An entropy preserving MOOD scheme for the Euler equations

Transcription

1 An entropy preservng MOOD scheme for the Euler equatons Chrstophe Berthon Laboratore de Mathématques Jean Leray, CNRS UMR 6629, Unversté de Nantes, 2 rue de la Houssnère, BP 92208, Nantes, France. Chrstophe.Berthon@unv-nantes.fr Vven Desveaux Laboratore de Mathématques Jean Leray, CNRS UMR 6629, Unversté de Nantes, 2 rue de la Houssnère, BP 92208, Nantes, France. Vven.Desveaux@unv-nantes.fr Abstract The present work concerns the dervaton of entropy stablty propertes to be satsfed by hgh-order accurate fnte volume methods. Such a stablty turns out to be crucal when approxmatng the weak solutons of hyperbolc systems of conservaton laws. In fact, several recent works propose some knd of dscrete entropy nequaltes assocated to hgh-order schemes. However, these entropy preservng schemes do not seem relevant to mpose that the converged soluton n the sense of the Lax-Wendroff Theorem satsfes the requred entropy nequaltes. We llustrate such a falure by exhbtng numercal schemes that, from one hand, satsfy entropy stablty and, from the other hand, do not prevent numercal blowup. Here, we recall the expected hgh-order dscrete entropy nequaltes to be certan that the approxmate soluton converges to an entropy soluton. Equpped wth these suffcent numercal entropy stablty, we propose to extend the recently ntroduced hgh-order MOOD scheme to satsfy the requred hgh-order entropy nequaltes. In fact, the MOOD approach s based on an a posteror estmaton and t seems mpossble to mpose a posteror the whole set of dscrete entropy nequaltes. We solve ths dffculty by consderng a fnte volume scheme, whch nvolvesat least one dscrete entropy nequaltes wth a numercal transport property. From one selected numercal transport dscrete entropy nequalty, we establsh that all the needed dscrete entropy nequaltes are satsfed. Argung ths specfc numercal transport entropy, we derve the expected a posteror entropy condton to get an entropy preservng hgh-order MOOD scheme. Numercal experments llustrate the relevance of the suggested numercal procedure.

2 Key words : Euler equatons, numercal approxmaton, fnte volume methods, hgh-order approxmaton, dscrete entropy nequaltes. 1 Introducton The present work concerns the dervaton of entropy preservng hgh-order numercal schemes to approxmate the weak solutons of the Euler equatons gven by t ρ+ x ρu = 0, t ρu+ x ρu 2 +p = 0, 1 t E + x E +pu = 0, where the pressure s gven by a perfect gas law: p = γ 1 E ρ u2 2 for a gven adabatc coeffcent γ 1,3]. To shorten the notatons, let us ntroduce the conservatve unknown state vector w : R R + Ω and the flux functon f : Ω R 3 defned as follows: w = t ρ,ρu,e and fw = t ρu,ρu 2 +p,e +pu, 2 wth Ω the convex set of admssble states gven by: } Ω = {w R 3 ; ρ > 0,ew = E ρ u2 2 > 0. Here, the functon e : Ω R + denotes the nternal energy. Because system 1 s well-known to be hyperbolc, the solutons may contans shock dscontnutes for nstance, see [27, 39, 43, 20] and references theren. In order to rule out unphyscal dscontnuous solutons, the system under consderaton must be endowed wth entropy nequaltes see [37, 38, 43] for further detals: t ρflns+ x ρflnsu 0 wth s = p ργ, 3 where F : R R s a smooth functon such that, defnes a convex map. To shorten the notaton, we set w Sw = ρflns 4 Gw = ρflnsu. 5 After Tadmor [47] see also [30, 43], the functon F must satsfy F y < 0 and F y < γf y for all y R. 6 Internatonal Journal on Fnte Volumes 2

3 Next, we consder the numercal approxmaton of the weak solutons of 1. Numerous numercal strateges can be found n the lterature as soon as frst-order fnte volume methods are nvolved. For nstance, the reader s referred to[28, 39, 48, 29, 9] wherethe usual numercal technques are detaled. By denotng a tme step and = x +1/2 x 1/2 a constant cell sze, the approxmaton of wx,t n + s gven as follows: w n+1 = w n f w n,w n +1 f w 1,w n n, 7 where f : Ω Ω R 3 s a Lpschtz-contnuous numercal flux functon whch s consstent: f w,w = fw. Here, the tme step s restrcted accordng to a CFL condton: max Z λ± w n,w n , 8 where λ ± w n,wn +1 represent some wave speeds assocated to the consdered numercal flux functon f w n,wn +1. The numercal flux functon defnton can be supplemented by addtonal robustness and stablty propertes. Concernng the robustness, the method must preserve the postveness of both densty and nternal energy. Hence, as soon as the sequence w n Z belongs to Ω, the adopted scheme must satsfy w n+1 Ω. In ths work, the stablty of the scheme s understood at the entropy level. Dscrete entropy nequaltes are mposed n order to exclude, at the dscrete level, undesrable unphyscal solutons. These reached dscrete entropy nequaltes read as follows: 1 Sw n+1 Sw n + 1 G w n,w n +1 G w 1,w n n 0, 9 where Sw s defned by 4 and G : Ω Ω R denotes the entropy numercal flux functon whch must be consstent: G w,w = Gw. The most common frst-order fnte volume methods 7 are proven to satsfy robustness and/or stablty propertes. For nstance, we cte[31] for the HLL scheme, [26, 25, 14] for the extenson to smple approxmate Remann solvers, [9, 15, 6] for relaxaton schemes, [9, 49, 48, 2] for HLLC scheme, [42, 31, 23, 8, 13] for the Roe and the extenson VFRoe schemes. Of course the prevous lst s not exhaustve. Now, numerous strateges have been proposed to ncrease the order of accuracy. One of the most popular, and adopted n the present paper, s based on a sutable reconstructon of the state vector on each sde of the nterfaces located at x +1/2. Indeed, n 7, f w n,wn +1 s nothng but a frst-order evaluaton of the flux functon at the nterface x +1/2. The space second-order or hgh-order extenson s obtaned by nvolvng a second-order or hgh-order evaluaton of the flux now gven by f w +1/2,w+ +1/2, Internatonal Journal on Fnte Volumes 3

4 where w ± +1/2 denote reconstructed states. Technques to derve w± +1/2 are wdely studed n the lterature and t s here mpossble to refer all the papers devoted to such a topc. Let us just menton the MUSCL reconstructon [50, 39, 5, 10, 34, 40, 35, 16, 19], the knetc second-order approaches [40, 35], the ENO/WENO reconstructon [41, 55, 54], the PPM reconstructon [52], the MOOD reconstructon [17, 21], and plenty of extensons... In fact, these hgh-order fnte volume methods, whch now read as follows: w n+1 = w n f w +1/2,w+ +1/2 f w 1/2,w+ 1/2, 10 nvolve dffcultes to derve robustness and stablty propertes. The Ω-preservng property to be satsfy by 10 s now well studed. It s obtaned by ntroducng a sutable lmtaton procedure nsde the reconstructon technque. We refer to [39, 9] where basc MUSCL reconstructons are consdered, and to [5, 7] where robustness of more sophstcated approaches are studed. In [41], the requred robustness s establshed wthn the WENO reconstructon framework. Let us underlne that these procedures to enforce the needed Ω-preservng property nvolve a pror lmtaton technques. Put n other words, these lmtatons are global and, sometme, turn out to be too strong. As a consequence, such usual lmtatons may be too dffusve. To correct ths loss of accuracy, the MOOD method has been recently presented n [17, 21]. It suggests to ntroduce an a posteror lmtaton technque. Hence, the lmtaton s just local n space to reduce the numercal vscosty and to ncrease the accuracy of the method. The dffcultes turn out to be very dstnct as soon as stablty propertes must be proven for hgh-order schemes gven by 10. Several attempts are proposed n the lterature. One proposed strategy s based on the Generalzed Remann Problem [3, 12, 11]. Unfortunately, the solutons of the GRP assocated wth 1 are very dffcult to be exhbted, and ths makes poorly attractve the resultng scheme. In [19, 18], the authors suggest to adopt new projecton technques but the obtaned numercal methods are, n general, sophstcated and extensons to more complex problems seem delcate. In thesame sprt, we cte the work by Bourdaras et al. [10] but, as specfed by the authors, the derved scheme cannot be easly mplemented. More recently, n [5], dscrete entropy nequaltes are obtaned but for a specfc entropy tme dervatve dscrete operator see also [10]. Moreover, these stablty results are unluckly obtaned by enforcng strong lmtaton procedures and thus by enforcng a lot of numercal vscosty. In addton, the relevance of the unusual entropy tme dervatve dscrete operator, accordng to the well-known Lax-Wendroff Theorem, s not establshed. Put n other words, we are not able to prove up to our knowledge that the consdered dscrete entropy nequaltes converge, n a sense to be prescrbed, to the expected entropy nequaltes 3. In [5] see also [55, 54, 35], an addtonal stablty crteron s obtaned by enforcng an entropy maxmum prncple [47]. However, ths stablty condton s weaker than the usual dscrete entropy nequaltes and, as a consequence, such a maxmum prncple s not consdered n the present work. In order to derve robust and entropy preservng hgh-order schemes, we here adopt the a posteror MOOD technque [17, 21] see also [33] for a related method. Internatonal Journal on Fnte Volumes 4

5 In the next secton, we gve our man motvatons by brefly studyng the convergence behavor of the dscrete entropy nequaltes as stated n [5, 10]. These motvatons are completed by numercal experments performed wth standard MUSCL schemes over very fne meshes. It turns out that these numercal approaches are not stable at all as soon as the mesh sze s small enough. As a consequence, we suggest to modfy the usual MUSCL schemes, or equvalently the usual hgh-order reconstructons, by ntroducng an a posteror lmtaton accordng to only one dscrete entropy nequalty. Indeed, an a posteror entropy evaluaton cannot be performed by consderng the whole space of convex entropy functons and we have to deal wth one partcular dscrete entropy nequalty. Hence, n Secton 3, consderng one specfc dscrete entropy nequalty, we prove that all the reached dscrete entropy nequaltes can be restored. Equpped wth such a result, Secton 4 s devoted to the dervaton of the e-mood scheme by ntroducng nsde the adopted ntal hgh-order scheme here, MUSCL scheme to smplfy, an a posteror restrcton gven by the preservaton of the relevant dscrete entropy nequalty. Ths procedure s llustrated by several numercal experments n the last secton. 2 Man motvatons The objectve of the present paper s to derve hgh-order accurate entropy preservng schemes to approxmate the weak solutons of 1. One of the man problem arsng when dealng wth hgh-order schemes concerns the dervaton of sutable dscrete entropy nequaltes. Let us just recall that the dscrete entropy nequaltes are derved so that the converged soluton s entropy preservng. Put n other words, the consdered dscrete entropy nequaltes must converge, n the sense of the wellknown Lax-Wendroff Theorem [36] see also [39, 24], to the expected contnuous entropy nequaltes 3. In fact, several hgh-order MUSCL entropy nequaltes have been derved n the recent lterature for nstance, see [5, 10]. But, t s not convncng that these dscrete nequaltes satsfy the expected convergence behavor. The purpose of the present secton s to brefly study the behavor of the usual hgh-order entropy nequaltes nsde the convergence regme. In fact, at the end of ths secton, we wll present several numercal experments to exhbt the falures of MUSCL schemes to restore 3, based on the usual reconstructon technques and both frst- and hgh-order tme dscretzaton. We do not rgorously justfy these falures but some arguments are here gven. Frst, for the sake of completeness, we recall the Lax-Wendroff Theorem for hgh-order space and tme accurate conservatve schemes. It s the opportunty to report the expected hgh-order entropy nequaltes to be satsfed so that the converged soluton s entropy preservng accordng to3. Next, we brefly revew the usual dscrete entropy nequaltes comng from hgh-order space and tme accurate schemes. We wll show that these usual dscrete entropy nequaltes concde wth the requred one wthn the Lax-Wendroff Theorem up to a postve measure. Put n other words, the usual dscrete entropy nequaltes seem nsuffcent to ensure that the converged soluton s entropy preservng. We llustrate ths negatve result wth Internatonal Journal on Fnte Volumes 5

6 numerous numercal experments. 2.1 Lax-Wendroff theorem for hgh-order schemes We approxmate the weak solutons of a hyperbolc system of conservaton laws n the shortened form: { t w + x fw = 0, wx,t = 0 = w 0 x, 11 where the state vector and the flux functon are gven by 2, and supplemented by the entropy nequaltes 3. We adopt a general m-step Runge-Kutta tme scheme wrtten as follows: w n,0 = w n, w n,l w n+1 = w n l 1 = w n,m. c l,j f n,j +1/2 fn,j 1/2, l = 1,,m, The coeffcents c l,j are assumed to satsfy the followng consstency condtons: 12 c l,j 0, m 1 c m,j = We mpose the scheme to be space hgh-order. To address such an ssue, we consder a numercal flux functon dependng on a large stencl: f n,j +1/2 = fs w n,j s+1,,wn,j +s, 14 where f s : Ω2s R 3 s contnuous and consstent: f s w,,w = fw. As usual, the ntal data s here approxmated as follows: w 0 = 1 x+1/2 x 1/2 w 0 xdx. For the sake of smplcty n the forthcomng statements, we ntroduce the followng pecewse constant functons: w x,t = w n, for x,t [x 1/2,x +1/2 [t n,t n +, w,l x,t = w n,l, for x,t [x 1/2,x +1/2 [t n,t n +. Theorem 2.1 Lax-Wendroff Assume that the sequence tends to zero wth a constant postve rato /, and assume there exsts a compact K Ω such that, for all 0 l m, w,l belongs to K, Internatonal Journal on Fnte Volumes 6

7 the sequence w converges n L 1 loc R R+ ;Ω to a functon w. Then w s a weak soluton of 11. In addton, let us assume the exstence of an entropy numercal flux G s : Ω2s R, whch s Lpschtz-contnuous and consstent: G s w,,w = Gw, where G s defned by5, such that we have the followng dscrete entropy nequalty: 1 Sw n+1 Sw n + 1 m 1 c m,j G n,j +1/2 Gn,j 1/2 0, 15 wth G n,j +1/2 = Gs w n,j s+1,,wn,j +s. Then w s an entropy soluton of Here, we pont out the dffcultes comng from establshng the dscrete entropy nequaltes 15. In fact, several frst-order schemes n the form 7, lke Godunov scheme, HLL and HLLC schemes, relaxaton schemes, Osher scheme [10, 6, 9, 4, 28] are proven to satsfy such requred dscrete entropy nequaltes. Unfortunately, by extendng these frst-order entropy preservng schemes to get hgh-order numercal methods n the form 12, we do not recover hgh-order dscrete entropy nequaltes gven by 15. Our purpose s now to exhbt hgh-order dscrete entropy nequaltes nhertng from tme and space hgh-order extensons, and to consder ther convergence behavor. The proof of the Lax-Wendroff Theorem 2.1 s classcal and several versons can be found n [36, 27, 39, 24]. However, because of the hgh-order dscrete entropy nequaltes 15, up to the authors knowledge, no complete proof can be found n the lterature. Although the proof s standard, for the sake of completeness, we detal t n Appendx A. 2.2 Space hgh-order dscrete entropy nequaltes From a gven frst-order scheme n the form 7 satsfyng frst-order dscrete entropy nequaltes 9, numerous methods have been ntroduced to ncrease the order of accuracy for nstance see [1, 3, 17, 51, 44]. In the present work, we restrct ourselves to MUSCL reconstructon technques known to gve second-order space accurate schemes. However, the reader s referred to [8, 53] where hgh-order MUSCL reconstructons are suggested. We recall that the MUSCL approach s based on a vector state reconstructon on each sde of the nterface located at x +1/2 as follows: w +1/2 = wn µn and w + +1/2 = wn µn The ncrement µ n Z s defned by a lmter functon to read: µ n = L w n wn 1,wn +1 wn, 17 Internatonal Journal on Fnte Volumes 7

8 where L : R 3 R 3 R 3 s a Lpschtz-contnuous functon, whch satsfes: Lv,v = v v R 3, 18 M > 0; Lv 1,v 2 M max v 1, v 2, v 1,v 2 R Precse defntons of L are wdely studed n the lterature for nstance, see [39] and references theren. From now on, let us underlne that the usual lmter functons mnmod, superbee, MC,... satsfy the requrements Next, from the frst-order scheme 7, we get a space second-order scheme n the form 10. Concernng the second-order dscrete entropy nequaltes assocated wth 10, several strateges have been recently proposed. For nstance, n [5], ndependently from the lmter choce L, one get the followng dscrete entropy nequaltes: 1 S w n S w + +1/2 +S w+1/2 + G w +1/2,w+ +1/2 G w 1/2,w+ 1/ A second example can be found n [10] where a specfc MUSCL procedure s ntroduced to get 1 S w n+1 1 x+1/2 w n + x x µn dx + 1 x 1/2 S G w +1/2,w+ +1/2 G w 1/2,w+ 1/ Immedately, we notce that the dscrete tme dervatve nvolved n both 20 and 21 does not concde wth the requred one gven by 15. Our purpose s now to llustrate that these varants of the dscrete entropy nequaltes are not effcent and are not relevant to get an entropy converged soluton. In the sequel, t wll be useful to unfy the notatons. We rewrte 20 and 21 as follows: 1 S w n+1 Sw n + 1 G n +1/2 Gn 1/2 1 Pn Sw n, 22 wherep n = P S w n,µn,fndsanmmedatedefntonwthp S benganoperator assocated wth an entropy functon S. Indeed, f we consder 20, we obtan: P S w,µ, = Next, f we consder 21, we obtan: P S w,µ, = 1 Sw µ/2+sw +µ/ /2 /2 S w + x µ dx. 24 Let us emphasze that snce S s a convex functon, we have n both defntons 23 and 24 as long as P s well defned P S w,µ, Sw 0, w Ω. Internatonal Journal on Fnte Volumes 8

9 In fact, P can be understood as a projecton to approxmate the entropy evaluated on w n. We mpose the exstence of a postve constant C such that 0 P S w,µ, Sw C 2 S µ Ths property s easly satsfed by both defnton 23 and 24. Now, we wll see that 1 Pn Sw n converges to a postve measure to make unsutable the dscrete entropy nequaltes 22. In order to provde a complete llustraton of the falure of22, we propose to extend these space hgh-order dscrete entropy nequaltes by consderng tme hgh-order accurate schemes. 2.3 Tme hgh-order dscrete entropy nequaltes In order to ncrease the tme order of accuracy, we here adopt the usual Runge- Kutta tme scheme to consder a numercal approxmaton gven by 12. To wrte the dscrete entropy nequaltes assocated wth 12, we adopt a reformulaton of 12 ntroduced by Shu and Osher [45, 46]. It conssts n wrtng 12 as a convex combnaton of tme frst-order schemes. We skp the computaton detals gven n [45, 46], but we just recall that, for all postve parameters α l,j 1 l m such that 0 j l 1 l 1 α l,j = 1, for all 1 l m, the m-step Runge-Kutta tme scheme 12 can be equvalently rewrtten as follows: w n,0 = w n, w n,l l 1 = w n+1 = w n,m, where the coeffcents β l,j are gven by α l,j w n,j β l,j f n,j α l,j +1/2 1/2 fn,j, 26 The sequence α l,j 1 l m 0 j l 1 β l,j = c l,j l 1 k=j+1 α l,k c k,j. s chosen n order to enforce the postveness of the parameters β l,j. Now, snce the parameters α l,j and β l,j are postve, we note that the ntermedate states w n,l are nothng but a convex combnaton of frst-order tme schemes wth tme steps respectvely gven by β l,j α l,j. Next, we establsh the dscrete entropy nequaltes satsfed by the tme hghorder scheme 12, or equvalently 26. Let us emphasze that the followng result turns out to be ndependent from the adopted space order of accuracy. Internatonal Journal on Fnte Volumes 9

10 Lemma 2.2 Let us consder a tme frst-order scheme gven by w n+1 = w n f+1/2 n fn 1/2, f+1/2 n = fs w s+1, n,w+s, n supplemented by dscrete entropy nequaltes as follows: 1 Sw n+1 Sw n + 1 G n +1/2 Gn 1/2 δw n, 27 where G n +1/2 = Gs wn s+1,,wn +s and δwn s a postve perturbaton. Assume that the parameters α l,j > 0 are defned such that the parameters β l,j are nonnegatve. Then the scheme 12 satsfes the followng dscrete entropy nequaltes: 1 Sw n+1 Sw n m c m,j G n,j m 1 +1/2 Gn,j 1/2 α m,j δ w n,j Before to prove ths result, let us comment the role played by the perturbaton δw ncenteredonwn, butwhchmaydependsonotherstates. Assoonasastandard space and tme scheme, n the form 7, satsfes dscrete entropy nequaltes 9, we get a vanshng perturbaton, δw = 0 for all w n Ω. As a consequence, nequaltes 28 exactly concde wth the requred dscrete entropy nequaltes 15. More generally, ths means that, as long as the frst-order scheme s entropy preservng wth δw n = 0, then the tme hgh-order Runge-Kutta scheme remans entropy preservng. Now, the stuaton turns out to be dstnct whenever δw n 0, and the rght hand sde n 28 must be carefully studed. Proof Let us ntroduce the ntermedate states as follows: w n,j = w n,j β l,j f n,j α l,j +1/2 fn,j 1/2. Snce β l,j α l,j 0, the state w n,j s nothng but the evoluton state by a tme frst-order scheme. As a consequence of 27, the ntermedate states w n,j satsfes a dscrete entropy nequalty gven by 1 S w n,j S w n,j + β l,j G n,j α l,j +1/2 Gn,j 1/2 δ From the equvalent formulaton 26, let us notce that w n,l l 1 = α l,j w n,j. j=1 Next, snce S s a convex functon, we obtan S w n,l l 1 α l,j S w n,j, w n,j Internatonal Journal on Fnte Volumes

11 to mmedately deduce S w n,l l 1 α l,j S w n,j β l,j G n,j +1/2 1/2 Gn,j + l 1 α l,j δ w n,j. 29 Now, nvolvng a standard proof by nducton, we establsh the followng nequalty: S w n,l Sw n l 1 c l,j G n,j +1/2 Gn,j 1/2 + l 1 α l,j δ w n,j, 1 l m. 30 For l = 1, we mmedately have α 1,0 = 1 and c 1,0 = β 1,0. Then 30 s deduced from 29. Next, let us assume that 30 holds true for all j such that 1 j l 1 and let us establsh the equalty for l. From 29 and substtutng the estmaton 30, we obtan S w n,l l 1 α l,j Sw n j 1 k=0 β l,j G n,j +1/2 1/2 Gn,j + Sw n l 1 α l,j δ l 1 β l,j + w n,j, l 1 k=j+1 S w n,j c j,k G n,k +1/2 Gn,k 1/2 l 1 α l,k c k,j α l,j δ w n,j, 1 j l 1 by G n,j +1/2 Gn,j 1/2 + and 30 s stated. Snce w n+1 = w n,m, by nvolvng 2.3, the proof s acheved. Equpped wth the above result, we are now able to exhbt the dscrete entropy nequaltes assocated wth both space and tme hgh-order accurate schemes Indeed, snce the assocated tme frst-order scheme comes wth dscrete entropy nequaltes gven by 22, we easly get dscrete entropy perturbatons gven by δw n = 1 Pn Sw n. Internatonal Journal on Fnte Volumes 11

12 As a consequence, we are dscussng about the followng hgh-order dscrete entropy nequaltes: 1 S w n+1 Sw n m 1 c m,j + G n,j +1/2 Gn,j 1/2 m 1 1 α m,j P n,j S w n,j. 31 Under the assumptons stated n Theorem 2.1, we easly obtan the weak convergence of the left-hand sde to t Sw+ x Gw. Regardng the rght-hand sde, we set a x,t = m 1 1 α m,j P n,j S w n,j, x,t [x 1/2,x +1/2 [t n,t n Now, let us ntroduce the nonnegatve measure δ defned as the weak-star lmt of the sequence a. Hence, n the lmt of and to zero wth a constant rato /, the nequalty 31 reads t Sw+ x Gw δ. We suggest to compare the measure δ to the entropy dsspaton measure β, whch s defned as the weak-star lmt of the followng sequence: b x,t = m 1 α m,j 1 w n,j w n,j 1 2, x,t [x 1/2,x +1/2 [t n,t n+1. The entropy dsspaton measure β was studed by Hou and LeFloch [32] see also DPerna[22] n the scalar case and wth a frst-order tme scheme. They conjectured that ths measure s concentrated on the curves of dscontnuty of w. In the followng statement, we establsh that both measure δ and β have the same behavor. Theorem 2.3 The measure δ s absolutely contnuous wth respect to the entropy dsspaton measure β. Proof Let φ be a nonnegatve test functon wth compact support K, and we set φ n = φx,t n. Snce P satsfes the property 25, we have,n P n,j S w n,j φ n C 2 S L K µ n,j,n 2 φ n, where 2 S s bounded over K, and µ n,j defned by 17. denotes the reconstructed ncrements Internatonal Journal on Fnte Volumes 12

13 By nvolvng 17 and 19, we get P n,j S w n,j φ n,n O1,n O1,n w n,j w n,j w n,j 1 w n,j w n,j +1 wn,j 2 φ n 1 +φn. 2 φ n, Snce the rato / remans constant, we deduce,n m 1 α m,j P n,j S O1,n m 1 w n,j α m,j w n,j φ n w n,j 1 2 φ n +φ n 1. Passng to the lmt, we get φdδ O1 φdβ, and the proof s completed. To conclude ths secton, let us emphasze that we have establshed the absolute contnuty of the measure δ wth respect to β, whle one may expect the equvalence between these two measures. In fact, the numercal results presented n Secton 2.4 wll confrm such an assumpton. Nowadays we are not able to establsh the absolute contnuty of β wth respect to δ. Moreover, the dscrete entropy nequaltes 31 cannot ensure the requred entropy stablty. 2.4 Numercal tests We turn consderng the numercal llustraton of the above results. More precsely, our objectve s here to numercally evaluate the measure δ ntroduced prevously. Accordng to the work by Hou and LeFloch [32], ths measure must vansh as long as the soluton s contnuous. Reversely, whenever the soluton admts shock dscontnutes, the evaluaton of δ must gve δ > 0. All the presented numercal experments are based on the same strategy. We adopt a space frst-order numercal flux functon f w L,w R gven by the wellknown HLLC scheme [48, 49]. The beneft of such a numercal flux functon s to exactly know the robustness and the dscrete entropy nequaltes [9, 6, 4, 15]. The space second-order accuracy s obtaned by a MUSCL reconstructon 16 where the lmter functon 17 s the mnmod functon, the van Albada 1 functon, the van Leer functon, the monotonzed central-dfference MC functon or the Superbee functon see [39] where all the lmter functons are detaled. Concernng the tme dscretzaton, both frst- and second-order accuracy are adopted. Internatonal Journal on Fnte Volumes 13

14 Accordng to [39, 9, 5], the tme ncrement s restrcted by the followng CFL condton: λ max ± w, λ Z +1/2,w+ ± +1/2 w + 1/2,w +1/ After [5], ths tme restrcton makes the consdered scheme robust and preserves the dscrete entropy nequaltes 31. The relevance of each compared scheme s evaluated by calculatng the L 1 -error: E = Z ρ N ρ ex x,t N, where w ex : R R + Ω denotes the exact soluton. In addton, we evaluate the measure δ by computng ts total mass: I = N a x,t n. n=0 Z Two numercal experments are performed. Both are devoted to the approxmaton of the soluton of Remann problems. Hence, the ntal data s made of two constant states separated by a dscontnuty located at x = 0: w 0 x = { wl f x < 0, w R f x > In the frst test, left and rght states are gven by ρ L = 1, ρ R = , u L = 1, u R = 1, p L = 1.5, p R = , 34 so that the exact soluton s made of a contnuous 1-rarefacton. In Tables 1 and 2, we gve respectvely the evaluaton of E and I obtaned by consderng a tme frst-order scheme wth several lmter functons. Frst of all, we note that van Leer, MC and Superbee are not stable enough and a numercal blowup appears wth very fne mesh. Concernng mnmod and van Albada 1 lmter functons, the behavor s better because both schemes seem to converge snce E goes to zero as tends to zero. At ths level, we may suspect that the blowups are consequences of some compresson phenomena, whle the mnmod lmter and the van Albada 1 lmter seem dffusve enough to avod such a falure. Accordng to the work by Hou and LeFloch [32], snce the converged soluton s contnuous, the entropy dsspaton measure I goes to zero and thus the measure δ s equal to zero. Fgure 1 llustrates the results stated n Tables 1 and 2. Next, Tables 3and 4andFgure 2 aredevoted to theresults obtaned wth a tme second-order scheme. Excepted wth the superbee lmter, all consdered schemes seem to converge lke the measure δ, whch tends to zero. Internatonal Journal on Fnte Volumes 14

15 Nb cells mnmod van Albada van Leer MC superbee E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-3 Table 1: L 1 error E for the 1-rarefacton usng frst-order tme dscretsaton Nb cells mnmod van Albada van Leer MC superbee E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-4 Table 2: Total mass I of the rght-hand sde of the proven entropy nequalty for the 1-rarefacton usng frst-order tme dscretsaton Superbee MC van Leer van Albada 1 mnmod Superbee MC van Leer van Albada 1 mnmod L 1 error 10 2 I Nb cells Nb cells Fgure 1: 1-rarefacton wth frst-order tme scheme: L 1 error left and total mass I of the rght-hand sde of the proven entropy nequalty rght Internatonal Journal on Fnte Volumes 15

16 Nb cells mnmod van Albada van Leer MC superbee E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-3 Table 3: L 1 error E for the 1-rarefacton usng second-order tme dscretsaton Nb cells mnmod van Albada van Leer MC superbee E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-0 Table 4: Total mass I of the rght-hand sde of the proven entropy nequalty for the 1-rarefacton usng second-order tme dscretsaton Superbee MC van Leer van Albada 1 mnmod L 1 error 10 4 I Superbee MC van Leer van Albada 1 mnmod Nb cells Nb cells Fgure 2: 1-rarefacton wth second-order Runge-Kutta tme scheme: L 1 error left and total mass I of the rght-hand sde of the proven entropy nequalty rght Internatonal Journal on Fnte Volumes 16

17 The second proposed numercal experment s devoted to approxmate shock solutons. Once agan, we consder a Remann Problem where the ntal left and rght states are defned as follows: ρ L = 1, ρ R = 1, u L = 10, u R = 10, p L = 1, p R = 1, 35 to obtan an exact soluton made of two shock dscontnutes propagatng wth opposte veloctes. The results obtaned wth a tme frst-order dscretsaton are reported Tables 5 and 6 and Fgure 3. We notce that van Leer, MC and Superbee lmter functons nvolve a numercal blowup. In fact, t seems that mnmodand vanalbada1lmters arealsonot stablebuttheblowupneedsextremely fnemeshes. Moreover, tsworth mentonng that the behavor of the measure δ, gven by I, seems to concde wth a postve value before a numercal blowup. In Tables 7 and 8 and Fgure 4, we present the convergence behavor of the L 1 -error and the measure δ by consderng tme Runge-Kutta second-order schemes. Only superbee lmter nvolves a numercal blowup whle the other schemes converge or seem to converge. However, we remark that the measure δ does not converge to zero but to a postve value accordng to [32]. As a consequence, the known dscrete entropy nequaltes 31 for nstance, gven by [10, 5] turn out to be not suffcent to ensure that the converged soluton s entropy preservng n the sense of the Lax-Wendroff Theorem Theorem Superbee MC van Leer van Albada 1 mnmod L 1 error 10 3 I Superbee MC van Leer van Albada 1 mnmod Nb cells Nb cells Fgure3: Shock-shockwthfrst-ordertmescheme: L 1 errorleftandtotal massi of the rght-hand sde of the proven entropy nequalty rght To conclude these numercal llustratons, the dscrete entropy nequaltes 31 s clearly unsutable snce t does not prevent nstabltes. 3 From one to all dscrete entropy nequaltes From the above results, an entropy preservng hgh-order scheme must satsfy the entropy condton 15, whle non-standard dscrete formulaton of the tme derva- Internatonal Journal on Fnte Volumes 17

18 Nb cells mnmod van Albada van Leer MC superbee E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-3 Table 5: L 1 error E for the shock-shock usng frst-order tme dscretsaton Nb cells mnmod van Albada van Leer MC superbee Table 6: Total mass I of the rght-hand sde of the proven entropy nequalty for the shock-shock usng frst-order tme dscretsaton Nb cells mnmod van Albada van Leer MC superbee E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-4 Table 7: L 1 error E for the shock-shock usng second-order tme dscretsaton Internatonal Journal on Fnte Volumes 18

19 Nb cells mnmod van Albada van Leer MC superbee Table 8: Total mass I of the rght-hand sde of the proven entropy nequalty for the shock-shock usng second-order tme dscretsaton Superbee MC van Leer van Albada 1 mnmod Superbee MC van Leer van Albada 1 mnmod L 1 error 10 3 I Nb cells Nb cells Fgure 4: Shock-shock wth second-order Runge-Kutta tme scheme: L 1 error left and total mass I of the rght-hand sde of the proven entropy nequalty rght Internatonal Journal on Fnte Volumes 19

20 tve may ntroduce unsutable entropy nequaltes ncludng a postve measure. In order to derve a hgh-order scheme able to restore 15, we wll adopt an a posteror technque based on the dscrete entropy nequaltes satsfacton. Let us recall that the expected stablty nequaltes 15 must be satsfed by all entropy pars ρflns, ρflnsu where F s a smooth functon such that 6 holds true. A posteror estmatons are relevant whenever a fnte number of estmatons are consdered, whle we here have an nfnte number of dscrete entropy nequaltes to be satsfed. The purpose of the present secton s to detal arguments to derve all the requred dscrete entropy nequaltes from just one. To address such an ssue, we frst reformulate the entropy pars as follows: Lemma 3.1 The entropy pars S, G, defned by 4-5, rewrtes Sw = ρψr, Gw = ρψru, where we have set and ψ denotes a smooth ncreasng convex functon. r = p1/γ ρ, 36 From now on, let us underlne that ths result s not essental n the sequel, but t makes easer several developments. Indeed, we wll see that consderng entropes Sw parameterzed by a monotone convex functon ψ wll be more convenent than consderng entropes parameterzed by a functon F wth the property6. However, we emphasze that all the followng scheme dervatons can be performed by adoptng the usual entropy pars gven by 4-5. Proof Frst, let us notce that the specfc entropy, defned by 3, wrtes r = s 1/γ. Now, let us consder two functons, S and G such that we have Sw = ρψr and Gw = ρψru, where ψ s a smooth ncreasng convex functon. By ntroducng Flns := ψ s 1/γ, we get to wrte and F y γf y = 1 γ Fy = ψ e y/γ, y R, F y = 1 γ ψ e y/γ < 0 ψ e y/γ +ψ e y/γ < 0. As a consequence, the smooth functon F satsfes 6 and the par S, G s thus an entropy par. Internatonal Journal on Fnte Volumes 20

21 Conversely, let us consder an entropy par S, G = ρflns, ρflnsu, where F satsfes 6. Snce we have we set to wrte the followng relatons: Snce 6 s satsfed, we easly obtan Flns = Fγln r, ψr := Fγln r, Sw = ρψr and Gw = ρψru. ψ r = γ r F γln r > 0 and ψ r = γ r 2 F γln r γf γln r > 0. As expected, ψ s an ncreasng convex functon, and the proof s completed. Argung the above result, we now establsh condtons so that a fnte volume method s entropy preservng as soon as just one relevant dscrete entropy nequalty s satsfed. Let us consder a conservatve scheme gven by w n+1 = w n f+1/2 n fn +1/2, 37 where f+1/2 n = t f ρ +1/2,fρu +1/2,fE +1/2 stands for the consstent numercal flux functon, accordng to 7 or more generally to 14. Theorem 3.2 Under the CFL condton 8, assume the scheme 37 s Ω- preservng: for all w n Ω, we have w n+1 Ω, for all Z. Assume the followng specfc dscrete entropy nequalty: ρ n+1 r n+1 ρ n rn f ρ +1/2 rn +1/2 fρ 1/2 rn 1/2 38 s satsfed, where we have set r n = pn 1/γ ρ n and r n +1/2 = { r n +1 f f ρ +1/2 < 0, r n f f ρ +1/2 > Moreover, assume the followng addtonal CFL lke condton holds: max 0,f ρ +1/2 mn 0,f ρ 1/2 ρ n. 40 Then the scheme 37 s entropy preservng: for all smooth ncreasng convex functon ψ, we have ρ n+1 ψ r n+1 ρ n ψr n f ρ +1/2 ψn +1/2 fρ 1/2 ψn 1/2, Internatonal Journal on Fnte Volumes 21

22 wth ψ n +1/2 defned as follows: { ψ r n ψ+1/2 n = +1 f f ρ +1/2 < 0, ψr n f fρ +1/2 > From now on, let us emphasze the partcular form of the numercal entropy flux functon nvolved n 38. In fact, we mpose to the entropy r to satsfy a transport lke property. Such a condton s clearly more restrctve than usual. For nstance, the HLL scheme s an entropy preservng scheme see [31] whch does not satsfy 38. But there exsts schemes preservng ths restrcton, lke the Sulcu relaxaton scheme [9, 15] or equvalently the HLLC scheme [49, 48], whch are frstorder entropy preservng schemes nvolvng entropy numercal flux functon gven by f ρ +1/2 ln F s n +1/2 where s n +1/2 = { p n +1 /ρ n +1 γ f f ρ +1/2 < 0, p n /ρn γ f f ρ +1/2 > 0. As a consequence, by ntroducng r n = s n 1/γ, such schemes are able to preserve the nequaltes Proof By defnton of the numercal entropy flux functon comng from 38, and argung the defnton of r +1/2 gven by 39, the followng relaton easly holds: f ρ +1/2 r +1/2 = f ρ r n +rn +1 +1/2 2 We plug ths relaton nto 38 to get r n+1 a ρ n+1 r n 1 + b ρ n+1 where we have set a = f ρ f 2 1/2 + ρ 1/2, b = ρ n f ρ f 2 +1/2 + ρ c = f ρ 2 +1/2 f ρ +1/2. Now, let us notce that +1/2 f ρ +1/2 r n + c ρ n+1 rn +1 rn. 2 f ρ f 1/2 + ρ a+b+c = ρ n f ρ +1/2 fρ 1/2, = ρ n+1 > 0. r n +1, 42 1/2 We easly see that a and c are nonnegatve. Moreover, the addtonal CFL lke condton 40 enforces the coeffcent b to be nonnegatve. As a consequence, we have establshed that the rght-hand sde of 42 s nothng but a convex combnaton of r n 1, rn and r n +1., 43 Internatonal Journal on Fnte Volumes 22

23 Accordng to Lemma 3.1, let us now consder an entropy par gven by S,G = ρψr,ρuψr, wth ψ a smooth ncreasng convex functon. The functon ψ beng ncreasng, from the nequalty 42 we get ψ r n+1 a ψ r 1 n + b r n + c r+1 n. ρ n+1 ρ n+1 ρ n+1 By argung the well-known dscrete Jensen nequalty, we deduce ψ r n+1 a ρ n+1 ψ r 1 n b + ρ n+1 ψr n + c ρ n+1 ψ r n +1. Next, by substtutng the coeffcents a, b and c by ther exact value gven by 43, we obtan ρ n+1 ψr n+1 ρ n ψr n 2 f ρ +1/2 ψrn +ψr n +1 f ρ +1/2 ψrn +1 ψr n f ρ 1/2 ψrn 1+ψr n + f ρ 1/2 ψrn ψr 1 n, whch can rewrte as follows: ρ n+1 ψr n+1 ρ n ψr n f ρ +1/2 ψn +1/2 fρ 1/2 ψn 1/2, where ψ +1/2. s defned by 41. The proof s thus acheved. 4 The e-mood scheme for the Euler equatons In ths secton, we derve space hgh-order numercal schemes, gven by 10, whch satsfy the requred dscrete entropy nequaltes 15. For the sake of smplcty n the forthcomng theoretcal developments, we restrct ourselves to tme frst-order numercal methods. However, after Lemma 2.2, the space hgh-order scheme, now detaled, wll easly extend by Runge-Kutta procedure to obtan tme hgh-order schemes, whch reman entropy preservng. Tme hgh-order extensons wll be used to perform numercal experments. To mpose the expected nequaltes 15, we now suggest to ntroduce an addtonal a posteror lmtaton when reconstructng both states w + 1/2 and w +1/2 on the cell x 1/2,x +1/2. Ths a posteror lmtaton technque was recently ntroduced by Clan et al. [17, 21], the so-called MOOD schemes. The MOOD technque allows to extend any frst-order scheme, whch satsfes some requred propertes, to get space hgh-order scheme preservng the same propertes. Its basedon an teratve processto determne, locally on each cell, thebetter reconstructon accordng to the mposed propertes here, robustness and stablty. Let us consder a frst-order conservatve scheme gven by 7. Under a standard CFL-lke condton 8, we assume that ths frst-order scheme satsfes the needed robustness and stablty propertes: Internatonal Journal on Fnte Volumes 23

24 Robustness If all thental states w n aren Ω thenthe evolved states w n+1 reman n Ω. Stablty For all Z, the followng dscrete entropy nequalty s satsfed: ρ n+1 r n+1 ρ n r n f ρ w n,w+1 n r n +1/2 f ρ w n 1,w n r n 1/2, 44 wth r+1/2 n defned as follows { r+1/2 n r = n +1 f f ρ w n,w+1 n < 0, r n f f ρ w n,w+1 n 45 > 0. Once agan, let us emphasze that such a frst-order scheme exsts. For nstance, the reader s referred to the HLLC scheme or the Sulcu relaxaton scheme [15, 48]. Next, we adopt a reconstructon procedure gven by 16. If the ncrement reconstructon µ n, s defned by 17, we stay wthn the standard MUSCL procedure, but µ n can be assocated to hgher accurate reconstructon approaches. In the sequel, the reconstructon s mposed to be Ω-preservng: w + 1/2 = wn 1 2 µn Ω and w +1/2 = wn µn Ω for all Z. We notce that the reconstructon satsfes the followng property: w n = 1 2 w+ 1/ w +1/2. 46 It s possble to avod ths restrcton on the reconstructon. Indeed, nvokng arguments stated n [5], we can consder a reconstructon such that w n s not a convex combnaton of w + 1/2 and w +1/2 : w n αw + 1/2 +1 αw +1/2, α 0,1. However, the relaton 46 makes easer to obtan the robustness requrements. As a consequence, for the sake of smplcty, we adopt the relaton 46. We are now able to present the suggested e-mood scheme. 1. Reconstructon step: For all n Z, on each sde of the nterface x +1/2, we evaluate hgh-order states, gven by w +1/2 = wn µn Ω and w + +1/2 = wn µn +1 Ω Evoluton step: The reconstructed approxmate soluton s evolved as follows: w n+1, = w n f w +1/2,w+ +1/2 f w 1/2,w+ 1/ A posteror lmtaton step: We have the followng alternatve. Internatonal Journal on Fnte Volumes 24

25 If for all Z, we have ρ n+1, r n+1, ρ n rw n f ρ f ρ w +1/2,w+ +1/2 w 1/2,w+ 1/2 r n +1/2 r n 1/2, 49 where r+1/2 n s defned by 45, then the soluton s vald and the updated approxmaton at tme t n + s defned by w n+1 = w n+1,, Z. Otherwse, for all Z such that 49 s not satsfed, we set and we go back to step 2. w + 1/2 = wn and w +1/2 = wn, Before we establsh the robustness and stablty propertes satsfed by the above e-mood scheme, we underlne several mportant ponts comng wth ths numercal procedure besde the ntal MOOD ntroduced n [17]. Frst of all, we recall that the ntal MOOD schemes consder an teratve procedure over the order of accuracy nvolved n the reconstructon step. In [17, 21], the authors adopt a sequence of reconstructons ndexed by the degree 0 d d max of the polynomal reconstructon, where µ n = 0 as soon as d = 0. It s worth notcng that the degree d s locally defned over the cell x 1/2,x +1/2. Next, durng the a posteror lmtaton step, f the property here, entropy preservng property s not satsfed, the order of accuracy s decreased and the MOOD technque s once agan performed but for a smaller value of d. Ths teratve procedure on d stops wth d = 0 snce a frst-order scheme s recovered, and by assumpton, ths frst-order scheme must preserve the expected property. For the sake of smplcty n the e-mood presentaton, we have stopped the teratve procedure at the end of the frst teraton. Of course, t s possble to adopt a procedure made of several teraton from d = d max to d = 0. The robustness and stablty results, stated below, wll be preserved. The second pont to be emphaszed concerns the effectve order of accuracy. Indeed, the e-mood scheme, but also the ntal MOOD scheme, substtutes the hgh-order scheme by a frst-order method as soon as the requred propertes are not satsfed. Clearly, f the mposed property s too strong, the lmtaton wll be actve over the whole doman of computaton and the resultng approxmaton wll turn out to be frst-order accurate. In practce, we have consdered a reconstructon step gven by a usual MUSCL approach and the resultng numercal mprovements are obvous. However, t seems mpossble to rgorously prove the order of accuracy excepted frst-order. From our pont of vew, the derved e-mood scheme must be understood as a stablzaton technque and not only as a space hgh-order procedure. The last concern s devoted to the choce of the a posteror lmtaton and ts practcal consequence. Indeed, the ntal MOOD scheme [17, 21] consders an a posteror lmtaton based on the robustness and on a maxmum prncple. In Internatonal Journal on Fnte Volumes 25

26 fact, consderng a maxmum prncple, several dffcultes arse see [5, 7, 55, 54] assocated to detecton of local extrema. Here, the entropy a posteror lmtaton turns out to be very easly mplemented. In addton, t s mportant to notce that n the orgnal MOOD method, enforcng a constant reconstructon on a cell x 1/2,x +1/2.e. d = 0 s not suffcent to ensure that the maxmum prncple s satsfed on ths cell. Indeed, all the states nvolved n the evoluton step 48 have to be constant reconstructons. Ths ncludes the states w 1/2 and w+ +1/2 whch have respectvely to be equal to wn 1 and w+1 n. In practce, ths mples that the orgnal MOOD method needs two dfferent reconstructons on each cell: one for each nterface. Ths s not the case for the e-mood scheme, snce as soon as the a posteror lmtaton procedure has been actvated on a cell, the evolved state on ths cell satsfes the requred robustness and stablty propertes, regardless of the reconstructon used on the neghbourng cells. Indeed, nsde a cell x 1/2,x +1/2 where the a posteror lmtaton procedure has been actvated, the e-mood scheme rewrtes w n+1 = w n f w n,w R f w L,w n. We underlne that ths frst-order scheme satsfes 44-45, ndependently of the defnton of w L and w R. Ths remark s essental snce t makes the method very attractve and computatonally costless when compared to the ntal MOOD scheme. Now, we are able to state the robustness and the stablty propertes satsfed by the e-mood scheme. Theorem 4.1 Assume the tme step satsfes the two followng CFL lke condtons: λ max ± w +, λ Z +1/2,w ± +1/2 w + 1/2,w +1/2 1 4, 50 max0,f ρ+1/2 mn0,fρ 1/2 ρ n. 51 Assume that w n and all the reconstructed states w ± +1/2, defned by 47, belong to Ω for all n Z. Then the updated state w n+1, gven by the e-mood belongs to Ω for all n Z. Moreover, for all smooth ncreasng convex functon ψ, the e-mood scheme satsfes 1 ρ n+1 ψr n+1 ρ n ψr n + 1 f ρ w +1/2,w++1/2 ψrn +1/2 fρ w 1/2,w+ 1/2 ψrn 1/2 0, 52 where r+1/2 n s defned by 45. As a consequence the e-mood scheme s entropy preservng. Proof Frst, we establsh the robustness of the e-mood scheme. Snce no a posteror lmtaton s devoted to enforce w n+1 to stay n Ω, we have to prove that, defned by 48, belongs to Ω for all n Z. If the lmtaton was actvated w n+1, Internatonal Journal on Fnte Volumes 26

27 on the cell x 1/2,x +1/2, then we have w + 1/2 = wn and w +1/2 = wn, so the reconstructed states are n Ω. If the lmtaton was not actvated, then the states w + 1/2 and w +1/2 are obtaned by the reconstructon procedure whch s assumed to preserve Ω. In both cases, the states w + 1/2 and w +1/2 are n Ω. Let us defne the two followng ntermedate states: w n+1,+ 1/2 = w+ 1/2 f w + /2 1/2,w +1/2 f w 1/2,w+ 1/2, w n+1, +1/2 = w +1/2 /2 f w +1/2,w+ +1/2 f w + 1/2,w +1/2. In fact, we notce that both ntermedate updated states, w n+1,+ and wn+1, 1/2 +1/2, are evaluated by nvolvng a frst-order scheme wth a mesh sze gven by /2. Snce the frst-order scheme s Ω-preservng, we mmedately get w n+1,+ 1/2 and wn+1, +1/2 n Ω as long as the CFL-lke condton 50 s satsfed. Now, by nvolvng 46, we have w n+1, = 1 2 wn+1,+ 1/ wn+1, +1/2, to mmedately deduce that w n+1. belongs to the convex set Ω. Next, by defnton of the e-mood scheme, the followng dscrete entropy nequalty s satsfed for all Z: 1 ρ n+1 r n+1 ρ n rn 1 + f ρ w +1/2,w+ +1/2 rn +1/2 fρ w 1/2,w+ 1/2 rn 1/2 0, where r+1/2 n s defned by 45. Under the CFL condton 51, we can apply Theorem 3.2 and the e-mood scheme satsfy all the requred entropy nequaltes 52. The proof s thus acheved. To conclude ths secton, let us underlne that the robustness of the e-mood schemes comes from the CFL condton 50 and the relaton 46. In fact, f 46 s not satsfed, after [5], addtonal CFL restrctons and reconstructon lmtatons can be mposed to enforce the requred robustness. Ths Ω-preservng property, naturally satsfed by the e-mood scheme, s another understandng of the ntal MOOD method [17] where an addtonal a posteror lmtaton s mposed to satsfy the expected robustness property. 5 Numercal experments For the sake of consstency, the numercal experments now detaled follow the same strategy as mposed n Secton 2.4. To valdate the e-mood scheme, we adopt a numercal flux functon nvolved n48 gven by the HLLC scheme[48, 49]. Concernng the e-mood reconstructon step 47, MUSCL lmters are consdered. Here, we only deal wth mnmod and superbee lmter functons. Indeed, after Tables 1-8, 53 Internatonal Journal on Fnte Volumes 27