Construction of second-order accurate monotone and stable residual distribution schemes for steady problems

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1 Journal of Computatonal Physcs 195 (2004) Constructon of second-order accurate monotone and stable resdual dstrbuton schemes for steady problems Rem Abgrall *,1, Mohamed Mezne Mathematques Applquees de Bordeaux, Unverste Bordeaux I, 351 cours de la Lberaton, Talence Cedex, France Receved 25 Aprl 2003; receved n revsed form 9 September 2003; accepted 10 September 2003 Abstract After havng recalled the basc concepts of resdual dstrbuton (RD) schemes, we provde a systematc constructon of dstrbuton schemes able to handle general unstructured meshes, extendng the work of Sdlkover. Then, by usng the concept of smple waves, we show how to generalze ths technque to symmetrzable lnear systems. A stablty analyss s provded. We formally extend ths constructon to the Euler equatons. Several test cases are presented to valdate our approach. Ó 2003 Elsever Inc. All rghts reserved. AMS: 65C99; 65M60; 76N10 Keywords: Compressble flow solvers; Resdual schemes; Unstructured meshes; Multdmensonal up-wndng 1. Introducton The numercal smulaton of compressble flows s generally done va some generalzaton of the onedmensonal Lax Wendroff scheme. It s well known that ths scheme s stable n the energy norm, but does not have any stablty property n the maxmum norm. The smulaton of flows wth strong dscontnutes can only be performed wth schemes havng propertes n the maxmum norm, because we want solutons wthout numercal oscllaton. Ths goal can be reached by modfcatons of the Lax Wendroff scheme, ether by addng dsspatve and fourth-order terms montored by complex ad hoc, problem dependent, sensors, or va more automatc methods comng from the theory of scalar nonlnear schemes, see e.g. [14,15] and the numerous references theren. In each case, the approxmaton reles strongly on the structure of the one-dmensonal problem. If the flow s represented by the values of the conservatve unknowns at the mesh ponts, a consequence s, for * Correspondng author. Tel.: ; fax: E-mal addresses: abgrall@math.u-bordeaux.fr (R. Abgrall), mezne@math.u-bordeaux.fr (M. Mezne). 1 Insttut Unverstare de France /$ - see front matter Ó 2003 Elsever Inc. All rghts reserved. do: /.cp

2 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) example, that the accuracy of the soluton degrades when gong to multdmensonal problems manly wth rregular meshes. The analyss va Taylor expanson and the equvalent equaton, that enables to study the formal accuracy of a scheme, s strctly vald only for one-dmensonal problems. It can be extended to multdmensonal problems only f the structure of the mesh allows specal algebrac combnatons resultng from the symmetres of the mesh. If the flow s represented by the averaged values of the conservatve unknowns on control volumes, there exsts technques to develop hgh order accurate schemes, for example the ENO or WENO schemes [1,13,16,17,27]. In that case the analyss va the equvalent equaton s dfferent, but the man problem becomes a reconstructon problem, on varables that have to be chosen carefully [4]. The prce to pay s a very large extenson of the computatonal stencl. Even n that case, standard technques use Remann solvers, so the fluxes are stll computed n a one-dmensonal sprt, resultng n large errors as analyzed by van Leer [26]. Hence the overall qualty of the scheme may be qute dsappontng, n many cases. In order to tackle these two problems compactness of the stencl, effectve accuracy of the soluton at least two classes of methods have emerged n recent years, the dscontnuous Galerkn schemes (DG), and the resdual dstrbuton schemes (RD). Though dfferent n sprt, they have a common core at least n ther unstablzed versons, the resdual property. Ths property that we dscuss below n the framework of RD schemes, allows to show the formal accuracy of the scheme on a very general mesh. The DG schemes use a dscontnuous polynomal representaton of the unknowns that s a generalzaton of what s done n fnte volume schemes. The soluton s updated va evaluaton of fluxes, and the stablzaton mechansm s obtaned by very smlar technques as n classcal fnte volume. The net effect of ths s to loose the resdual property. On the contrary, the RD schemes use a pontwse representaton of the soluton, lke n fnte dfference schemes. The unknowns are updated by evaluatng the amount of resdual sent to the vertces, and the stablzaton mechansm can be smlar to artfcal vscosty, as n the SUPG-lke fnte element method [18,20], or nspred by the nonlnear technques of the so-called hgh resoluton schemes [15,19]. One can also take nto account the genunely multdmensonal structure of the problem [23]. In ths paper, we consder schemes of the RD class. Our goal s to propose a systematc constructon of robust, hgh order and compact schemes on general meshes. The schemes are derved n such a way that all the decsons are made on elements only, the neghborng elements play no role n the way the resduals are sent to the element vertces. In that respect, the schemes we consder are the most compact possble. Ths s a very pleasant property for parallelsm ssues. Ths type of schemes has receved recently a lot of nterest, one may quote the poneerng work of Roe, Sdlkover, Deconnck and ther co-workers [11,22,23], and also more recently [2,3,10,25]. However, the solutons that are proposed n these contrbutons are not fully satsfactory, n partcular, they may be not robust enough n some stuatons. In ths paper, we propose to reconsder the approach descrbed n [22] for a very partcular scheme, the frst-order scalar N scheme. From t, one can construct StrusÕ PSI scheme [11] that s second-order. These schemes have the specal property that, for a trangular mesh, on each element, at least one resdual vanshes, allowng very smple algebra. We show how to extend ths approach to more general schemes where we do not need to assume a specal structure of the underlyng frst-order scheme. Ths approach, va a stablty analyss usng smple waves, enables to extend the method to lnear symmetrzable PDEs. Only steady problems are consdered here, the unsteady case s dscussed n [6]. The format of the paper s the followng. We frst recall the formalsm of RD schemes, n partcular we recall what s necessary to get second-order accuracy. Then we analyze the scalar problems and show how to construct second-order schemes. Gong to system problems, we show some specal propertes of the Lax Fredrchs scheme and the N scheme for symmetrzable systems. These stablty propertes enable to ustfy formally our constructon. Several numercal tests, for lnear and nonlnear PDEs demonstrate the qualtes of our approach.

3 476 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) The resdual dstrbuton schemes Let us consder the steady equaton of flud mechancs dv FðW Þ¼0; x 2 X; t > 0; ð1þ supplemented by boundary condtons at the nflow, outflow and sold boundares. The vector of conservatve varables W s defned by W ¼ðq; qu; EÞ; where q s the densty, u s the velocty, and E ¼ q þ 1 2 qu2 s the total energy, beng the nternal energy per unt volume. The flux s defned by 0 1 qu F qu u þ pid A: uðe þ pþ Lastly, the pressure s defned by the perfect gas equaton of state p ¼ðc 1Þ E 1 : 2 qu2 In the sequel, the rato of specfc heats c s assumed to be constant. In order to approxmate (1), we consder a trangular mesh where the elements are denoted by ft t g t¼1;nt, the vertces are denoted by fm s g s¼1;ns. Strctly speakng, the vertces of T have to be ndexed n the lst fm s g s¼1;ns, namely M 1, M 2, M 3. When there s no ambguty, we denote them by 1 ; 2 ; 3 or more smply 1; 2; 3. The vector ~n s the scaled nward vector normal to the boundary of T, opposte to the vertex,.e. ~n ¼ 2T rk ; where K s the barycentrc coordnate at M. The followng teratve RD scheme approxmates (1): C W nþ1 Dt W n þ X T ;M 2T U T ¼ 0; ð2þ and we consder the lmt, f t exsts, of W n when n!þ1. In (2), W n s an approxmaton of W, soluton of (1) at ðm ; t n Þ, C s the area of the dual control volume assocated to M, Dt s the (pseudo-) tme step, and U T stands for the resdual sent by the element T to the vertex M. The resduals satsfy the followng conservaton relatons: X Z ¼ dv F h dx :¼ U T : ð3þ U T ¼1;3 T In (3), F h s a contnuous nterpolaton of the flux F that converges n L 1 loc to F. In [5], we show that under the classcal assumptons of the Lax Wendroff theorem, the lmt soluton of (2) are weak solutons of (1). An example s gven by fnte volume type schemes on trangular meshes, another one by the SUPG scheme, see [2] for detals. Besdes the conservaton relaton (3), several other requrements are needed: the scheme has to be stable and accurate. The stablty s generally met by usng a monotoncty preservng scheme. We brefly recall ths concept for a scalar problem, we come back to t later n the system case where thngs are much less clear.

4 2.1. Monotoncty preservng schemes R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) In practce, all the known RD schemes can be wrtten as U T ¼ X M 2T ;M 6¼M c T ðu u Þ: ð4þ For ths scheme to be L 1 stable, t s enough that c T P 0 for all ; : ð5þ The stablty s obtaned thanks to a CFL-lke condton [11]. Ths s the so-called monotoncty preservng condton Accuracy: the lnear preservng (LP) condton We brefly recall the analyss of [2]. It s shown that a converged RD scheme produces a formally secondorder accurate soluton of the steady problem (1) under the followng three requrements: 1. The mesh s regular. 2. The approxmaton F h s second-order accurate on smooth solutons. 3. For any smooth soluton of (1), U T ðw Þ¼Oðh3 Þ for any vertex M and any trangle T such that M 2 T. In most cases, the thrd condton s met by mposng that there exsts a famly of unformly bounded coeffcents (or matrces for system problems) b T such that U T ¼ b T UT : Ths s the LP condton ntroduced n [11] whch s satsfed by the SUPG scheme and the PSI scheme of Strus [11] that we recall later. It s known that t s not possble to have a lnear scheme that s both monotoncty preservng and lnearty preservng: ths s GodunovÕs theorem [11]. The schemes that satsfy both requrements must be nonlnear. The constructon of such schemes s the topc of the next secton. 3. Dscretzaton of scalar equatons Several constructons of monotoncty preservng scheme exst, so we start by ndcatng some motvatons for revstng the problem. Then we provde a general constructon of LP schemes startng from a monotone frst-order scheme, and then provde numercal examples Problem statement and relatons to prevous constructons Defnng hx; y as the the dot-product of the vectors x and y, we consder the problem h ~ k; ru ¼0; x 2 X; t > 0; u ¼ g on C ; ð6þ where C s the nflow boundary of C ¼ ox. If the unknown u s pecewse lnearly nterpolated, the total resdual U T s gven by U T ¼ X3 ¼1 k u ;

5 478 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) where k ¼ 1 2 h~ k;! n : We notce that P 3 ¼1 k ¼ 0. Here, and untl the end of the paper, we have dentfed the vertces of T wth the ndces ¼ 1; 2; 3 because there s no ambguty. Smlarly, we drop the superscrpt T n U T U : We provde three examples of monotoncty preservng schemes. The frst one s the Rusanov scheme, U ¼ 1 3 U a X! ðu u Þ ð7þ 6¼ wth a P max k, so that c ¼ 1 ða k 3 Þ P 0. We have clearly P U ¼ U. Another example s gven by the N (narrow) scheme [11]. It can be wrtten as U ¼ k þ ðu euþ; ð8þ where eu s obtaned by recoverng the conservaton,.e.! 1! X eu ¼ k u : X k The scalar n :¼ ð P k Þ 1 s always defned unless ~ k ¼ 0. 2 Ths scheme can be consdered as a conservatve method of characterstcs. It s monotone under a CFL-lke condton because U ¼ X k þ nk ðu u Þ; hence c ¼ k þ nk P 0. A last example s provded by the classcal upwnd scheme. None of these scheme s lnear preservng. In order to get a monotoncty preservng LP scheme, several constructons exst, but all of them use the scalar N scheme as a base scheme. One may quote the PSI scheme of Strus [9], SdlkoverÕs constructon of the same scheme [22]. Another method s the hybrdzaton technque of [2,24] whch conssts n blendng a frst-order monotone scheme (resdual U ð1þ ) and a second-order LP scheme (resdual U ð2þ ), U ¼ U ð1þ þð1 ÞU ð2þ : One may also quote the B scheme by Deconnck et al. [24]. It s not monotoncty preservng, even though t s not easy to produce an oscllatory counter example. In both cases, the frst-order scheme s the N scheme, and the second-order scheme s the LDA scheme. In [2], we show that a specal choce of leads to the PSI scheme, but other choces are possble. The soluton provded by the blendng technque seems nterestng because t may be thought at frst glance that any monotone frst-order scheme fu ð1þ g and any LP scheme fu ð2þ g mght be blended together, leadng to a rcher class of schemes. Ths s not true because the two schemes must have some compatblty relatons otherwse the blended scheme mght ndeed reduce to the frst-order one,.e. 1. An example where 1 much too often s the blendng of the N scheme and the Lax Wendroff scheme U ¼ U Dt 3 2 k U: The problem here s that the N scheme s upwnd, whle the Lax Wendroff s not, so that we mght have U ¼ 0 for the N scheme and U 6¼ 0 for the Lax Wendroff one. Snce s defned by a relaton of the type 2 Snce for any, U! 0 unformly f ~ k! 0, there s no defnton problem n the case ~ k ¼ 0.

6 ¼ max uðr Þ; ¼1;3 where r s the rato of the second-order resdual versus the frst-order one and u a real valued functon the graph of whch satsfes some geometrcal constrants smlar to what happens n the TVD framework (see [2] for detals), we have n general ¼ 1. In some cases, the LP condton s not so clear, for example, the blendng of the Rusanov and the Lax Wendroff scheme s monotoncty preservng by constructon but ts LP property s not clear. Ths s why we have tred to develop another constructon, partally nspred from Sdlkover [22], but n our opnon more powerful snce non-trangular elements can be consdered. The constructon bascally reles on the exstence of a frst-order monotone scheme, and provdes a systematc way of defnng an LP monotone scheme Constructon We start from a frst-order monotoncty preservng scheme U ¼ X c ðu u Þ; c P 0: To smplfy the notatons, we present the technque on trangular elements, but the extenson to more general elements s obvous though tedous. As Sdlkover [22], we want to construct a resdual U such that: 1. The scheme defned by U s monotoncty preservng. 2. The scheme s conservatve,.e. P U ¼ P U ¼ U. 3. The scheme s LP, more precsely, we want b :¼ U =U to be bounded. The frst condton can be wrtten 1 l ¼ U P 0; U.e. l 6 1. The second condton s wrtten X l U ¼ 0: R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) Before gong further, let us examne the L 1 stablty condton. The stablty condton of the frst-order scheme s (we put back temporarly the superscrpt T ) P 2T Dt max ct 6 1; T 3 T.e. Dt 6 Dt ð1þ :¼ mn max T 3 P 2T ct T 1 : ð9þ Thus, the natural stablty condton of the new scheme s Dt 6 Dt ð2þ :¼ mn max T 3 ð1 l Þ P 2T ct T 1 : ð10þ The monotoncty constrant s l 6 1 only: l s allowed to be negatve. It s clear that f l s too negatve, we mght have Dt ð2þ Dt ð1þ. If we want that the maxmum tme step of the frst-order scheme s of the order

7 480 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) of that of the new scheme, t s better that 1 l s not too large, that s l s not too negatve. For ths reason, we ask 0 6 l 6 1: ð11þ Usng the prevous notatons, the LP condton s b ¼ð1 l Þ U U ðboundedþ: For that reason, we demand l ¼ 1 as often as possble. To summarze, the problem s: fnd fl g ¼1;N 2½0; 1Š N such that P N ¼1 l U ¼ 0; fl g ¼1;N 2½0; 1Š N ; l ¼ 1 ðas often as possbleþ: Assume now N ¼ 3 for smplcty. We are lookng for solutons where l ¼ 1 (.e. U possble, and for whch the new scheme depends contnuously on the parameters. We may assume U 1 6¼ 0. The equaton n (12) becomes l 1 ¼ l 2 U 2 þ l U 3 U 3 : 1 U 1 ð12þ ¼ 0) as often as The solutons we are seekng are those for whch l 1 ¼ 1 as often as possble, wthout volatng the monotoncty condton. (1) If ð U 2 =U 1 Þð U 3 =U 1 Þ > 0. The lne defned by 1 ¼ l 2 U 2 þ l U 3 U 3 1 U 1 has ether an empty ntersecton wth ½0; 1Š 2, or has two ntersecton ponts. For symmetry reasons, we assume l 1 ¼ l 2 ¼ l, hence l ¼ U 1 U 2 þ U 3 : (a) If l < 0, then we take l 2 ¼ l 3 ¼ l 1 ¼ 0 and then U ¼ U, ¼ 1; 3. (b) If 0 < l < 1, then U 1 ¼ 0; U 2 ¼ U 2 U; U 3 U 2 þ U ¼ U 3 U: 3 U 2 þ U 3 (c) If l > 1, then U 1 ¼ U; U 2 ¼ 0; U 3 ¼ 0: (2) If ð U 2 =U 1 Þð U 3 =U 1 Þ < 0, the lne 1 ¼ l 2 U 2 þ l U 3 U 3 1 U 1

8 cuts the boundary 0 6 l 1 6 1or06 l at most one pont. The ntersecton ponts are l 2 ¼ 1; l 3 ¼ U 2 þ U 1 and l U 3 ¼ 1; l 2 ¼ U 3 þ U 1 : 3 U 2 (a) If l 2 ¼ 1; l 3 ¼ ðu 2 þ U 1 Þ=U 3 < 0, we set U 1 ¼ U 1 þ U 2 ; U 2 ¼ 0; U 3 ¼ U 3: (b) If l 2 ¼ 1; l 3 ¼ ðu 2 þ U 1 Þ=U 3 2½0; 1Š, we set U 1 ¼ 0; U 2 ¼ 0; U 3 ¼ U: (c) If l 3 ¼ 1; l 2 ¼ ðu 3 þ U 1 Þ=U 2 2½0; 1Š, we set U 1 ¼ 0; U 2 ¼ U; U 3 ¼ 0: (d) If l 3 ¼ 1; l 2 ¼ ðu 3 þ U 1 Þ=U 1 < 0, we set U 1 ¼ U 1 þ U 3 ; U 2 ¼ U 2; U 3 ¼ 0: In each case, one can easly check that the b 2½0; 1Š and depends contnuously on the data. In ths constructon, the node ¼ 1 plays a specal role, so the scheme s numberng dependent. Ths s why a better soluton s to average the three solutons obtaned by makng the vertces ¼ 1;...; 3 play a pvot r^ole. Ths scheme s now contnuous and ndependent of the numberng of the mesh ponts. A smpler soluton s to make the constructon on the pvot ndex 0 the ndex for whch U s maxmum. Ths s the soluton we use n all the numercal llustratons. The prevous soluton gves results that are ndstngushable from those obtaned by ths one (see Fg. 1) Numercal llustratons R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) We apply the above constructon to 1 ou 2 2 ox þ ou oy ¼ 0; ðx; yþ 2½0; 1Š2 ; uðx; 0Þ ¼1:5 x; uð0; yþ ¼1:5; uð1; yþ ¼0:5: ð13þ Fg. 1. Some possble confguratons n the soluton of (12) for N ¼ 3.

9 482 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) The PDE (13) s nonlnear. The lnk between (13) to a lnear PDE s obtaned va a conservatve lnearsaton [9],.e. we determne for each trangle u such that Z 1 ou 2 þ ou Z Z ou dxdy ¼ u T 2 ox oy T ox dxdy þ ou T oy dxdy: ð14þ P In the left-hand sde of (14), u s pecewse lnearly nterpolated. The obvous soluton s u ¼ 1 3 ¼1;3 u. The total resdual Z U ¼ u T ou ox dxdy þ Z T ou oy dxdy Fg. 2. Mesh for the problem (14). s dstrbuted by mean of any of the three schemes descrbed above, the Rusanov scheme, the N scheme and the frst-order upwnd scheme. The upwnd scheme uses the classcal one-dmensonal Murman Roe scheme adapted to (13), rewrtten n the dstrbuton framework as n [2]. For comparson purposes, we dsplay the mesh on Fg. 2 and the soluton obtaned by a standard second-order ENO scheme on unstructured meshes. The scheme constructed from the Rusanov scheme (resp. the N scheme, the one-dmensonal upwnd scheme) s denoted by L-Rusanov (resp. PSI and L-upwnd). The solutons are plotted on Fgs. 3 and 4. We plot cross-sectons n the dscontnuty n Fg. 5. On Fg. 6, we have dsplayed cross-secton plots n the fan. We see that the qualty of the results s always better for the second-order dstrbuton schemes than for the second-order ENO scheme. Ths s a consequence of the LP property. Among them, there s a herarchy: the PSI scheme s the best, the results for the upwnd scheme are almost dentcal, followed by those for L-Rusanov. For the latter scheme, the results seem a bt wggly n the fan. Ths s not a stablty problem, snce the results are converged. We beleve that snce the Rusanov scheme s very dsspatve, the lmtaton mechansm that we propose s probably too over-compressve. The way to remedy to ths drawback s not known. 4. Gong to lnear hyperbolc systems The man dffculty to step from scalar to symmetrzable systems of PDEs of the type A ou ox þ B ou oy ¼ 0 ð15þ

10 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) (a) (b) (c) Fg. 3. (a) N scheme, (b) upwnd scheme, (c) Rusanov scheme. Fg. 4. (a) PSI, (b) L-Rusanov, (c) L-upwnd, (d) second-order ENO scheme.

11 484 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) N scheme FV scheme Rusanov scheme ENO scheme Lmted FV scheme Lmted Rusanov scheme PSI scheme Frstorder Second order Fg. 5. Cross-sectons n the shock for the problem (14) N scheme FV scheme Rusanov scheme 1 N scheme FV scheme Rusanov scheme Eno2 scheme Frst order second order Fg. 6. Cross-secton n the fan for the problem (14). supplemented by boundary condtons, s that, n general, the two Jacoban matrces A and B do not commute. The consequence s that there exsts no bass of common egenvectors to the matrces A and B. However, the analyss of the Cauchy problem ou ot þ A ou ox þ B ou ¼ 0 ðsupplemented by boundary and ntal condtonsþ ð16þ oy shows that the soluton s pecewse smooth, wthout hgh frequency oscllatons at the dscontnutes when the ntal and boundary condtons are pecewse smooth and the Jacoban matrces are symmetrc.

12 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) Unfortunately, systems (15), (16) are well posed only n L 2 or n Hlbert spaces whch use n the analyss of a numercal scheme seems very complex [7,8]. It seems legtmate, and all the numercal experments support ths, to seek for a stablty crteron that has a L 1 flavor n order to control the oscllatons of an approxmatng scheme for (15). In the followng, we frst recall some partcular dstrbuton schemes. We show some of ther propertes, n partcular when appled to smple waves. These propertes ustfy, n our opnon, the heurstc arguments we use to construct stable LP schemes. In order to smplfy the text, we assume n the followng that A and B are symmetrc. The dscusson can easly be generalzed to symmetrzable systems by changng the canoncal dot-product to the one assocated to the symmetrzaton varables Some RD schemes for (15) If one nterpolates U lnearly n each trangle, the total resdual U assocated to (15) s Z U ¼ A ou þ B ou dxdy ¼ X K U ; ð17þ ox oy ¼1;3 T where U denotes the conservatve varables at the vertces of T and the matrces K are K ¼ n x A þ n y B; where n x and n y represent the components of ~n. We denote by K n the matrx K n ¼ n x A þ n y B where ~n ¼ðn x ; n y Þ. Thanks to these notatons, the Rusanov scheme s U ¼ 1 3 U a X! ðu U Þ ð18þ wth a P max kk k. Another example s provded by the upwnd fnte volume scheme formulated as a RD scheme. The system N scheme [25] can be wrtten as U ¼ K þ ðu eu Þ; ð19þ where eu s computed to recover the conservaton property! X K eu ¼ X K U : The matrx P K s nvertble whenever A and B have no common egenvectors [2]. When there exst common egenvectors, P K may be no longer nvertble but the matrces NK have a meanng and n each case, the scheme (19) s well defned [2]. These three examples satsfy the conservaton relaton P U ¼ K U. Lastly, we notce that the scheme (2) can be rewrtten as U nþ1 ¼ X T U C e nþ1 ð20þ wth T ;M 2T

13 486 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) M M Fg. 7. Defnton of M 0 n (A.4). eu nþ1 ¼ U n Dt T UT : Ths enables to localze the analyss on each of the trangles of the mesh. ð21þ 4.2. Stablty analyss by smple waves We call smple wave a soluton of the type UðxÞ ¼C þ bh~n; xr; where C s a constant vector, x s any pont, r s a normalzed egenvector of the matrx K ~n and b 2 R s arbtrary. The functon U s lnear on T, ts nodal values are stll denoted by U. Lastly, we stll denote u ¼hU ; r,.e. U ¼ u r þ C. We notce that A ou ox þ B ou oy ¼ K ~nr ¼ kðrþr; where kðrþ s the egenvalue assocated wth r, hence U s proportonal to r. Our am s to ustfy the expermental fact that the Rusanov, N and fnte volume schemes are monotoncty preservng,.e., there s no creaton of numercal oscllaton Wave decomposton The frst remark s that, locally n a trangle, U n s the sum of smple waves. In fact, n T, UðxÞ ¼ X U K ðxþ wth K ðxþ ¼ h~n ; x þ C : 2T

14 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) Callng fr n g an orthogonal bass of egenvectors of K ~n, we may wrte UðxÞ ¼ 1 X 2T ¼1;3 X hu ; r n h~n ; xr n þ C: n ð22þ Ths shows that a pecewse lnear U s a sum of smple waves. The decomposton s not unque. Dependng on the scheme, we mght need adapted wave decomposton to prove our clam, but the central dea s contaned n (22) Results of the stablty analyss Wthn a trangle T, the pecewse lnear nterpolaton of U can be decomposed as a sum of smple waves UðxÞ ¼ X u r ðxþr r ; ð23þ r: wave where u r s of the form u r ðxþ ¼a r h~n r ; xþc r wth a r 2 R, ~n r s a untary vector and r r s an egenvector of K ~nr. The three schemes consdered here are lnear, so the resduals sent to node M s the sum of the resdual sent to ths node by U r ðxþ ¼u r ðxþr r, namely U ¼ X UðU r ðxþþ r:wave wth some abuse of language. The analyss carred out n Appendx A for the Rusanov and the N schemes can be extended wthout dffculty to the fnte volume scheme. We show that for any smple wave U r, the updated quanttes eu defned by eu ¼ U r ðm Þ Dt 3T U ðu r Þ satsfy k eu k 6 max M 2T ku rðm Þk ð24þ under a CFL-type condton. Gatherng the propertes and relatons (20), (21), (23) and (24), we say that the scheme s stable and monotone. We are not able to exhbt any norm for whch a relaton lke ku n k 6 CðU 0 Þ would be true n general under a CFL-lke condton. From [7,8], the classcal L p norms are not sutable, maybe the less standard L p;a norms of [8] for whch the Cauchy problem s well posed. But we have not found a way to use them n a practcal way. From an expermental pont of vew, the condtons (20), (21), (23) and (24) seem suffcent; ths s why we call ths stablty.

15 488 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) Constructon of LP schemes for systems We present now a method whch, startng from a stable monotone scheme, enables the constructon of monotone second-order schemes at steady state. These LP schemes have the resdual U ¼ B U; where B s a matrx. They are oscllaton free. We show that they also satsfy the stablty requrement descrbed n Secton Constructon The dea of the constructon s the followng. Startng from a monotone scheme, t s possble to decompose the soluton as a sum of smple waves ndexed by r. The resdual can then be spltted nto a sum of resduals, U r, each of them actng on a sngle smple wave. These sub-resduals can be wrtten as a postve weghted sum of dfference, U r ¼ X! c r u r u r r r : ð25þ Here, the states U are descrbed by mean of a sum of smple waves U ¼ X u r r r: r: wave Ths s a geometrcal property of the scheme: t states that smple waves evolve n a non-oscllatory manner. Ths non-oscllatory behavor of the scheme should be ndependent of the way we choose to descrbe t. Thus, the dea s to choose an orthonormal bass ft l g l, to notce that X hu ; t l ¼hU; t l and to nterpret the coeffcents hu ; t l as scalar resduals to whch we apply the lmtaton technque of Secton 3. We construct resduals u l; such that P u l; ¼hU; t l; u l; ¼ bl hu; t ð26þ l; wth b l 2½0; 1Š. The resdual U ¼ X l u l; t l ð27þ can be rewrtten as U ¼ B U; ð28þ where the matrx B s unformly bounded by constructon. In the next secton, we show the scheme preserves the monotoncty Analyss Consder an orthonormal bass ðt l Þ l. We construct the lmted scheme by U ¼ B U;

16 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) that s U ¼ X r B Uðu r ðxþr r Þ: The matrx B s constructed by hu ; t l¼b l hu; t l wth b l 2½0; 1Š. We set k ¼ Dt=3T : We have ku n ku k¼ X l hu n ; t l kb l hu; t l 2 : ð29þ In Appendx B, we show that f b l 2½0; 1Š, relaton (29) mples, for smple waves, the followng nequalty: ku n ku k 6 max M 2T ku rðm Þk: ð30þ 4.4. Example of the Cauchy Remann equatons We consder the Remann problem ou ot þ A ou ox þ B ou ¼ 0; t > 0; ð31þ oy supplemented by Remann data per quadrant. The matrces A and B are A ¼ and B ¼ 0 1 : 1 0 We test the scheme of (26) (28) on the followng Remann data U ¼ðu; vþ: f x > 0 and y > 0; 1 f x > 0 and y > 0; >< >< 1 f x < 0 and y > 0; 1 f x < 0 and y > 0; u ¼ and v ¼ 1 f x > 0 and y < 0; 1 f x > 0 and y < 0; >: >: 1 f x < 0 and y < 0; 2 f x < 0 and y < 0: ð32þ The soluton s self smlar, Uðx; y; tþ ¼ eu ðx=t; y=tþ. The functon eu satsfes n ou ou m on om þ A ou on þ B ou om ¼ 0 ð33þ wth the boundary condtons at nfnty gven by the Remann data (31), (32) at tme t ¼ 1. The problem s solved by a tme marchng technque. The computatonal doman s ½ 2; 2Š½ 2; 2Š. The soluton of (33) corresponds to the soluton of the Remann problem (31), (32) at tme t ¼ 1. The PDE (33) s ellptc n n 2 þ m and hyperbolc n n 2 þ m 2 P 1: the boundary condtons can be easly computed by solvng one dmensonal Remann problems.

17 490 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) Fg. 8. Solutons of the Remann problem for the Cauchy Remann equatons. N scheme, rght: u, left: v. Fg. 9. Solutons of the Remann problem for the Cauchy Remann equatons. Lmted N scheme, h ¼ 0, rght: u, left: v. In the method, the choce of the orthogonal bass s free. In R 2, they can be ndexed by the angle h. We have chosen the egenvectors of cos ha þ sn hb. The results are presented for the lmted N scheme that reduces to the PSI one for scalar problems. Of course, they wll depend on the choce of h: two dfferent angles gve two dfferent schemes. What we want to check numercally s that, frst, the non-oscllatory behavor of the results s ndependent of h, and second, that ther global qualty s the same whatever h. From Fgs. 8 11, we see that the lmted solutons are much more accurate than the frst-order scheme. The qualty of the soluton does not depend on the angle, as conectured, even f the angle s soluton dependent as on Fg Case of the Euler equatons Frst, as n [9], the nonlnear problem s replaced by a lnearzed one n each trangle. It s well known that the state vector and the Euler fluxes are quadratc n

18 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) Fg. 10. Solutons of the Remann problem for the Cauchy Remann equatons. Lmted N scheme, h ¼ 45, rght: u, left: v. Fg. 11. Solutons of the Remann problem for the Cauchy Remann equatons. Lmted N scheme h ¼ arctanðu=vþ, rght: u, left: v. p Z ¼ ffffff q ð1; u; HÞ T ; where H s the enthalpy of the flud. For example, we have W ¼ 1 DðZÞZ where DðZÞ s lnear n Z. For ths 2 to be true, one needs that the rato of specfc heats c be constant. Ths s what we assume. Thanks to the lnearsaton, the problem reduces to ow ot þ A oz ox þ B oz oy ¼ 0; where the Jacobans A and B are functons of the average of Z on T, see [9]. Once ths s done, we consder the N scheme and the lmted scheme as descrbed n Secton 4.3. For example the N scheme s C W nþ1 W n þ P Dt T ;M 2T UT ¼ 0; U T ¼ P M 2T Kþ NK ew n ew n : Here, ew ¼ 1 2 Dð ZÞZ, andk s evaluated at the state W ¼ 1 2 DðZÞ Z.

19 492 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) In the smulatons, we consder two types of boundary condtons: wall and nflow/outflow condtons. They are approxmated as n [2]. More precsely, the nflow/outflow condtons are obtaned usng Steger Warmng flux modfed as n [12], and wrtten n fluctuaton form. The wall condton s mposed weakly. Here, as n [12], we have chosen to mpose the flux pn F ¼ x B pn y A 0 wrtten n fluctuaton form. Other solutons are possble, such as the ones descrbed n PallereÕs thess [21]. The lmted N scheme has been tested aganst numerous test cases n subsonc, transonc and supersonc stuatons. We only present the most sgnfcant examples. The NACA 0012 s very classcal and well documented. The sphere problem s fully subsonc, so one can check the amount of numercal dsspaton. The bow shock problem enables to check the robustness of the scheme. Lastly the scramet problem enables to check the behavor of the schemes on complex waves, and ther nteractons. In each problem, the ntal condton s a unform flow gven by the condtons at nfnty. The scheme s mplct, the mplct phase s provded by an approxmate lnearsaton of the frst-order Roe solver. Our am s not to have a maxmum effcency, but to show the accuracy of the new scheme, as well as the robustness of the method Flow around a NACA 0012, M 1 ¼ 0:85, h ¼ 1 The mesh s plotted n Fg. 12. We plot the velocty for the N scheme and the lmted N scheme on Fg. 13. Ths shows the mprovement of the slp lne at the tralng edge of the arfol. We also present the pressure solnes n Fg. 14. The pressure coeffcent along the arfol s provded n Fg. 15. Fg. 16 presents the entropy devaton; there s a clear mprovement. In Fg. 15, we compare the pressure coeffcent obtaned by the scheme of [2] and the present one. We see that the new scheme provdes oscllaton free solutons, whereas the blendng of the N and LDA scheme 3 of [2] was provdng slght oscllatons. The comparson of the entropy devaton between the new scheme and the blended scheme of [2] also shows a clear mprovement Hypersonc flow, M 1 ¼ 8 It was not possble to run ths test case wth the scheme of [2], because very quckly negatve pressure problems were occurrng. A zoom of the mesh s gven n Fg. 17. The Mach number for the N scheme and the lmted N scheme are gven n Fg. 18. We also gve cross-sectons of the densty (Fg. 20), of the Mach number (Fg. 19) and the entropy devaton (Fg. 21) along the symmetry axs. Our results are clearly oscllaton free. The boundary condtons are better taken nto account wth the lmted N scheme. 3 The LDA scheme s defned by U ¼ NK þ U.

20 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) Fg. 12. Zoom of the mesh for the NACA 0012 problem. Fg. 13. Velocty solnes Subsonc cylnder, M 1 ¼ 0:35 We have run ths subsonc test case wth the N scheme, the lmted N scheme, the LDA scheme and the blended N/LDA scheme of [6]. We dsplay the Mach number n Fg. 22 and the entropy devaton n Fg. 23. In both case, we have plotted the same solnes. The mesh s smlar as n Fg. 17. As expected the best results are obtaned wth the LDA scheme. Ths s partcularly clear from the entropy devaton and Mach number solnes. The Mach solnes are symmetrc wth respect to the axs orthogonal to the velocty at nfnty, as t should be. As expected, the worst results are obtaned for the N scheme. The results for the lmted N and blended scheme are of smlar qualty. The entropy devaton s better for the blended LDA/N scheme, but the symmetry of the Mach number solnes s better respected wth the lmted N scheme.

21 494 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) Fg. 14. Pressure solnes Blended LDA/N scheme N scheme Lmted N scheme Fg. 15. Plot of cp along the arfol Scramet, M 1 ¼ 3:5 We have run the N scheme, the LDA scheme, the blended scheme of [6], Deconnck et al. B scheme [10] and the lmted N scheme on a scramet-lke case. Our mplementaton of the B scheme s the followng. We frst compute the N and LDA resduals denoted, respectvely, U N and U LDA for ¼ 1;...; 3. Then we consder the rght and left egenvectors assocated to the flow drecton (the result s rather ndependent of

22 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) Blended LDA/N scheme N scheme Lmted N scheme Fg. 16. Entropy devatons along the arfol. Fg. 17. Mesh for the blunt body problem. the choce, and qualtatvely ndependent of ths choce). We denote them by r l and l, l ¼ 1; 4. Then we compute h l ; U N and h l ; U LDA. Next we ntroduce the blendng parameters l l, l ¼ 1;...; 4, l l ¼ P 3 ¼1 h l; U N P 3 ¼1 h l; U N þ ; ¼ : Then the B scheme can be wrtten as U B ¼ X4 l¼1 h l ; U LDA þ l l h l ; U N U LDA r l :

23 496 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) Fg. 18. Iso Mach lnes, lmted N scheme N scheme Lmted N scheme Fg. 19. Cross-secton of the Mach number. The geometry has been found on the Mach number at nfnty s set to 3.5. A zoom of the mesh s gven n Fg. 24. Ths case s nterestng because t provdes a good example of the dfference between the schemes. The Mach number s dsplayed n Fg. 25. We have used the same solnes. The most dsspatve scheme s once more the N scheme. The least dsspatve s the LDA scheme, but t s very oscllatory. The three other schemes are monotone. The blended scheme of [6] s the most dsspatve among the three (ths s clear from the structure of the reflected shocks). The B scheme has a better behavor

24 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) N scheme Lmted N scheme Fg. 20. Cross-secton of the densty N scheme Lmted N scheme Fg. 21. Cross-secton of the entropy devaton. wth respect to ths crteron, but the best s the lmted N scheme. Ths s confrmed by the Mach number dstrbuton on the lne y ¼ 0:3 n the throat, see Fg. 26. Ths shows that the choce of the blendng parameters s mportant (the B scheme has a rcher structure than the scheme of [6]), but the constructon presented n ths paper seems the most effcent.

25 498 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) Fg. 22. Mach number solnes: (a) N scheme (mn ¼ 0, max ¼ 0.756), (b) LDA scheme (mn ¼ 0.001, max ¼ 0.84), (c) lmted N scheme (mn ¼ 0.002, max ¼ 0.83), (d) blended LDA/N scheme (mn ¼ 0.002, max ¼ 0.82). 6. Conclusons We have presented and analyzed a stable and monotone method for the computaton of compressble flows. The schemes we develop are formally second-order accurate on regular unstructured meshes. The capabltes of the schemes are presented on several subsonc, transonc and supersonc flows. The results are good. Compared wth any scheme constructed by blendng two schemes, as n [2] or [10], the computatonal complexty s reduced by a half, snce only one frst-order scheme has to be evaluated. We have used very crude boundary condtons n ths paper, n partcular the wall boundary condtons could be mproved along the lnes of PallereÕs thess, where t becomes easy to ncorporate the lmtaton technque developed n the paper. Ths s done n [6]. The man problem s n the nonlnear convergence hstory. The scalar results are converged: the L 2 resdual of our results s below Gong to systems, the stuaton s much less clear. We have been able to

26 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) Fg. 23. Entropy devaton solnes: (a) N scheme (mn ¼ 0, max ¼ 0.007), (b) LDA scheme (mn ¼ )0.0002, max ¼ ), (c) lmted N scheme (mn ¼ 0, max ¼ ), (d) blended LDA/N scheme (mn ¼ 0, max ¼ ). drop the L 2 and L 1 resdual for the Cauchy Remann problem below 10 7, but the convergence hstory depends very much on the mesh, and on the angle h n Secton 4.4. In partcular the convergence hstory s much smoother when h s the same throughout the mesh. In some other tests, we were not able to drop the resdual below In the Euler case, except for very coarse meshes, we were never able to drop the resdual below However, n each case, most of the mesh ponts are converged, only very few of them have an erratc convergence hstory. The reasons of ths behavor are not understood and wll be nvestgated elsewhere. Lastly, an extenson of these methods to unsteady flow felds s presented n [6].

27 500 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) Fg. 24. Zoom of the mesh for the scramet case. Fg. 25. Mach number solnes: (a) N scheme (mn ¼ 1.97, max ¼ 3.6), (b) LDA scheme (mn ¼ 1.44, max ¼ 6.47), (c) lmted N scheme (mn ¼ 1.85, max ¼ 3.6), (d) blended LDA/N scheme (mn ¼ 1.87, max ¼ 3.6), (e) B scheme (mn ¼ 1.88, max ¼ 3.6).

28 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) B scheme Blended LDA/N scheme Lmted N scheme Fg. 26. Mach number dstrbuton along y ¼ 0:3 n the throat. Acknowledgements Ths work has been performed under a CEA grant through the Laboratore correspondant du Cea LRC-03. The frst author also acknowledge the hosptalty of RIACS, NASA Ames Research Center, where part of ths work has been performed. Appendx A. Stablty analyss for frst-order schemes A.1. Case of the Rusanov scheme In ths case, we have U ¼ 1 3 U a X! ðu U Þ wth U proportonal to r. Hence, U s proportonal to r. Ths shows that U ¼hU ; rr wth hu ; r ¼ X 1 3 hðk aidþr; rðu u Þ :¼ X c R ðu u Þ and, by defnton of a,

29 502 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) c R :¼ 1 3 hðk aidþr; r P 0: We consder then the teratve scheme eu nþ1 ¼ U n Dt C eu nþ1 ¼ u Dt C X T ;M 2T U : For a smple wave t reduces to! X X c R ðu u Þ r: ða:1þ T ;M 2T M 2T Ths shows that the smple wave evolves proportonally to r, and eu nþ1 6 max M2T U n under a CFLtype condton, and a sngle tme step. Then, thanks to (20), ku nþ1 k 6 max M neghbor of M ku n k: Ths does not show that f U n s globally a smple wave, so wll be U nþ1. But t shows that the profle of the soluton remans monotone. We see that n general, eu nþ1 s a lnear combnaton of terms lke (A.1) that are averaged accordng to (20). Ths s a strong ndcaton that there s no creaton of spurous oscllatons. A.2. Case of the system N scheme The system N scheme can be wrtten as X U ¼ K þ NK ðu U Þ ða:2þ wth N ¼ð P K Þ 1. We start by reducng the problem to the case where N ¼ Id usng a (local) change of varable. We wrte X K þ NK ðu U Þ¼K þ M X K ðu U Þ ¼ M 1=2 ðm 1=2 K þ M 1=2 Þ X ðm 1=2 K M1=2 ÞðM 1=2 U M 1=2 U Þ wth M ¼ N > 0. We make the change of varable e K ¼ M 1=2 K M 1=2, f K ¼ M 1=2 K M 1=2, and V ¼ M 1=2 U.Weget ek ¼ M 1=2 K M1=2 and f K ¼ M 1=2 K M1=2 ; because the matrces are symmetrc. Hence, the resdual becomes U ¼ M 1=2 X fk þ ek ðv V Þ; and the scheme (A.2) becomes

30 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) ! "! # V nþ1 ¼ V n Dt K T f X þ ek ðv V Þ ¼ V n Dt K T f X þ ek V þ V ða:3þ for whch N ¼ Id and the matrces K e ; K f are symmetrc. Ths s why, n the followng, we assume that N ¼ Id. From now on, we wrte wth the form (A.3) of the scheme, even though, strctly speakng, t s not any longer the N scheme. As we have seen before, t s mportant to wrte the vector X ek V þ V as a sum of smple waves. The matrx f K plays the role of a combnaton of proectors. We consder V, the lnear nterpolaton of V on T. We see that VðxÞ ¼V 0 þ X3 ¼1 h~n ; x 2T V ; where ~n s the nward normal unt opposte to the node of T and V 0 s a constant vector. The second step s to rewrte ths equalty wth the rght (resp. left) egenvectors r l (resp. l )ofk (wth h k l ; r l ¼dk ), VðxÞ ¼V 0 þ X3 ¼1 X 4 l¼1 h~n ; x h l 2T ; V r l : The last step s to ntroduce a pont M 0 on the sde opposte to the vertex, see Fg. 7, and to rewrte VðxÞ as VðxÞ ¼V 0 0 þ X3 ¼1 X 4 l¼1 ƒ! h~n ; M 0 x h l 2T ; V r l :¼ V 0 þ X3 ¼1 X 4 l¼1 h l ; V u l ðxþr l ; ða:4þ.e., as a sum of smple waves. We notce, thanks to the defnton of the ponts Ml 0, l ¼ 1;...; 3, u l ðm Þr l ¼ 0; ul ðm kþr l ¼ 0; ul ðm Þr l 6¼ 0: Snce the N scheme s lnear, t s enough to evaluate the N scheme on each of the smple waves. Let us consder the wave that vanshes for ¼ 2 and 3. It s proportonal to r 1 l : V ¼ au 1 l ðm Þr 1 l : We may assume that a ¼ 1. We have V nþ1 ¼ u 1 l ðm Þr 1 l Dt T f K þ ðu 1 l ðm Þr 1 l þ e VÞ wth ev ¼ u 1 l ðm 1Þ K 1 r1 l ¼ k l u1 l ðm 1Þr 1 : Hence, u 1 l ðm Þr 1 l þ e V ¼ðu 1 l ðm Þþk l u1 l ðm 1ÞÞr l ; where k l s the egenvalue of K 1 assocated wth r 1 l, and then

31 504 R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) V nþ1 ¼ u 1 l ðm Þr 1 Dt l T u 1 l ðm Þ þ k l u1 l ðm 1Þ f K þ r 1 l : Let us ntroduce! p the left egenvector of K f and l p the correspondng egenvalue. We have! p ðvnþ1 Þ¼ u 1 l ðm Þ Dt ðl p Þ þ u 1 l T ðm Þ þ k l u1 l ðm 1Þ! p ðr1 l Þ; that s! p ðvnþ1 Þ¼ 1 Dt ðl p Þ þ u 1 l T ðm Þ Dt ðl p Þ þ k l T u1 l ðm 1Þ! p ðr1 l Þ: Snce k l 6 0and ð lp Þ þ P 0, ths shows the stablty of the scheme under a CFL-lke condton, 8 and p; Dt T ðlp Þ þ 6 1: ða:5þ Appendx B. Stablty analyss for the lmted scheme Usng the notatons of Secton 4.3.2, we show here that f b l 2½0; 1Š, we have for smple waves, ku n ku k 6 max M 2T ku rðm Þk: For a smple wave, we have ðh eu nþ1 ; t l Þ 2 ¼ðu r hr r ; t l b l khu ; t l Þ 2 ; whch s convex n b l,so ðu r hr r ; t l b l khu ; t l Þ 2 6 maxððu r hr r ; t l Þ 2 ; ðu r hr r ; t l khu ; t l Þ 2 Þ: Hence, we can rewrte X ðh eu nþ1 ; t l Þ 2 ¼kPU n k2 þkqðu n ku Þk 2 ; l where P (resp. Q) s the orthogonal proector onto the subspace generated by the vectors of ft l g for whch maxððu r hr r ; t l Þ 2 ; ðu r hr r ; t l khu ; t l Þ 2 Þ¼ðu r hr r ; t l Þ 2 (resp. maxððu r hr r ; t l Þ 2 ; ðu r hr r ; t l khu ; t l Þ 2 Þ¼ðu r hr r ; t l khu ; t l Þ 2 Þ: We consder the examples of the Rusanov scheme and the system N scheme for whch we have shown that (30) s true. B.1. Case of the Rusanov scheme For a smple wave, we have shown that eu nþ1 k eu nþ1 k 6 max ku M 2T rðm Þk s proportonal to r, and

32 so QðU n ku Þ s proportonal to QðrÞ and we have kpu n k2 þkqðu n ku Þk 2 6 u r ðm Þ 2 kprk 2 þ max ku M 2T rðm Þk 2 kqrk 2 6 max M 2T ku rðm Þk 2 ðkprk 2 þkqrk 2 Þ¼max M 2T ku rðm Þk 2 ; because P and Q are two orthogonal proectors. Ths ends the proof n the case of the Rusanov scheme. B.2. Case of the system N scheme For the sake of smplcty, we only consder the 2 2 case of the Cauchy Remann system. The proof s smlar n the general case usng the arguments of Appendx A.2. The Cauchy Remann system reads ou ot þ R. Abgrall, M. Mezne / Journal of Computatonal Physcs 195 (2004) ou ox þ ou oy ¼ 0: Consderng a drecton ~n, the egenvalues of K ¼ K ~n are k~n k, and the normalzed orthogonal egenvectors are denoted by r. An easy calculaton shows that kk k¼k~n kid, hence X 3 ¼1 K 2 ¼ P 3 ¼1 k~n Id :¼ aid: k Usng ths remark, we see that for a general smple wave U ¼ u r ðm Þr, we have eu ¼ N X3 ¼1 K U! ¼ a X3 ¼1 and then, wth c ¼ ak~n kk~n khr; r 2 P 0, k~n ku r ðm Þhr; r 2!r; eu nþ1 ¼ U n k X3 ¼1 c u r ðm Þ u r ðm Þ hr þ ; rr þ because (settng u r ðm Þ1), að P 3 ¼1 k~n khr; r 2 Þ¼1. Another way of statng ths result s ( h eu nþ1 ; r þ ¼ u r ðm Þ k X3 c u r ðm Þ u r ðm Þ!) hr; r þ :¼ A hr; r þ h eu nþ1 ; r ¼u r ðm Þhr; r : We have ¼1 kqð eu nþ1 Þk 2 ¼kQðAhr; r þ r þ Þk 2 þkqðu r ðm Þhr; r r Þk 2 6 max ku M 2T rðm Þk 2 fkqðhr; r þ r þ Þk 2 þkqð½hr; r Šr Þk 2 g ¼ max ku M rðm Þk 2 kqðrþk 2 2T

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