1. Introduction. The present work is devoted to the numerical approximations of weak solutions of the Euler equations: (1.1)

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1 STABILITY OF THE MUSCL SCHEMES FOR THE EULER EQUATIONS CHRISTOPHE BERTHON Abstract. The second-order Van-Leer MUSCL schemes are actually one of the most popular hgh order scheme for flud dynamc computatons. In the frame work of the Euler equatons, we ntroduce a new slope lmtaton procedure to enforce the scheme to preserve the nvarant regon: namely the postveness of both densty and pressure as soon as the assocated frst order scheme does t. In addton, we obtan a second-order mnmum prncple on the specfc entropy and secondorder entropy nequaltes. Ths new lmtaton s developed n the general framework of the MUSCL schemes and the choce of the numercal flux functons remans free. The proposed slope lmtaton can be appled to any change of varables and we do not mpose the use of conservatve varables n the pecewse lnear reconstructon. Several examples are gven n the framework of the prmtve varables. Numercal D and D results are performed usng several fnte volume methods. Key words. Euler equatons, MUSCL schemes, nvarant regons, mnmum prncple on the specfc entropy, entropy nequaltes AMS subject classfcatons. 65M6 76N99 35L65. Introducton. The present work s devoted to the numercal approxmatons of weak solutons of the Euler equatons:. t ρ + x ρu =, t ρu + x ρu + p =, t E + x E + pu =, where the pressure s gven by the perfect gas law:. p = γ E ρ u, γ, 3]. For the sake of smplcty n the notatons, t wll be convenent to rewrte the system. as follows:.3 t W + x fw =, wth the state vector W : R R + Ω and the flux functon fw : Ω R 3 where the set Ω of the admssble states s defned by: }.4 Ω = {W R 3 ; ρ >, u R, ew = E ρ u >. Because of the shock waves, the system. must be supplemented by an entropy nequalty see Lax [7] but also Godlewsky-Ravart [] or Toro [6] to further detals:.5 t ρfln s + x ρfln su, s := sw = p ρ γ, MAB, UMR 5466, LRC M3, Unversté Bordeaux I, 35 cours de la lbératon, 334 Talence, France. INRIA Futurs, projet ScAlApplx, Domane de Voluceau-Rocquencourt, B.P. 5, 7853 Le Chesnay Cedex, France.

2 C. BERTHON where the functon W ρfln s s assumed to be convex. It s well known that such a convex property s satsfed see Godlewsky-Ravart [], Tadmor [4] as soon as the functon F verfes:.6 F y < and F y F y <, y R. γ To smplfy the notatons, we ntroduce Fy = Fln y. By defnton of F, let us note that F s a decreasng functon for all y >. In addton, as proved by Tadmor [4], the specfc entropy s satsfes the followng mnmum prncple:.7 sx, t + h mn {sy, t; y x u h}. The most usual and basc approach to approxmate the solutons of. s based on a pecewse constant approxmaton at the tme t n : W n x = W n, x x x +, where x Z denotes the mesh nodes. We have set x + = x + x + x /. For the sake of smplcty n the present work, we wll assume that the mesh s unform wth the sze x. All the results stated n ths paper easly extend to non-unform grds. The sequence W n Z s defned by the followng conservatve scheme see Harten- Lax-Van Leer[4], Godlewsky-Ravart [], Toro [6], LeVeque []:.8 W n+ = W n t FW n x, Wn + FWn, Wn, where the functon Ω Ω FW L, W R denotes the Lpschtz consstent numercal flux functon see [4,,, 6]. The tme step t s assumed to satsfy the CFL lke condton:.9 t x max λ, λ + Z + +, where λ ± are the numercal acoustc waves assocated wth the numercal flux functon under consderaton. The reader s refereed to [,, 6] to complementary + detals about the fnte volume methods for hyperbolc system of conservaton laws. Addtonal stablty propertes can be proved for several schemes. The well-known Godunov scheme [3, ], but also the Lax-Fredrchs scheme see Lax [8] or Tadmor [4] but also [, 6], the knetc scheme as proposed by Perthame see Khobalatte- Perthame [6], Perthame [] and Perthame-Qu [3] or the Sulcu relaxaton scheme Bouchut [5], Coquel-Perthame [], Berthon [4] for nstance, satsfy the postveness of the densty and the nternal energy, a set of dscrete entropy nequaltes.5 and a dscrete formulaton of the mnmum prncple on the specfc entropy.7. In the sequel, the scheme.8 wll be assumed to satsfy the followng propertes: P: ρ n+ > and e n+ > whenever ρ n > and en >.

3 STABILITY OF THE MUSCL SCHEMES 3 P: The conservatve dscrete entropy nequaltes ρ n+. Fs n+ ρ n Fs n + t {ρ x Fsu}W n, Wn + {ρ Fsu}W n, Wn, for all functons F = F ln wth F satsfyng.6, where {ρ Fsu}W L, W R denotes the numercal entropy flux functon. P3: The mnmum prncple on the specfc entropy. s n+ mns n, s n, s n +. Several strateges have been proposed to ncrease the accuracy of the numercal solutons. The hgh-order verson of the frst-order robust schemes s a very attractve subject. The Godunov-type scheme certanly denotes the man class of second-order scheme. It was largely studed n the last twenty years [, 3, 6, 8, 9, 5, 6, 9,,, 3]. Unfortunately, these procedures are, n general, based on the generalzed Remann problem see Ben-Artz-Falcovtz [3] or Bourgeade-LeFloch-Ravart [7]. The solutons of such problems are used wth some benefts to establsh stablty propertes lke entropy nequaltes see Coquel-LeFloch [9]. The computatons of these solutons turn out to be dffcult and make the scheme poorly attractve. Several procedure have been proposed to approxmate the Generalzed Remann Problem. Actually, the most celebrate second-order scheme s the MUSCL scheme see Van Leer [9] whch s systematcally used n the ndustral numercal smulatons based on the fnte volume methods. These schemes extend any frst-order scheme nto a second-order approxmaton usng a very smple numercal procedure. It uses a better reconstructon than a pecewse constant functon snce pecewse lnear functon n the form. W n x, t n = W n + σn x x, x x, x +, are consdered [, 6]. The value σ n denotes the slope of the lnear functon on the cell x, x + see Fgure.. Several choce for the slope σn are proposed n [,,, 6]. Some of these choces wll be dscussed latter on. In fact, the conservatve varables W are not the most used and several papers see [6, 3, ] consder a pecewse lnear reconstructon on a relevant change of varables n the form:.3 κw n x, t n = κw n + σn x x, x x, x +, where κ denotes a smooth change of varables. For nstance, the so-called prmtve varables ρ, u, p or the varables ρ, u, s enter ths framework. For the sake of smplcty n the notatons, the varables ρ, u, s wll be denoted entropc varables n the present work. We consder the nner approxmaton n the cell see Fgure. located at x = x and x = x +. These approxmatons are denoted Wn,± W n,± = W n x ±, tn. For the sake of smplcty n the remander of the paper, we set and are defned by.4 W n,± = W n + Wn,±,

4 4 C. BERTHON where the ncrement W n,± s defned by W n,± = ± x σn, n the case of lnear functon based on the conservatve varables.. If we consder lnear functons based on a change of varables.3, W n,± fnds the followng defnton: W n,± = κ κw n ± x σn W n W n,+ W n W n,+ W n, W n W n, + W n + x x x + Fg... Pece-wse lnear MUSCL reconstructon.5 In the sequel, we wll denote conservatve slope the followng case: or equvalently W n,.6 W n, + W n,+ W n, + W n,+ = W n =, whle the general slope wll be denoted by + W n,+ W n, or equvalently W n, + W n,+. Both stuatons wll be consdered n the present paper. In general, works devoted to stablty propertes of the MUSCL schemes see Perthame [] or Khobalatte-Perthame [6] for nstance solely consder the conservatve slope.5. The space second-order scheme wrtes:.7 W n+ = W n t x FW n,+, W n, + FWn,+, Wn, where F s the assocated frst-order flux functon ntroduced n.8. The man dffculty les on the constructon of the vector ncrement W n,±. A large lterature s devoted to ths subject but essentally for the scalar conservaton laws and t s based on the Total Varaton Dmnshng crteron [9]. As emphaszed by Coquel-LeFloch [9], the total varaton of a soluton of., n general, s not a dmnshng functon of tme. Thus, t s necessary to focus on propertes.5 and.7. In the framework of the Euler equatons., the MUSCL scheme.4-.7 s used but, n general, the stablty propertes are not ensured. However, Khobalatte- Perthame [6] and Perthame-Qu [3] exhbt conservatve slope lmtatons n order,

5 STABILITY OF THE MUSCL SCHEMES 5 to preserve the nvarant regon; namely the postveness of both densty and nternal energy. Ths result s establshed n the framework of the knetc scheme wth a relevant lke CFL restrcton. In Khobalatte-Perthame [6] a new lmtaton based on the specfc entropy s proposed to conjecture a dscrete entropy mnmum prncple.. In fact, t wll be seen that ths scheme enters the general approach developed n the present work see secton. devoted to the conservatve slope. The scope of ths paper s to extend the propertes P, P and P3 to the secondorder MUSCL schemes. Argung a relevant CFL lke condton, n the next secton, we develop an easy slope lmtaton to preserve the nvarant regon by the MUSCL scheme In addton, we establsh second-order entropy nequaltes. and a second-order entropy mnmum prncple. These results are proved n the general context of slope lmtatons.6 and next n the specfc case of conservatve slope.5. In fact, our lmtaton wll be understood as a correcton of the usual lmtaton functons. Concernng the choce of the numercal flux functon, we just assume that t satsfes the frst-order stablty propertes P, P and P3. The presentaton of the stable lmters s concluded by the ntroducton of an hybrd approach. Ths hybrd procedure wll be seen less restrctve concernng the slope lmtaton but for a more severe CFL restrcton. Moreover, some remarks are gven concernng the second-order tme accuracy. The thrd secton s devoted to the slope lmtaton tself. More precsely, we detal the procedure appled to the slope to enforce the stablty propertes. Several pont of vew are consdered: prmtve varables, entropc varables but also conservatve varables. The hybrd procedure s detaled. In the last secton, we llustrate the nterest of the method. Several numercal tests are performed n D and D and comparson wth the usual MUSCL schemes are proposed. We conclude the paper wth a summary and some remarks.. Stablty propertes for second-order schemes. In the present secton, we develop a general approach to ensure the man stablty propertes of the MUSCL scheme Actually, we do not specfy the choce of the vector ncrement W n,± whch wll be done n the next secton. Ths secton s solely devoted to a space lmtaton such that f W n,± satsfes these lmtatons then the stablty propertes are enforced. Put n other words, we do not mpose a pecewse lnear reconstructon on the conservatve varables W and any change of varables turns out to be admssble, prmtve varables for nstance, for the lnear reconstructon. We conclude ths secton wth a tme second-order accurate scheme whch preserves the stablty propertes... The general slope lmtatons. Our approach s entrely based on the followng remark: The MUSCL scheme.7 can be understood as the average over the cell of three values obtaned by the frst-order scheme. Indeed, let us fx a cell and a set of postve coeffcents α, α, α+ such that α + α + α + =. We ntroduce an ntermedate state W n, α Wn, + α W n, unquely defned as follows: + α + Wn,+ = W n. The role played by the three states W n,, W n, and W n,+ s dsplayed n the fgure.. Now, we propose to evolve each state wth the frst-order scheme.8 as follows: W n+, = W n, t α x FW n,, W n, FW n,+, Wn,,.a

6 6 C. BERTHON W n+ t n+ W n+, W n+, W n+,+ W n,+ W n, W n, W n,+ W n, + t n x α x α x α+ x x + W n W n W+ n Fg... Interpretaton of the MUSCL scheme as an average of frst-order scheme W n+, W n+,+ = W n, t α x = W n,+ t α + x FW n, FW n,+, W n,+ FW n,, W n,,.b, W n, + FWn,, W n,+..c We mmedately deduce that the updated soluton by the MUSCL scheme.4-.7 s nothng but the average of the three above updated states:. W n+ = α Wn+, + α W n+, + α + Wn+,+. Ths new formulaton of the scheme s central to establsh the expected stablty propertes. Frst, we show that the scheme.7 preserves the nvarant regon as soon as the assocated frst order scheme does t. Indeed, assume the CFL lke condton.3 t x max λ ±,, Z λ±,, λ±,+ mn Z α, α, α +, where λ ±,± denotes the numercal acoustc waves accordng to the ntermedate states W n,, W n, and W n,+. Then, the property P can be appled to each updated partal state vectors W n+,, W n+, and W n+,+. As a consequence, as soon as W n,± Ω for all Z, we have W n+,± Ω. Snce Ω s a convex doman, we mmedately deduce that W n+, gven by., belongs to Ω. We have just establshed Theorem.. Let us consder a frst-order scheme whch preserves the nvarant regon. Assume that W n Ω and Wn,± satsfy for all Z: W n,± = W n + Wn,± Ω,.4 W n, = W n α α W n, α+ α W n,+ Ω. Assume the CFL restrcton.3. Then the MUSCL scheme.4-.7 preserves the nvarant regon: ρ n+ > and E n+ ρu n+ /ρ n+ >. From now on, let us note that the lmtaton on the nner approxmaton W n,± s, n general, mposed. Indeed, f W n,± does not belong to Ω, the numercal flux

7 STABILITY OF THE MUSCL SCHEMES 7 functon s not defned. The actual novelty les on the lmtaton on the ntermedate state W n,. In the next sectons, we wll see that ths new lmtaton s not badly restrctve and does not make complex the resultng scheme. Now, we extend the above result wth a mnmum prncple on the specfc entropy and entropy nequaltes. Theorem.. Let the frst-order scheme satsfy the propertes P, P and P3. Assume that W n Ω and assume that W n,± satsfes.4. Assume the CFL condton.3. Then the second-order scheme.4-.7 satsfes the followng mnmum prncple:.5 s n+ = sw n+ mns n,+, sn,±, s n, +, sn,± = sw n,±. In addton, the followng entropy nequaltes are satsfed.6 t x ρ n+ Fs n+ ρ Fs n + {ρ Fsu}W n,+, W n, n,+ {ρ Fsu}W +, Wn,, for all functons F = F ln wth a functon F satsfyng.6, where ρ Fs n as follows: s defned.7 ρ Fs n = α ρn, Fs n, + α ρn, Fs n, + α + ρn,+ Fs n,+. Proof. By defnton of the partal updated states W n+,±, defned by., and snce the frst-order scheme satsfes., we mmedately deduce the followng sequence of nequaltes:.8 s n+, s n+, s n+,+ mns n,+, sn,, s n, mns n,+ mns n,, s n,, s n,+ mns n,+ mns n,, s n,+, s n, + mnsn,+, sn,±, sn,±, sn,±, s n,, s n,, s n, +, +, +. Now, we apply the convex property of the entropy functon W ρ Fs = ρfln s where F satsfes.6, to obtan ρ n+ Fs n+ α ρn+, Fs n+, + α ρn+, Snce F s a decreasng functon, we deduce from.8: ρ n+ Fs n+ α ρn+, ρ n+ Fmns n,+ + α ρn+,, sn,± Fs n+, + α + ρn+,+ Fs n+,+. + α + ρn+,+ Fmns n, sn, sn +,, s n, +. Once agan, argung the fact that F decreases, we mmedately obtan the expected mnmum prncple on the specfc entropy.5. Concernng the proof of the entropy nequaltes.6, we note that entropy nequaltes are satsfed by each partal state W n+,± : ρ n+, ρ n+, Fs n+, ρ n, t α x {ρ Fs n+, ρ n, Fs n, + Fsu}W n, Fs n, +, W n, {ρ n,+ Fsu}W, Wn,,

8 8 C. BERTHON ρ n+,+ Fs n+,+ t α x {ρ ρ n,+ t α + x {ρ Fsu}W n, Fs n,+ + Fsu}W n,+, W n,+ {ρ n, Fsu}W, W n,,, W n, n, {ρ Fsu}W, W n,+. Indeed, these nequaltes are drectly deduced from the entropy nequaltes satsfed by the frst-order scheme. Owng the convex property of the functon W ρ Fs, the sum of the three above nequaltes gves: ρ n+ Fs n+ α ρn, + t α x {ρ Fs n, + α ρn, Fsu}W n,+ + Fs n, + α + ρn,+ Fs n,+, W n, n,+ {ρ Fsu}W +, Wn,, whch completes the proof. Ths concludes the presentaton of the man propertes satsfed by the MUSCL schemes.4-.7 as long as the frst-order assocated scheme s stable... The conservatve slope. We propose to focus our attenton on the specfc case.5. The conservatve slope enters the above approach but smplfed results can be establshed as soon as the condton.5 s satsfed. Of course, ths equalty s satsfed as long as the conservatve varables are consdered for the slope reconstructon procedure: W n,+ = W n, = W n, W n,± = W n ± W n. However, as proved n [6], a slope reconstructon usng entropc varables ρ, u, s s also possble. Now, we follow the dea ntroduced by Perthame []. We set W n+, W n+,+ = W n, t x/ = W n,+ t x/ The updated soluton W n+ FW n, FW n,+, W n,+ FW n,+, Wn,,, W n, + FWn,, W n,+. gven by the MUSCL scheme.7 thus rewrtes: W n+ = Wn, + Wn+. Argung the same notatons as ntroduced n.3, let us assume the followng CFL lke restrcton:.9 t x/ max Z λ±,, λ±,+. Now, the arguments used n the general approach gve smlarly Theorem.3. Let us consder a frst-order scheme whch preserves the nvarant regon. Assume that W n and W n,± are n Ω for all Z. Assume the CFL condton.9. Then the MUSCL scheme.4-.7 preserves the nvarant regon.

9 STABILITY OF THE MUSCL SCHEMES 9 In addton, assume that the frst-order scheme satsfes the propertes P, P and P3. Then the scheme.4-.7 satsfes the mnmum prncple on the specfc entropy.5 and the entropy nequaltes.6 hold wth the followng defnton of ρ Fs n :. ρ Fs n = α ρn, Fs n, + α + ρn,+ Fs n,+. The second-order MUSCL knetc scheme proposed by Khobalatte-Perthame [6] ly enters the present framework of conservatve slope..3. The hybrd lmtaton. We propose a thrd approach denoted as hybrd. Indeed, n the framework of the general slope, we wll prove a stablty result where we solely restrct the nner approxmatons W n,± smlarly to the case of the conservatve slopes. The restrcton wll not use the ntermedate state W n,. In fact, the restrcton wll be done on the CFL condton to obtan the expected result. Ths approach s based on the followng result where we prescrbe a specfc choce of the parameters α, α+ and α = α + α + : Lemma.4. Let W n and W n,± be n Ω for all Z. Then, there exsts α and n, such that α +. W n, = W n α α W n, α+ α W n,+ Ω. Proof. Assume α = α + = then we have W n, = W n Ω. By a contnuty argument, we mmedately obtan the result. By vrtue of ths lemma, we have Theorem.5. Let us consder a frst-order scheme whch preserves the nvarant regon. Assume that W n Ω and Wn,± Ω for all Z. Let the parameters α and α + be gven by the lemma.4. Assume the CFL condton.3. Then the MUSCL scheme.4-.7 preserves the nvarant regon. In addton, assume that the frst-order scheme satsfes the propertes P, P and P3. Then the scheme.4-.7 satsfes the mnmum prncple on the specfc entropy.5 and the entropy nequaltes.6. Proof. Snce the parameters α and α + are gven by the lemma.4, we mmedately deduce that all the assumptons of the theorem. are satsfed and the proof s completed. In ths procedure, the slope lmtaton s less restrctve than for the general slope approach. Indeed, no lmtatons are mposed concernng the ntermedate state W n,. However, the modfed values of the parameters α and α + may ntroduce a very restrctve CFL condton. Indeed, n vew of the formula.3, the tme ncrement t decreases proportonally wth the smaller coeffcent α ±..4. The tme dscretzaton. Concernng the tme dscretzaton, we propose a basc method whch ensures both second-order accuracy n tme and stablty propertes. To evolve n tme from the date t n to t n + t, we consder the followng scheme:. W = W x FW n,+, W n, + FWn,+, Wn,, W = W t x F W +, W + F W +, W, W n+ = t t t + t W + t t t + t W n.

10 C. BERTHON where we have set t = t t t + t. The scheme. s second-order accuracy n tme. We note a very dscrepancy wth the usual approach where t = t = t. In the present method, the tme ncrement t s chosen n order to ensure the CFL restrcton.3 wth W n Z and the second tme ncrement t must satsfy the CFL restrcton.3 for W Z. As soon as the slope W n,± and W ± satsfy the lmtaton.4, we mmedately deduce that W Ω for all Z. Snce Ω s convex and W n+ s defned by a convex sum of W n and W, we have W n+ n Ω. As a consequence, the tme and space second-order accurate scheme. preserves the nvarant regon. In the usual case t = t = t, let us note that the tme ncrement t does not satsfy, n general, the CFL condton.3 for W Z. 3. The lmtaton procedure. In ths secton, we propose to detal the use of our lmtaton.4. We gve several examples of lmtatons whch appears as the most frequently used. In addton, we detal a example for the conservatve and the hybrd slope lmtatons. Let us wrte the condton.4 as follows: 3. { ρ n, >, ρ n, >, ρ n,+ >, p n, >, p n, >, p n,+ >. The lmtatons.4 and 3. are equvalent. 3.. Prmtves varables. The frst varables we propose to lmt are certanly the most usual choce: the prmtve varables. Then, we set 3. ρ n,± u n,± p n,± = ρ n ± ρ = u n ± u = p n ± p In the present secton, we propose a lmtaton procedure to be appled to the ncrements ρ, u and p n order to satsfy 3.. Presently, the ncrements are assumed to be known and computed by usual slope lmters see the next secton to several examples: mnmod, superbee... Now, we show how modfy ρ, u and p to satsfy 3.. The proposed procedure s not unque and dstnct approach can be developed. Frst, for the sake of smplcty, we fx the parameters α ± : α + = α = α+ = 3. Then, we note that ρ = ρn 3.3 >. The lmtatons on ρ thus reads: ρ ρ n <. Concernng the lmtaton on p, we begn wth the lmtaton on p n,± 3.4 p p n <. whch rewrtes:

11 STABILITY OF THE MUSCL SCHEMES We fx ρ and p such that 3.3 and 3.4 are satsfed. To conclude the lmtaton, we have to mpose the condton on p n,. After a straghtforward computatons, we have p, ρ = p n γ + ρ n u. Wth a fxed ρ, the ncrement u must satsfy ρ n 3.5 u < γ ρ n p n + ρ ρ n. To llustrate the lmtaton procedure, we propose to gve an example based on the mnmod lmter. Let us set δρ = mnmod ρ n ρn, ρn + ρn, δu = mnmod u n un, un + un, δp = mnmod p n pn, pn + pn, where the mnmod functon s defned as follows: mnmoda, b = max, mna, b + mn, maxa, b. Instead of the mnmod functon, other lmter functon can be used. secton, we wll consder both mnmod and superbee functons. By solvng 3.3, 3.4 and 3.5, the ncrements read: ρ = ρ n max, mn, δρ ρ n, u = sgnδu mn δu, γ ρ n p = p n max, mn, δp p n. p n + ρ ρ n In the next Such a slope reconstructon satsfes the stablty condton 3.. Moreover, t s clear that ths corrected lmtaton does not make more complex the scheme than the orgnal MUSCL scheme. 3.. Entropc varables. We propose a lnear reconstructon based on the entropc varables ρ, u, s. We set 3.6 ρ n,± u n,± s n,± = ρ n ± ρ, = u n ± u, = s n ± s,

12 C. BERTHON wth α = α = α+ = 3. Once agan, we assume that the ncrements ρ, u and s are ntally computed by a classcal slope lmter. Now, we focus on the procedure to modfy these ncrements to satsfy the restrctons 3.. The condton on ρ reads: 3.7 ρ ρ n <, whle the condton assocated to s n,± gves 3.8 s s n <. Concernng the last condton p n, computatons, we have: s n, = 3s n γ + ρ ρ n ρ n γ >, equvalently, we consder s n, ρ + u γ s n + s Then, we deduce the followng nequalty: 3.9 γ ρ n + γ 3s n + ρ ρ n ρ n ρ u < ρ n ρ ρ n γ s n + s ρ ρ n We note that ths nequalty can be solved f and only f 3. γ s n s. γ s n s. 3s n + ρ γ ρ n s n + s ρ γ ρ n s n s >. >. After the To enforce ρ and s such that 3. s satsfed, we propose to chose ρ such that 3. 3s n + ρ γ ρ n s n ρ γ ρ n s n >. Then, wth a fxed ρ, the ncrement s s reduced to satsfy 3.. We summarze the lmtaton procedure as follows:. The ncrements ρ, u and s are computed by a usual approach.. ρ s reduced to satsfy 3.7 and s s reduced to satsfy 3.8 and u s reduced to satsfy 3.9.

13 STABILITY OF THE MUSCL SCHEMES 3 Let us gve an example once agan based on the mnmod functon. We set δρ = mnmod ρ n ρ n, ρ n + ρ n, δu = mnmod u n u n, u n + u n, δs = mnmod s n s n, s n + s n. By solvng 3.7 and 3., the value of ρ s gven by ρ n ρ = max, mn, δρ ρ, f γ ln 3 n ln. ρ n max ξ, mnξ +, δρ, otherwse, where ξ +, respectvely ξ denotes the postve, resp. negavte, root of the equaton ρ n ξ γ ξ γ =. Solvng 3. and 3.9, we obtan the followng values for s and u: s = γ γ s n max, mn 3 + ρ ρ ρ n ρ n γ γ, s + ρ ρ ρ s n, f ρ <, n ρ n γ γ s n max 3 + ρ ρ ρ n ρ n γ γ, mn, s ρ ρ + ρ s n, f ρ <, n ρ n s n, max mn, s, f ρ =, s n γ γ u = sgnδu mn 3s n + ρ δu ρ s n, n + s ρ ρ s n n s γ ρ + ρ. n γ We do not clam that ths procedure s optmum but t yelds to slopes whch satsfy the lmtaton 3.. Moreover, the modfed lmtaton functon does not nvolve very complex lmtaton when consderng the entropc varables ρ, u, s for nstance, see [6] where equatons smlar to 3. are solved Conservatve varables. We propose to consder the specfc case of the conservatve slope.5. The reader s referred to the work of Khobalatte-Perthame [6] where a conservatve slope lmtaton s proposed on the bass of entropc varables. In the present work, we just consder a lnear reconstructon based on the conservatve varables: 3.3 ρ n,± ρu n,± E n,± = ρ n ± ρ, = ρu n ± ρu, = E n ± E. Once agan, the ncrements ρ, ρu and E are assumed to be known. We modfy these ncrements such that W n,± Ω. ρ n

14 4 C. BERTHON 3.4 The condton ρ n,± > reads: ρ ρ n <. The second lmtaton devoted to p n,± 3.5 E n E n,± reads: ρun,± ρ n,± The sum of the nequaltes 3.5 wrtes: ρu n, 3.6 ρ n, >. + ρun,+ ρ n,+ >. To fnd ρ and ρu satsfyng 3.6, we propose to consder ρ such that E n ρu n ρ n ρ + ρun 3.7 ρ n + ρ >. Next, for a fxed ρ, ρu s chosen to satsfy 3.6. We summarze the conservatve slope lmtaton as follows:. The ncrements ρ, ρu and E are computed by a usual approach.. ρ s reduced to satsfy 3.4 and ρu s reduced to satsfy E s reduced to satsfy 3.5. To propose an example of conservatve slope reconstructon whch satsfes 3., we consder say the mnmod functon and we set δρ = mnmod ρ n ρn, ρn + ρn, δρu = mnmod ρu n ρu n, ρu n + ρu n, δe = mnmod E n E n, E n + E n. The followng ncrement defnton, obtaned by solvng , satsfes the stablty condton 3.: ρ = ρ n max E n ρn u n E E n, mn n ρn u n E n, δρ ρ n, ρu = maxξ, mnξ +, δρu, E = max E n ρun + ρu ρ n + ρ where ξ ± are defned as follows:, mn ξ ± = u n ρ ± ρ n ρ ρ n E n E n ρun ρu ρ n ρ, δe, ρn u n.

15 STABILITY OF THE MUSCL SCHEMES Prmtves varables for the hybrd approach. The last lmtaton procedure we gve s devoted to the hybrd procedure. We just consder the case of prmtve varables and we set = ρ n ± ρ, ρ n,± u n,± p n,± = u n ± u, = p n ± p, where the ncrements are computed by a usual method. In the hybrd procedure, we have to enforce the restrctons.4 whch rewrte n the followng form: ρ n,± >, p n,± >. These two condtons are satsfed as soon as the ncrements ρ and p verfy the nequaltes 3.3 and 3.4. Now, the hybrd approach mposes to satsfy the lemma.4. So, we are searchng for parameters α and α + such that the condton. s satsfed. For the sake of smplcty, we propose to consder parameters n the form: α = α + = α, /. As a consequence, we obtan after computatons: ρ n, = ρ n, p n, = p n α α γ ρn ρ α + α u. Wth fxed ncrements ρ, u and p, there exsts α, / such that p n, >. The parameter α s soluton of a second order nequalty not detaled here. In vew of the CFL condton.3, the relevant value of α may reduce drastcally the tme ncrement t see the numercal test. For numercal smulatons, the coeffcent α s chosen n order to maxmze the value of mnα, α, α+ ; the better choce of α s gven by /3. 4. Numercal results. In ths secton, we llustrate our numercal procedure wth several D and D tests. Concernng the D numercal tests, they are performed usng the same strategy. The mesh s assumed to be unform and made of cells. The CFL number s fxed to accordng to the CFL lke restrcton.3. Wth α ± 4. = /3, let us note that the CFL condton reads t x max Z λ±,± 6. Two dstnct frst-order fnte volume schemes are consdered: the knetc scheme [6,,, 3] and the Sulcu relaxaton scheme [, 4, 5, ]. Concernng the computaton of the slopes, we consder two of the most popular formulas: the mnmod and the superbee functons for nstance, see [,,, 6] to further detals. These lmtatons are thus modfed accordng to the above theory see the secton 3 and the examples theren. ρ n

16 6 C. BERTHON densty velocty.5 soluton standard MUSCL frst order soluton standard MUSCL frst order 5!.5 prmtve a! prmtve 5!.5 entropc b! entropc 5!! Fg. 3.. Case : Relaxaton MUSCL scheme, mnmod slope computaton: a modfed mnmod lmtaton for prmtve varable reconstructon, b modfed mnmod lmtaton for entropc varables reconstructon. The numercal results are systematcally compared wth the standard MUSCL approach. In the present work, the standard MUSCL scheme s gven by.8 where the reconstructon s performed on the prmtve varables ρ, u, p. Concernng the CFL condton, we adopt the followng restrcton: 4. t x max Z λ±,± 6.

17 STABILITY OF THE MUSCL SCHEMES 7 densty velocty.5 soluton sandard MUSCL frst order soluton standard MUSCL frst order 5!.5 prmtve a! prmtve 5!.5 entropc b! entropc 5!! Fg. 3.. Case : Relaxaton MUSCL scheme, superbee slope computaton: a modfed superbee lmtaton for prmtve varable reconstructon, b modfed superbee lmtaton for entropc varable reconstructon. In general, the CFL number s fxed to. To be consstent wth our CFL condton 4., we consder a CFL number equal to /6 for the standard MUSCL scheme. The frst test corresponds to a Remann soluton made of a shock wave and a rarefacton wave separated by a contact dscontnuty. The ntal data s made of two constant states defned as follows: ρ L = u L = p L = ρ R = u R = p R =.

18 8 C. BERTHON a b conservatve hybrd conservatve hybrd.5.5!! Fg Case : Relaxaton MUSCL scheme a modfed mnmod slope computaton for the conservatve and the hybrd approaches b modfed superbee slope computaton for the conservatve and the hybrd approaches. The left and rght states are separated by a dscontnuty located at x =. The soluton s dsplayed at the tme t =.3. The numercal results are dsplayed n the fgures 3. to 3.4. In the fgure 3., we present the results obtaned wth the Sulcu relaxaton scheme. To be compared, we gve the frst-order accurate result and the approxmate soluton obtaned wth the classcal second-order MUSCL scheme. The slope s computed wth the mnmod functon see [] and modfed accordng to.4. In ths fgure, also we dsplay the approxmate results obtaned nvolvng the prmtve varable reconstructon secton 3. and the entropc varable reconstructon secton 3.. In the next fgure 3., the same methods are consdered but for the modfed superbee functon accordng to the restrcton.4.. The numercal results are supplemented by approxmatons obtaned nvolvng conservatve varables secton 3.3 and the hybrd approach secton 3.4. These results are dsplayed n the fgure 3.3. The fgure 3.4 s devoted to the numercal results usng the knetc scheme. For the sake of clarty, we do not dsplay the hybrd approach. To conclude the dscusson about ths test, we emphasze that the new lmtaton procedure does not provde the oscllatons. The fgure 3.5 llustrates ths purpose. For two mesh refnements, we dsplay the result obtaned wth the classcal superbee MUSCL scheme and wth the modfed superbee and modfed mnmod slope reconstructons. In addton, we dsplay the result obtaned wth the centered slope, known to nvolve large oscllatons. We just recall that the ntermedate ncrement δρ, δu and δp, ntroduced n secton 3 to modfy the lmtaton functons, reads as follows n the case of centered slopes: δρ = ρn + ρn, δu = un + un, δp = pn + pn. Two small oscllatons persst for a fne mesh but the approxmate soluton turns

19 STABILITY OF THE MUSCL SCHEMES 9.5 a soluton standard mnmod standard superbee frst order b! c.5 prmtve entropc conservatve.5 prmtve entropc conservatve!! Fg Case : Knetc MUSCL scheme a standard approach wth a prmtve varable reconstructon b modfed mnmod slope computaton for the prmtve, the entropc and the conservatve varables c modfed superbee slope computaton for the prmtve, the entropc and the conservatve varables. out to be n a very good agreement wth the soluton. To conclude the frst numercal experment, we note that the standard MUSCL approaches and the modfed MUSCL reconstructon gve the same level of accuracy. Ths remark s emphaszed n the fgure 3.6 where logarthmc L -error for both standard and modfed MUSCL reconstructon. Ths pont s crucal snce t confrms that our lmtaton does not reduce the order of the numercal method. In the second test, we consder a Remann soluton made of two rarefacton waves. The left and rght states whch made the ntal data are defned as follows: ρ L = u L = p L = ρ R = u R = p R = The soluton s dsplayed at the tme t =.. The fgure 3.7 s devoted to the numercal results performed wth the relaxaton scheme whle n the fgure 3.8, we dsplay the results completed wth the knetc scheme. We present the approxmatons based on the modfed mnmod and modfed superbee slope computatons. The results usng prmtve, entropc and conservatve varable reconstructon are gven. Let us note that the hybrd method nvolves very small tme ncrement and no results are performed by ths procedure for the test of two rarefacton waves. The last D test concerns a Remann soluton made of two shock waves. The

20 C. BERTHON a b soluton standard superbee nodes standard superbee 5 nodes soluton superbee nodes superbee 5 nodes.5.5!.5 c soluton mnmod nodes mnmod 5 nodes!.5 d soluton centered nodes centered 5 nodes!! Fg Case : Relaxaton MUSCL scheme wth prmtve varable reconstructon a classcal superbee slope computaton b modfed superbee slope computaton c modfed mnmod slope computaton d modfed centered slope computaton.!.5! standard superbee modfed superbee standard mnmod modfed mnmod!.5!3!3!.8!.6!.4!.!!.8 Fg Case : Logarthmc L -error versus log x for the relaxaton MUSCL scheme wth standard and modfed lmter for the prmtve varable reconstructon. ntal data for ths test s defned as follows: ρ L = 3 u L = p L = 573 ρ R = 3 u R = p R = 573 The numercal solutons are dsplayed at the tme t =. n the fgures 3.9 and 3..

21 STABILITY OF THE MUSCL SCHEMES.5 a soluton standard mnmod standard superbee frst order.5 b! prmtve entropc conservatve.5 c prmtve entropc conservatve!! Fg Case : Relaxaton MUSCL scheme a standard approach wth a prmtve varable reconstructon b mnmod slope computaton for the prmtve, the entropc and the conservatve varables c superbee slope computaton for the prmtve, the entropc and the conservatve varables. The same strategy of presentaton used for the second test s adopted here. We note that the approaches based on the superbee functon, standard or modfed, nvolve oscllatons and the better results are obtaned wth the mnmod functon. In the last smulaton, we propose a D Remann problem see Kurganov-Tadmor [5]. The ntal data s defned as follows: ρ = 33 p =.3 u =.6 v = ρ 3 =.38 p 3 =.9 u 3 =.6 v 3 =.6 ρ =.5 p =.5 u = v = ρ 4 = 33 p 4 =.3 u 4 = v 4 =.6 At the tme t =.3, the densty soluton s dsplayed n the fgure concluson. In ths paper, we propose a new verson of the celebrate MUSCL scheme to approxmate the solutons of the Euler equatons. We focus our attenton on the lnear reconstructon procedure to enforce several stablty propertes. The obtaned second-order MUSCL scheme ensures the numercal solutons to satsfy the postveness of the densty and the pressure but also a second-order entropy mnmum prncple. In addton, second-order entropy nequaltes are establshed. These results are obtaned argung a relevant CFL condton. We recall that no CFL condton s,

22 C. BERTHON.5 a soluton standard mnmod standard superbee frst order.5 b! prmtve entropc conservatve.5 c prmtve entropc conservatve!! Fg Case : Knetc MUSCL scheme a standard approach wth a prmtve varable reconstructon b modfed mnmod slope computaton for the prmtve, the entropc and the conservatve varables c modfed superbee slope computaton for the prmtve, the entropc and the conservatve varables. actually, establsh concernng the Euler equatons excepted n the work of Perthame et al. [6, 3]. In general, a CFL condton n the form 4., wth as CFL number, s consdered. As specfed n [6], ts use should be done wth cauton. In the case of conservatve reconstructon, the CFL restrcton we fnd s accordng to several works [6, 3] devoted to stablty of MUSCL schemes. These stablty results are obtaned ndependently of the choce of the numercal flux functon as long as several stablty propertes are satsfed by the numercal flux functon. Put n other words, the proposed MUSCL method preserves the stablty propertes of the assocated frst-order scheme. The present MUSCL method s shown to be a smple modfcaton of the standard approach and ts mplementaton s obtaned after few lnes of code see the examples gven n secton 3. To llustrate the method, numercal tests are proposed. They are performed usng two dstnct numercal flux functons. Several slope reconstructon are detaled. These slopes are based on conservatve and non-conservatve varables. The results llustrate the nterest of the method and establsh that the order of accuracy of the orgnal MUSCL scheme s preserved.

23 STABILITY OF THE MUSCL SCHEMES 3 7 a soluton standard mnmod standard superbee frst order 7 b! c 7 7 prmtve entropc conservatve prmtve entropc conservatve 7 7!! Fg Case 3: Relaxaton MUSCL scheme a standard approach wth a prmtve varable reconstructon b modfed mnmod slope computaton for the prmtve, the entropc and the conservatve varables c modfed superbee slope computaton for the prmtve, the entropc and the conservatve varables. Acknowledgments. The author thank B. Nkonga and B. Dubroca for ther helpful suggestons and comments. REFERENCES [] M. Baudn, C. Berthon, F. Coquel, R. Masson, H. Tran, A relaxaton method for twophase flow models wth hydrodynamc closure law, Numer. Math. 99, no 3, pp [] J. B. Bell, P. Colella, J. A. Trangensten, Hgher order Godunov methods for general systems of hyperbolc conservaton laws, J. Comput. Phys. 8, no., pp [3] M. Ben-Artz, J. Falcovtz, A second-order Godunov-type scheme for compressble flud dynamcs, J. Comput. Phys. 55, no., pp [4] C. Berthon, Inégaltés d entrope pour un schéma de relaxaton, C. R. Acad. Sc. I, Math. 34, pp [5] F. Bouchut, Nonlnear stablty of fnte volume methods for hyperbolc conservaton laws, and well-balanced schemes for sources, Fronters n Mathematcs seres, Brkhäuser 4. [6] F. Bouchut, Ch. Bourdaras, B. Perthame, A MUSCL method satsfyng all the numercal entropy nequaltes, Math. Comp. 65, no. 6, pp [7] A. Bourgeade, P. LeFloch, P.-A. Ravart, An asymptotc expanson for the soluton of the generalzed Remann problem. II. Applcaton to the equatons of gas dynamcs, Ann. Inst. H. Poncar Anal. Non Lnare 6, no. 6, pp [8] P. Colella, A drect Euleran MUSCL scheme for gas dynamcs, SIAM J. Sc. Statst. Com-

24 4 C. BERTHON a 5 soluton standard mnmod standard superbee frst order 5 b! c 5 5 prmtve entropc conservatve prmtve entropc conservatve 5 5!! Fg. 3.. Case 3: Knetc MUSCL scheme a standard approach wth a prmtve varable reconstructon b modfed mnmod slope computaton for the prmtve, the entropc and the conservatve varables c modfed superbee slope computaton for the prmtve, the entropc and the conservatve varables. put. 6, no., pp [9] F. Coquel, P. LeFloch, An entropy satsfyng MUSCL scheme for systems of conservaton laws, Numer. Math. 74, no., pp [] F. Coquel, B. Perthame, Relaxaton of energy and approxmate Remann solvers for general pressure laws n flud dynamcs, SIAM J. Numer. Anal. 35, no. 6, pp [] E. Godlewsky, P.A. Ravart, Hyperbolc systems of conservatons laws, Tome, SMAI Eds, Ellpse 99. [] E. Godlewsky, P.A. Ravart, Hyperbolc systems of conservatons laws, Tome, Appled Mathematcal Scences, Vol 8, Sprnger 995. [3] S. K. Godunov, A dfference scheme for numercal computaton of dscontnuous solutons of equatons of fluds dynamcs, Math. Sbornk, No 47, pp [4] A. Harten, P.D. Lax, B. Van Leer, On upstream dfferencng and Godunov-type schemes for hyperbolc conservaton laws, SIAM Revew, Vol 5, No, pp [5] A. Harten, S. Osher, Unformly hgh-order accurate nonoscllatory schemes, SIAM J. Numer. Anal. 4, no., pp [6] B. Khobalatte, B. Perthame, Maxmum prncple on the entropy and second-order knetc schemes, Math. of Comp., Vol 6, No 5, pp [7] P. D. Lax, Hyperbolc systems of conservaton laws and the mathematcal theory of shock waves, Conference Board of the Mathematcal Scences Regonal Conference Seres n Appled Mathematcs, No.. SIAM, Phladelpha 973. [8] P. D. Lax, Shock waves and entropy, Contrbutons to nonlnear functonal analyss, E. H. Zarantonello, Ed., pp [9] B. van Leer, Towards the ultmate conservatve dfference scheme. V. A second-order sequel

25 STABILITY OF THE MUSCL SCHEMES 5 Fg. 3.. D Test wth a 4x4 structured mesh: Relaxaton MUSCL scheme usng modfed superbee slope functon on the prmtve varables. to Godunov s method, J. comput. Phys. 3, pp [] R. J. LeVeque, Fnte volume methods for hyperbolc problems, Cambrdge Texts n Appled Mathematcs. Cambrdge Unversty Press, Cambrdge. [] B. Perthame, Boltzmann type schemes for gas dynamcs and the entropy property, SIAM J. Numer. Anal., Vol 7, No 6, pp [] B. Perthame, Second-order Boltzmann schemes for compressble Euler equatons n one and two space dmensons, SIAM J. Numer. Anal., Vol 9, No, pp [3] B. Perthame, Y. Qu, A varant of Van Leer s method for multdmensonal systems of conservaton laws, J. Comput. Phys., no., [4] E. Tadmor, A mnmum entropy prncple n the gas dynamcs equatons Appl. Numer. Math., no. 3-5, pp [5] A. Kurganov, E. Tadmor, Soluton of two-dmensonal Remann problems for gas dynamcs wthout Remann problem solvers, Numer. Methods Partal Dfferental Equatons 8, no. 5, pp [6] E. F. Toro, Remann solvers and numercal methods for flud dynamcs. A practcal ntroducton Second edton. Sprnger-Verlag, Berln 999.