the pressure in the Euler equations as found from (5) or (6). Therefore, (16) is replaced byp T ¼ p T Lþ q L ðS L u L ÞðS M u L Þ¼ p TRþ q R ðS R u R ÞðS M u R Þ: ð23ÞSince the normal velocity in the Riemann fan is assumed to be constant, the densities in the intermediatestates q a are given by the same as the HLLC Riemann solver, (17). In the case of B x = 0, the tangentialmomentum equations of MHD coincide with those of the Euler equations. Also, the tangential componentsof the magnetic field behave like passive scalars. Therefore, in this case, the intermediate states,v a ¼ v a;ð24Þw a ¼ w a;ð25ÞB y a¼ B yaS a u aS a S M; ð26ÞB z a¼ B zaS a u aS a S M; ð27Þe a ¼ ðS a u a Þe a p Tau a þ p T S M; ð28ÞS a S Mare easily obtained by the similar procedure for the HLLC solver. In the case of B x 6¼ 0, on the other hand,the intermediate states from (24)–(28) are not consistent with the jump condition on the contact discontinuity(5) in which the tangential components of the velocity and magnetic field must be continuous. Therefore,those are replaced by the HLL average (9) asv L ¼ v R ¼ ðqvÞq v ; ð29Þw L ¼ w R ¼ ðqwÞq w ; ð30ÞB y L¼ B y R¼ B y ; ð31ÞB z L¼ B z R¼ B z ; ð32Þe a ¼ ðS a u a Þe a p Tau a þ p T S M þ B x ðv a B a v B Þ: ð33ÞS a S MFinally, F aare computed from each jump condition. This solver is referred to as HLLC-G Riemann solverin this paper.It is found that the fluxes obtained by (29)–(33) are not reduced to those by (24)–(28) in the limit ofB x = 0, and partly correspond to the single-state HLL fluxes. Therefore, the HLLC-G Riemann solver cannotsimulate the structures related to the Alfvén and slow waves with high enough resolution although isolatedcontact discontinuities as well as isolated fast shocks can be resolved exactly.In order to resolve the discontinuities across the slow and Alfvén waves sharply, Gurski [13] also proposedanother method, as it were extended HLLC-type solver, where the intermediate states are obtained

which is identical to (14) except that p is replaced by p T . Since the normal velocity is assumed to be constantover the Riemann fan, the equalitiesu L ¼ u L¼ u R ¼ u R ¼ S Mð39Þare given. In addition to (39), our assumption will restrict the total pressure to constant asp T L¼ p T L¼ p T R¼ p T R¼ p T : ð40ÞThe average total pressure p Tin the Riemann fan should be evaluated consistent with the jump conditionsfor each wave. The present choice of S M (38) consistently gives the equalities that p T ¼ p T L¼ p T Rfrom thejump conditions of the normal momentum across S R and S L as indicated by (23) for the HLLC-type solver.More explicitly, (23) can also be rewritten asp T ¼ ðS R u R Þq R p TLðS L u L Þq L p TRþ q L q R ðS R u R ÞðS L u L Þðu R u L Þ: ð41ÞðS R u R Þq R ðS L u L Þq LThe other equalities p T L¼ p T R¼ p Tare also satisfied automatically as shown later. Note that contact, tangential,and rotational discontinuities can be formed in the Riemann fan even under the restriction (40).Once S M and p T are given, the states U a neighboring U a are obtained from the jump conditions acrossS a ,0 1 0q aq a S 1 0 1 01Mq aq a u aq a S Mq a S2 M þ p TB 2 xq q a v aq a S aq v a S a u aq a u 2 a þ p T aB 2 xM B x B y aq a v aq a v a u a B x B yaa w aq a w a S M B x B z a¼ S aq a w aq a w a u a B x B za; ð42ÞB y aB y aS M B x v aB yaB y au a B x v aB@ B C Bz aA @ B z aS M B x w C B C BCa A @ B za A @ B za u a B x w a Aðe a þ p T ÞS M B x ðv a B a Þðe a þ p TaÞu a B x ðv a B a Þe awhere a = L or R as used in the previous section. It is certain that the second equation of (42) is consistentwith our choice of S M and p T because p Titself is derived from this equation. The first equation of (42) givesS au aT. Miyoshi, K. Kusano / Journal of Computational Physics 208 (2005) 315–344 325q a ¼ q a ; ð43ÞS a S Mwhich is identical to (17) except for the expression of S M . Solving the third and fifth equations of (42) simultaneously,one obtains thatv a ¼ v S M u aa B x B ya ; ð44Þq a ðS a u a ÞðS a S M Þ B 2 xe aB qy a¼ B a ðS a u a Þ 2 B 2 xya : ð45Þq a ðS a u a ÞðS a S M Þ B 2 xAlso, from the fourth and sixth equations, we obtain thatw a ¼ w S M u aa B x B za ; ð46Þq a ðS a u a ÞðS a S M Þ B 2 xB qz a¼ B a ðS a u a Þ 2 B 2 xza : ð47Þq a ðS a u a ÞðS a S M Þ B 2 x

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.

328 T. Miyoshi, K. Kusano / Journal of Computational Physics 208 (2005) 315–344This solver is named as HLLD solver because, as shown in the next subsection, not only isolated contactdiscontinuities but also isolated all other discontinuities (not shocks) in the MHD system are exactly resolved,i.e., ‘‘D’’ represents all Discontinuities in MHD.It is found at once that the four-state HLL Riemann solver for MHD as illustrated in Fig. 3 is naturallyreduced to the two-state solver as in Fig. 2 for B x = 0 because, in this case, S L ¼ S R ¼ S M from (51). Therefore,the estimation of the inner intermediate states as (59)–(62) is not necessary for the case of B x = 0. Notealso that the HLLC solver for the Euler equations [2,29] is included in the HLLD solver for the MHD equationsas the limit of zero magnetic field.In order to obtain the exact upper and lower bounds of the signal speeds for the Riemann problem,complicated exact solutions for the MHD Riemann problem are needed [2]. Therefore, S L andS R may be estimated approximately by (12) or (13). Also, since the explicit expressions of the largestand smallest eigenvalues for MHD are known as in (3), other estimations may be utilized, forinstance such thatS L ¼ minðu L ; u R Þ maxðc f L ; c fR Þ; S R ¼ maxðu L ; u R Þþmaxðc f L ; c fR Þ; ð67Þor any adequate estimations (e.g., [16]).5.2. Exact resolution of isolated discontinuities and shocksAt an isolated tangential discontinuity (6),u L ¼ u R ¼ u; p TL¼ p TR¼ p T : ð68ÞSubstituting (68) into (38) givesS M ¼ ðS R uÞq R u ðS L uÞq L u¼ u: ð69ÞðS R uÞq R ðS L uÞq LThen, by substituting (69) into (43) to (48), it is obtained thatU L ¼ U L; U R ¼ U R: ð70ÞIn the case of which tangential discontinuities can be formed, i.e., B x = 0, the HLLD solver is reduced to thetwo-state HLL solver as stated previously. Thus, the HLLD solver gives the exact solution of an isolatedtangential discontinuity for a general x/t,U ¼ U L if x=t < u;ð71ÞU R if x=t > u:At an isolated contact discontinuity (5), in addition to (68),v L ¼ v R ¼ v; w L ¼ w R ¼ w; B yL ¼ B yR ¼ B y ; B zL ¼ B zR ¼ B z : ð72ÞFrom the condition (68), (70) is obtained in the same way. Also, by substituting (72) into (59)–(63) with(49),U L ¼ U L; U R ¼ U R:ð73ÞTherefore, an isolated contact discontinuity can be resolved exactly as (71) in the HLLD solver.Rotational discontinuities must satisfy the jump conditions of (4). Therefore, at an isolated rotationaldiscontinuity,q L ¼ q R ¼ q;ð74Þbesides (68). Therefore, the wave speed of an isolated rotational discontinuity is given by

s A ¼ u jB xjp ffiffiffi ; ð75Þqthat is, s A = k 2 or k 6 of (3). Without loss of generality, assume here that B x > 0 and s A > u. In this case, therelations,pffiffiffi pffiffiffiqð vR v L Þ ¼ B yR B yL ; qð wR w L Þ ¼ ðB zR B zL Þ; ð76Þmust be realized from the jump conditions. Similar to the discussion for an isolated contact or tangentialdiscontinuity, (70) is obtained from (68). Substituting (76) into (59)–(63) givesU L¼ U R ¼ U Lwith the help ofpffiffiffie R e L ¼ qð vR B R v L B L Þ: ð78ÞThe equalities (77) are retained also for B x < 0. On the other hand, when s A < u,U L ¼ U L; U L¼ U R ¼ U R ¼ U R:ð79ÞTherefore, in general,U ¼ U L if x=t < s A ;ð80ÞU R if x=t > s A :Thus it is shown that the HLLD solver also resolves an isolated rotational discontinuity exactly.The speed of an isolated fast shock s f is computed as the largest or smallest eigenvalue of the Roe matrix. Wemay consider an isolated shock corresponding to the largest eigenvalue only, i.e., s f ¼ k Roe7, because of the symmetryof the left and the right states. The left and the right states at an isolated fast shock are related bys f ðU R U L Þ ¼ F R F L : ð81ÞOnce the maximum signal speed S R in the HLLD solver is given by the exact shock velocity, (38) givesS M ¼ ðs f u L Þq L u L ðS L u L Þq L u L¼ u L ð82Þðs f u L Þq L ðS L u L Þq Lby using the jump conditions for the continuity equation and the normal component of the momentumequation in (81). Also,p T ¼ p T L;ð83Þby substituting (82) into (41) or equivalently (23). Then, substituting (82) and (83) into (43)–(48), some algebraicmanipulations with the help of (81) giveU L ¼ U R ¼ U L:ð84ÞTherefore, it is also obtained straightforwardly from (59)–(63) thatU L¼ U R ¼ U L:Similarly, when s f = S L ,U L ¼ U L¼ U R ¼ U R ¼ U R:Thus, the exact shock solution,U ¼ U L if x=t < s f ;U R if x=t > s f ;T. Miyoshi, K. Kusano / Journal of Computational Physics 208 (2005) 315–344 329ð77Þð85Þð86Þð87Þ

330 T. Miyoshi, K. Kusano / Journal of Computational Physics 208 (2005) 315–344is obtained in the HLLD solver. It is noted that the exact resolution of an isolated fast shock (87) relies onthe proper choices of S L and S R , and thus, the algorithm of (13) is appropriate in this particular viewpoint.On the other hand, the resolution of an isolated contact, tangential discontinuity (71) and rotational discontinuity(80) does not depend on the detailed estimation of S R and S L .5.3. Positivity of HLLD solverConsider the set of the physically realistic states with positive densities and positive pressures for theMHD equations such that8239qquqv>=G ¼qw; q > 0 and e2 qjvj2 2 jBj2 > 0 :B y6 74 B z 5>:>;eIn the MHD equations, and likewise in the Euler equations, the average states defined byU =(1 h)U 1 + hU 2 with 0 6 h 6 1 are physically realistic states if both U 1 and U 2 are physical [16]. SinceGodunov-type schemes update the subsequent states by averaging the states of the exact or the approximatesolution of the Riemann problem, a positively conservative scheme can be constructed in thoseschemes if all the states generated are physical. Thus, a positively conservative Riemann solver generatesstates in G if the preceding states are contained in G.We now consider the right outer and inner intermediate states, U Rand UR. The positivity preservingconditions becomeq R > 0;ð88Þq R > 0;ð89Þp R ¼ðc1Þ 1 1e R2 q R jv R j2 2 jB R j2 > 0; ð90Þp R ¼ðc1 11Þ eR2 q R jv R j2 2 jB R j2 > 0: ð91ÞIn order to simplify the subsequent arguments, the following variables are introduced:n S R u R ; g S R S M ; f S M u R : ð92ÞSince S R is the maximum signal speed in the Riemann system, both n and g are positive. On the other hand,f can be either positive or negative.The conditions of (88) and (89) are easily shown from (43) and (49) such thatq R ¼ q R ¼ n g q R > 0:ð93ÞTo show the inequality (90), let us consider the positivity of u defined and rearranged by some algebraicmanipulations as follows:

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),

332 T. Miyoshi, K. Kusano / Journal of Computational Physics 208 (2005) 315–344or (67), since the HLLD solver for MHD clearly satisfies (98), our solver ensures the positivity preservingproperty (90).Subsequently, if the pressure at the outer states is retained as positive, the positivity of the pressure at theinner states is shown straightforwardly by using (59)–(63) to bep R ¼ðc1Þ eR¼ p R þ c 1212 q R jv j 2 12 jB j 2pffiffiffiffiffi q R vR þ B R signðB xÞ 2pffiffiffiffiffi q R v þ B signðB x Þ 2¼ p R > 0:Also, it is noted from (99) that our solver is consistent with the continuous condition of the pressure acrossrotational discontinuities.The positivity of the density and pressure at the left outer and inner intermediate states, U Land UL ,isalso assured if the inequality,sffiffiffiffiffiffiffiffiffiffic 1S L < u L c fL ;ð100Þ2cis satisfied. Thus, the proof of the positivity is completed. It is obvious that (98) and (100) are stronger thanthe conditions by Gurski [13]. The present proof will also be applicable to the single-state HLL Riemannsolver since the one intermediate state of the HLL solver is given by the average of the HLLD intermediatestates contained in G.At the end of this subsection, we reconsider the prerequisite inequalities to obtain a stronger conditionfor the pressure positivity (95). We find at once that the prerequisite inequality as n P c fR is assured due tothe estimation (67) or something equivalent. On the other hand, it is not so easy to confirm whether theother inequality g P c fR is satisfied or not at any situation, because S M is evaluated not only from the rightstates but also from the left states. In order to simplify the discussion, we only pay attention to expansionwaves which are more problematic than shocks in terms of positivity. From the jump condition of the normalmomentum across S R , we obtain thatp Tg ¼ n þ p T R:q R nSince p TRis greater than p T under the condition u R u L > ðp TRp TLÞ=q L ðS L u L Þ, the inequalitiesg > n P c fR are satisfied at such strong expansion waves. Thus, both prerequisite inequalities for the pressurepositivity are satisfied in the most problematic situation. Notice that this fact does not mean that othersituations than strong expansion waves would necessarily break the prerequisite inequalities. Moreover,since the positivity preserving condition (98) is a rather weak and easily conquerable restriction, the HLLDsolver is thought to be a positively conservative scheme.ð99Þ6. Numerical testsWe present in this section results for several test problems computed by the HLLD Riemann solver. Forcomparison, the same problems are solved by the single-state HLL Riemann solver and the standard linearizedRiemann solver, the so-called Roe scheme [26], in which we adopt the Roe average by Balsara [1].Also, the HLLD solver is compared with the variants of the HLLC-type solver. In our tests, as an algorithmto estimate S L and S R , (67) is used for simplicity and efficiency because numerical solutions by(67) are indistinguishable from those by (13) in spite of not resolving isolate fast shocks exactly. Also, cis fixed to 5/3 throughout our numerical tests. In order to evaluate the essential abilities of the solver,

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

T. Miyoshi, K. Kusano / Journal of Computational Physics 208 (2005) 315–344 3351.6ρ1.51.41.3HLL solverHLLC-G solverHLLC-L solverHLLD solverRoe scheme0.0 0.1 0.20.60.20.1v0.50.40.3w0.0-0.1HLL solverHLLC-G solverHLLC-L solverHLLD solverRoe scheme0.0 0.1 0.20.20.10.0-0.1HLL solverHLLC-G solverHLLC-L solverHLLD solverRoe scheme0.0 0.1 0.21.5HLL solverHLLC-G solverHLLC-L solverHLLD solverRoe scheme0.80.7HLL solverHLLC-G solverHLLC-L solverHLLD solverRoe scheme0.6B y0.5B z1.40.0 0.1 0.20.0 0.1 0.2Fig. 6. Results of one-dimensional shock tube test with the same initial states as in Fig. 5. Numerical solutions of the HLL solver, theHLLC-type solver by Gurski (HLLC-G) [13], the HLLC-type solver by Li (HLLC-L) [19], the HLLD solver, the Roe scheme areplotted at t = 0.2. (Top) q, (middle left) v, (middle right) w, (bottom left) B y , (bottom right) B z are shown within 0.02 < x < 0.26.might be due to the underestimation of the signal speed of the rarefaction wave in the Roe scheme, which isinevitable for the linearized Riemann solver. Corresponding undershoots are not observed in the HLLDsolver.

336 T. Miyoshi, K. Kusano / Journal of Computational Physics 208 (2005) 315–3440.2v0.60.51st-HLLC-L solver2nd-HLLC-L solver1st-HLLD solver2nd-HLLD solver0.40.11st-HLLC-L solver2nd-HLLC-L solver1st-HLLD solver2nd-HLLD solver0.0 0.10.30.2w0.0 0.11.5B y0.80.71st-HLLC-L solver2nd-HLLC-L solver1st-HLLD solver2nd-HLLD solver0.61.41st-HLLC-L solver2nd-HLLC-L solver1st-HLLD solver2nd-HLLD solver0.0 0.10.50.4B z0.0 0.1Fig. 7. Results of one-dimensional shock tube test with the same initial states as in Fig. 5. Numerical solutions of the first- and secondorderHLLC-type solver by Li (1st- and 2nd-HLLC-L) [19], the first- and second-order HLLD solver (1st- and 2nd-HLLD) are plottedat t = 0.2. (Top left) v, (top right) w, (bottom left) B y , (bottom right) B z are shown within 0.02 < x < 0.1.Although the above two tests contain a number of different waves and are quite useful when comparingMHD solvers with one another, tests for isolated waves are also meaningful. Since information of the slowwaves is lost when constructing the HLLD solver, we concentrate on test problems for the slow waves inparticular. Following the test of a slow switch-off shock by Falle et al. [11], the initial states are given by(q, p, u, v, w, B y , B z ) = (1.368, 1.769, 0.269, 1, 0, 0, 0) for x < 0, (1, 1, 0, 0, 0, 1, 0) for x > 0, with B x =1.Fig. 9 shows the results at t = 0.2, where the tangential component of the magnetic field, B y , is switchedoff behind the shock. It is found that the HLLD solver can resolve the isolated slow shock equally tothe Roe scheme despite the loss of information on the slow waves while the HLL solver cannot resolvethe shock well. We note that an error caused by the jump of the initial states is observed in every solver.We also perform the test of a slow switch-off rarefaction given also by [11], where the initial states are(q, p, u, v, w, B y , B z ) = (1, 2, 0, 0, 0, 0, 0) for x < 0, (0.2, 0.1368, 1.186, 2.967, 0, 1.6405, 0,) for x > 0, withB x =1.Fig. 10 shows the results at t = 0.2, where B y is switched off over the head of the rarefaction. Alsoin this test, a startup error is observed near the tail of the rarefaction. It is thought that such errors may beinevitable unless the initial left and right states are connected continuously with a finite width. We find thatthe strong slow rarefaction can be calculated by the HLL and the HLLD solvers without any extra numericaldissipation. The isolated fast rarefaction [11] can also be calculated although the results are not shownin this paper. The success of these solvers may be because the solvers are bounded by the maximum and

T. Miyoshi, K. Kusano / Journal of Computational Physics 208 (2005) 315–344 3371.0ρHLL solverHLLD solverRoe scheme0.8HLLD solverRoe schemeρ0.3HLLD solverRoe schemeρ0.50.70.20.0-0.5 0.0 0.50.6-0.04 0.000.10.11 0.151.0pHLL solverHLLD solverRoe scheme0.5u0.50.0-0.5 0.0 0.50.0HLL solverHLLD solverRoe scheme-0.5 0.0 0.50.0-0.5v1.00.5B y-1.0-1.50.0-0.5HLL solverHLLD solverRoe scheme -1.0HLL solverHLLD solverRoe scheme-0.5 0.0 0.5-0.5 0.0 0.5Fig. 8. Results of one-dimensional shock tube test with the initial left states (q, p, u, v, w, B y , B z ) = (1, 1, 0, 0, 0, 1, 0), the right states(0.125, 0.1, 0, 0, 0, 1, 0), and B x = 0.75. Numerical solutions of the HLL solver, the HLLD solver, and the Roe scheme are plotted att = 0.1. (Top left) q, (middle left) p, (middle right) u, (bottom left) v, (bottom right) B y , (top middle) q around the slow compound wave,(top right) q around the slow shock are shown.minimum signal speeds in the Riemann fan and, as a result, include the effect of not only the slow rarefactionbut also the fast rarefaction properly. On the other hand, the Roe scheme fails in this test, where theentropy condition is broken at x = 0. It is found, however, that an additional entropy correction (e.g., [14])can cure the unphysical solution in this problem.

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

T. Miyoshi, K. Kusano / Journal of Computational Physics 208 (2005) 315–344 3391.0ρHLL solverHLLD solverRoe scheme1.0ρHLL solverHLLD solverRoe + E-fix0.50.50.0-0.5 0.0 0.50.0-0.5 0.0 0.53.0v1.5B y2.01.01.00.00.5HLL solverHLLD solverRoe scheme 0.0HLL solverHLLD solverRoe scheme-0.5 0.0 0.5-0.5 0.0 0.5Fig. 10. Results of one-dimensional shock tube test with the initial left states (q, p, u, v, w, B y , B z ) = (1, 2, 0, 0, 0, 0, 0), the right states(0.2, 0.1368, 1.186, 2.967, 0, 1.6405, 0), and B x = 1. Numerical solutions of the HLL solver, the HLLD solver, and the Roe scheme areplotted at t = 0.2. (Top left) q, (bottom left) v, (bottom right) B y , (top right) q by the Roe scheme with the entropy condition (Roe +E–fix) are shown.Evans and Hawley [10] can maintain the divergence free condition within machine accuracy, there are nogeneral guidelines to apply any one-dimensional MHD solver properly. Thus, the application of the CTmethod to the HLLD solver is beyond the scope of this paper. On the other hand, the other divergencecleaning method by projection [3], diffusion [23], or transport [8,25] of divergence errors can be directlyadded on to existing one-dimensional MHD schemes.Particularly, Dedner et al. [8] showed that some add-on divergence cleaning methods can be expressed bythe so-called generalized Lagrange multiplier (GLM) formulation of the MHD equations. They alsopointed out that the choice of the mixed hyperbolic/parabolic GLM-MHD gives excellent results sincethe divergence errors in the mixed GLM-MHD are transported out of the domain by two waves withthe maximal admissible speed even in stagnation points and are damped at the same time. The mixedGLM-MHD is also effective in other aspects; easy to implement on an existing code, fast due to the explicitapproximation, conservative for the physical quantities. Thus, keeping practical applications in mind, thedivergence cleaning method based on the mixed GLM-MHD is adopted in our multi-dimensional test.We perform, in particular, the so-called Orszag–Tang vortex problem [24] which is a standard twodimensionaltest for MHD schemes (e.g., [30] and many other references therein). The Orszag–Tang vortexproblem is thought to be appropriate for comparing the resolution of MHD schemes because complex

340 T. Miyoshi, K. Kusano / Journal of Computational Physics 208 (2005) 315–344010-110log ( ρ)10 0log ( ρ)10 10-110-210HLL solverHLLD solverRoe + E-fix-310-0.5 0.0 0.5-210HLL solverHLLD solver-310-0.5 0.0 0.53.02.0u3.02.0u1.01.00.00.0-1.0-1.0-2.0-3.0HLL solver-2.0HLLD solverRoe + E-fix -3.0HLL solverHLLD solver10 0-0.5 0.0 0.510 0-0.5 0.0 0.5-110log ( p T)10-110log ( p )10 T-210-210-310HLL solverHLLD solverRoe + E-fix-310HLL solverHLLD solver-0.5 0.0 0.5-0.5 0.0 0.5Fig. 11. Results of one-dimensional shock tube test with the initial left states (q, p, u, v, w, B y , B z ) = (1, 0.45, u 0 , 0, 0, 0.5, 0), the rightstates (1, 0.45, u 0 , 0, 0, 0.5, 0), and B x = 0. Numerical solutions of the HLL solver, the HLLD solver, and the Roe scheme with theentropy correction (Roe + E–fix) are plotted at t = 0.05. (Left) u 0 = 3, (right) u 0 = 3.1. (Top to bottom) log 10 q, u, log 10 p T are shown.Note that the solution of the Roe scheme is not plotted in the right panel because the Roe scheme fails after several time steps in thecase of u 0 = 3.1.

T. Miyoshi, K. Kusano / Journal of Computational Physics 208 (2005) 315–344 341Table 1Relative differences of the temperature from the solution of the Roe scheme with corresponding accuracy and resolution, dT RoeN 1st HLL 1st HLLD 2nd HLL 2nd HLLD50 0.0779 0.0054 0.0239 0.0040100 0.0579 0.0043 0.0143 0.0028200 0.0453 0.0034 0.0083 0.0027400 0.0353 0.0024 0.0046 0.0011interactions of several MHD shock waves are included in the evolution of the vortex. The initial conditionsof the problem are given by (q, p, u, v, w, B x , B y , B z )=(c 2 , c, siny, sinx,0, siny, sin2 x, 0) in a squaredomain 0 < x, y

342 T. Miyoshi, K. Kusano / Journal of Computational Physics 208 (2005) 315–344Fig. 12. Gray-scale images of the temperature distribution in the Orszag–Tang vortex problem at t = p for (left to right) the HLLsolver, the HLLD solver, the Roe scheme at N = 200, and the reference solution. The left half of the domain is shown.3.02.0HLL solverHLLD solverRoe scheme1.0HLL solverHLLD solverRoe scheme1.00.00 1 2 3 4 5 60.50 1 2 3 4 5 6Fig. 13. One-dimensional temperature distribution in the same problem as in Fig. 12 along (left) y = 0.64p, (right) y = p. for the HLLsolver, the HLLD solver, and the Roe scheme. The solid line shows the reference solution in each panel.7. ConclusionsWe have proposed the multi-state HLL approximate Riemann solver for the ideal MHD equations,named as HLLD approximate Riemann solver. The HLLD Riemann solver was constructed under theassumption that the normal velocity is constant over the Riemann fan. This assumption naturally concludedthat the four states approximation in the Riemann fan are appropriate for B x 6¼ 0, whereas theintermediate states are reduced to the two states when B x = 0. We showed that the HLLD solver can exactlyresolve all isolated discontinuities in the MHD system as well as isolated fast shocks. We also provedanalytically that the HLLD solver is positively conservative if proper inequalities for the maximum andminimum signal speeds are satisfied. Especially, it was shown that those conditions are quite similar to theconditions for the HLLC solver except for the difference of the expressions for the maximum and minimumeigenvalues. Thus, we considered that the HLLD solver for MHD is a natural extension of theHLLC solver for the Euler equations. Also, several numerical tests demonstrated that the HLLD solveris accurate enough in comparison with the linearized Riemann solver while keeping its robustness. From atypical two-dimensional test, it was considered that the extension to multi-dimensions in the HLLD solver

T. Miyoshi, K. Kusano / Journal of Computational Physics 208 (2005) 315–344 343may be applicable in a similar way as in the linearized Riemann solver. The ratios of CPU time s for theOrszag–Tang vortex problem at N = 400 in our codes are (s HLL :s HLLD :s Roe ) = (1:1.35:2.67) in the part ofthe flux calculation and (1:1.13:1.69) over the whole simulation run, although the actual efficiency for thewhole simulation much depends not only on the optimization of the code but also on the algorithm of thedivergence cleaning method, the higher-order extension method, the time integration method, and so on.Thus, the HLLD solver seems to be effective particularly in regard to robustness and efficiency rather thanthe Roe scheme, although exact positivity preserving property in multi-dimensions may not be necessarilyassured even in the HLLD solver. Also, the HLLD solver may be applicable to the modified MHD systemas subtracting off the background potential field first proposed by Tanaka [28] and applied by many others[18,22,25]. In this modified system, the HLLD solver will be constructed by replacing the total pressurep T with the fluctuating part of the total pressure ~p T ¼ p T jB 0 j 2 =2in(41) for example. A detailed algorithmwill be given in the future paper. These all results indicate that the HLLD Riemann solver for theideal MHD equations can be an alternative to the linearized Riemann solver for MHD.AcknowledgementsPart of this work was financially supported by the collaboration program of Japan Atomic EnergyResearch Institute (JAERI), that was arranged by Y. Kishimoto and Y. Ishii of NumericalEXperiment of Tokamak (NEXT) group at Naka Fusion Research Establishment, JAERI. We thankthe anonymous reviewers for constructive comments and suggestions improving the readability of thepaper.References[1] D.S. Balsara, Linearized formulation of the Riemann problem for adiabatic and isothermal magnetohydrodynamics, Astrophys. J.Supp. 116 (1998) 119.[2] P. Batten, N. Clarke, C. Lambert, D.M. Causon, On the choice of wave speeds for the HLLC Riemann solver, SIAM J. Sci.Statist. Comput. 18 (1997) 1553.[3] J.U. Brackbill, D.C. Barnes, The effect of nonzero $ Æ B on the numerical solution of the magnetohydrodynamic equations, J.Comput. Phys. 35 (1980) 426.[4] M. Brio, C.C. Wu, An upwind differencing scheme for the equations of ideal magnetohydrodynamics, J. Comput. Phys. 75 (1988)400.[5] P. Cargo, G. Gallice, Roe matrices for ideal MHD and systematic construction of Roe matrices for systems of conservation laws,J. Comput. Phys. 136 (1997) 446.[6] W. Dai, P.R. Woodward, An approximate Riemann solver for ideal magnetohydrodynamics, J. Comput. Phys. 111 (1994) 354.[7] S.F. Davis, Simplified second-order Godunov-type methods, SIAM J. Sci. Statist. Comput. 9 (1988) 445.[8] A. Dedner, F. Kemm, D. Kröner, C.-D. Munz, T. Schnitzer, M. Wesenberg, Hyperbolic divergence cleaning for the MHDequations, J. Comput. Phys. 175 (2002) 645.[9] B. Einfeldt, C.D. Munz, P.L. Roe, B. Sjögreen, On Godunov-type methods near low densities, J. Comput. Phys. 92 (1991) 273.[10] C.R. Evans, J.F. Hawley, Simulation of magnetohydrodynamic flows: a constrained transport method, Astrophys. J. 332 (1988)659.[11] S.A.E.G. Falle, S.S. Komissarov, P. Joarder, A multidimensional upwind scheme for magnetohydrodynamics, Mon. Not. R.Astron. Soc. 297 (1998) 265.[12] J. Gressier, P. Villedieu, J.M. Moschetta, Positivity of flux vector splitting schemes, J. Comput. Phys. 155 (1999) 199.[13] K.F. Gurski, An HLLC-type approximate Riemann solver for ideal magnetohydrodynamics, SIAM J. Sci. Comput. 25 (2004)2165.[14] A. Harten, J.M. Hyman, Self adjusting grid methods for one-dimensional hyperbolic conservation laws, J. Comput. Phys. 50(1983) 235.[15] A. Harten, P.D. Lax, B. van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAMRev. 25 (1983) 35.

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