Monotonization of flux, entropy and numerical schemes for conservation laws

Transcription

1 J Math Aal Appl wwwelsevercom/locate/jmaa Mootozato of flx, etropy ad mercal schemes for coservato laws Admrth a,, GD Veerappa Gowda a, Jérôme Jaffré b a TIFR Cetre, PO Box 134, Bagalore 56001, Ida b INRIA-Rocqecort, B P 105, Le Chesay Cedex, Frace Receved 31 March 008 Avalable ole 0 May 008 Sbmtted by V Radlesc Abstract Usg the cocept of mootozato, famles of two step ad -step fte volme schemes for scalar hyperbolc coservato laws are costrcted ad aalyzed These famles cota the FORCE scheme ad gve a alteratve to the MUSTA scheme These schemes ca be exteded to systems of coservato law Crow Copyrght 008 Pblshed by Elsever Ic All rghts reserved Keywords: Fte volmes; Fte dffereces; Rema solvers; Coservato laws; Mootozato 1 Itrodcto Let f : R R be a Lpschtz cotos fcto ad cosder the scalar coservato law t + f x = 0, t >0, x R, x, 0 = 0 x, x R, where 0 BVR L 1 loc R Ths problem has bee extesvely stded the last decade I geeral problem 11 ad 1 does ot admt reglar soltos becase of two reasos: The characterstcs do ot fll the etre space, gvg rase to empty regos where the soltos caot be determed by the tal data The characterstcs tersect ad lead to mltvaled soltos I order to overcome these dffcltes, the cocept of wea soltos s trodced To costrct sch soltos wth smple data, oe faces dffcltes ad I order to overcome, oe mae se of the varace property of Eq 11, that s t s varat der the trasformato x αx, t αt for ay α>0 Ths leads to a self smlar 11 1 * Correspodg athor E-mal addresses: adt@mathtfbgres Admrth, gowda@mathtfbgres GD Veerappa Gowda, jeromejaffre@rafr J Jaffré 00-47X/$ see frot matter Crow Copyrght 008 Pblshed by Elsever Ic All rghts reserved do:101016/jjmaa

2 48 Admrth et al / J Math Aal Appl solto the varable ξ = x/t s called the rarefacto wave Ths solto s ot determed by the tal data ad s sed to determe the solto the empty space I order to overcome, dscottes are trodced the regos where the characterstcs tersect Now sg the defto of wea soltos, t s possble to determe the les of dscotty called shocs Shocs mst satsfy the Rae Hgoot codto coectg the taget to the le of dscotty wth the jmps of the solto across t The ext problem s the oqeess of wea soltos Ths oqeess bascally comes from the way oe flls the empty rego by a solto Now by loog at the derlyg physcal pheomea a cocept of a Etropy crtero s trodced whch qely defes the solto the empty rego for smple data Usg the above cocepts, exstece of a qe etropy solto s proved for arbtrary data There are three fdametal methods sed to obta the exstece ad qeess of a etropy solto 1 The Hamlto Jacob method Here oe assmes that the flx f s C ad strctly covex ad cosders the Hamlto Jacob eqato: v t + fv x = 0, x R, t>0, 13 vx,0 = v 0 x, x R, 14 where v 0 x = x 0 0θ dθ Ths eqato has a qe vscosty solto ad a explct formla was gve by Hopf [] The lettg = x v, Lax ad Ole proved that s the qe etropy solto of Eqs 11, 1 [] The vashg vscosty method Here oe loos at the olear parabolc eqato t + f x = ε xx, x R, t>0, x, 0 = 0 x, x R Krzov [4] proves that there exsts a qe solto ε of Eqs 15, 16 covergg L 1 loc to a qe etropy solto of Eqs 11, 1 as ε 0 3 Nmercal schemes I ths method mercal schemes are derved sg space ad tme dscretzatos of Eqs 11, 1 These scheme calclate a approxmato solto whch coverges to the qe etropy solto Lax ad Fredrchs, Godov, Eqst ad Osher, Roe ad others cotrbted to ths approach I ths paper we cocetrate o mercal schemes to solve Eqs 11, 1 ad we wll costrct schemes whose mercal flxes ca be evalated by pot evalatos of the flx fcto f cotrarly to may mercal schemes whch mercal flx evalatos volve calclatos of tegrals, maxma or mma of f Ths property of sg oly pot evalatos of the mercal flx s crcal for extedg wthot too mch complexty a mercal scheme to systems I Secto we trodce the cocept of mootozato whch leads s to a ew defto of etropy solto Ths approach ca lead to the cocept of etropy for systems I Secto 3 we costrct a frst two step mootozato scheme whch s actally the Force scheme [5,6] I Secto 4 ths scheme s geeralzed to a famly of two step mootozato schemes ad aalyzed I Secto 5 these mercal schemes are exteded to a famly of -step mootozato schemes whch gves a alteratve to the MUSTA scheme [7] These schemes are easy to exted to systems Secto 6 Mootozato ad etropy A basc gredet stdyg a mercal scheme s the stdy of the Rema problem Let a,b,x 0 R ad let x = { a f x<x0, b f x x 0, 1 ad the solto set Ra,b,x 0 assocated to 1 s gve by Ra,b,x 0 = { ; s a wea solto of Eqs 11, 1 }

3 Admrth et al / J Math Aal Appl I geeral Ra,b,x 0 ca have more tha oe solto Ths set Ra,b,x 0 trs ot to be a mportat tool dervg fte volme schemes Ths s a well stded method for detals see [3] Let h>0, t > 0, = t/h ad dscretze the doma to dsjot rectagles R wth legth h ad breadth t That s for Z, x 0letR = I J where x +1/ = h, t = t, I =[x +1/,x +3/, J = [ t, + 1 t mst satsfy the CFL codto: let sp f θ 1, θ [ M,M] where M = 0 We dscretze 0 by { 0 } Z defed as 0 = 1 0 x dx h I ad we assme that for 0 j, { j } Z are ow I order to defe { +1 } Z, we choose a solto w R, +1,x +3/ for each Z ad defe wx,t = w x, t for x, t R The from the CFL codto, w s well defed ad s a solto of Eq 11 for x R, t<t<+ 1 t wth tal codto wx,t = for x I Now we defe { +1 } Z as +1 = 1 wx,t +1 dx h I Sce w satsfes 11 R t, + 1 t ad hece tegratg 11 over R to obta the followg formla +1 t +1 = 1 f wx +3/,t dt 1 +1 t f wx +1/,t dt 3 t t t The evalato of +1 from 3 depeds heavly o the choce of the Rema problem solto {w } Hece geeral dfferet choces of {w } gve rase to dfferet sets of { } I fact oe ca geerate ftely may L 1 -cotractve coverget schemes provded f has both creasg as well as decreasg parts [1] If the flx f s strctly mootoe the the characterstcs of Eq 11 do ot tersect the les x = x +1/ for t t<+ 1 t Hece for ay w R, +1,x +3/ { w x f f > 0, +3/,t= +1 f f < 0, ad scheme 3 redces to the stadard pstream scheme { +1 = f +1 f f f < 0, f f +1 f f 4 > 0 As a coseqece of ths, the scheme s depedet of the choce of the Rema data solto {w } ad coverges L 1 loc to the qe etropy solto If f s strctly covex ad the {w } are chose so that they satsfy the Lax Ole, Krzov etropy codto, the scheme 3 redces to the fte volme scheme +1 = F G +1, F G 1, 5 where F G a, b s the Godov flx defed by { F G mθ a,b fθ f a b, a, b = 6 max θ a,b fθ f a b t

4 430 Admrth et al / J Math Aal Appl I fact from a theorem of Ole see [3], eve f f s ot covex, the Godov scheme 5, 6 coverges L 1 loc to the qe etropy solto of problem 11, 1 Besdes ts optmal propertes terms of mercal dffso, a drawbac of the Godov scheme s that ts mercal flx 6 caot be wrtte terms of pot vales of the fcto f, a pot whch becomes crtcal whe extedg the method to systems O the other had we otce from scheme 4 that f the flx s mootoe, the the choce of the Rema problem solto s rrelevat ad the flx ca be calclated terms of pot vales of f Therefore we covert problem 11 to a problem havg a mootoe flx fcto Ths procedre s called mootozato Gve M>0, α R, deote f α by f α = f /α ad choose α sch that sp [ M,M] f < α Let be a solto of Eqs 11, 1 wth 0 M, ad cosder the chage of varables x = X + ατ, t = τ, x,t = vx,τ The v satsfes v τ + f 1/α v X = 0 for X R, τ>0, 8 vx,0 = 0 X for X R 9 Frthermore vx,0 = 0 x M ad from 7, f 1 v = fv αv s a strctly mootoe fcto α for v [ M,M] Hece the fte volme scheme for Eqs 8, 9 does ot deped o the choce of the Rema data solto Frthermore t prodces a solto v h whch coverges L 1 loc to the qe etropy solto v of problem 8, 9 Therefore x, t = vx αt,t s the qe etropy solto for problem 11, 1 Coseqetly we ca state the followg alteratve defto of the etropy solto Defto Etropy solto Let L 1 loc L be a wea solto of problem 11, 1 The s sad to be a etropy solto to ths problem, f x, t = vx αt,t for all α sch that f 1/α s strctly mootoe [ 0, 0 ] ad v s the qe solto of problem 8, 9 obtaed after covergece of the solto of the pstream fte volme scheme 4 appled to problem 8, 9 I the above aalyss we chage the varables so that the ew varables, the flx fcto becomes strctly mootoe Ths allows s to redce the fte volme scheme o rectaglar space-tme meshes to a smple pstream mercal scheme Now the ext secto we reverse the order, amely we eep the eqato as t s bt chage the rectaglar mesh to parallelogram meshes to obta a mercal scheme whch les betwee Godov ad Lax Fredrchs schemes terms of mercal vscosty 3 A frst two step mootozato scheme 31 Formlato of the two step mootozato scheme We assme aga that the tal data satsfes 0 [ M,M] for some M>0ad we trodce some more otato x = + 1/h, t +1/ = + 1/ t, p = x +1/, t, p +1/ = x,+ 1/ t, P +1/ = parallelogram wth vertces p,p +1,p+1/ +1,p +1/, P = parallelogram wth vertces p +1/,p +1 +1,p+1,p +1/ 1, asshowfg31 We assme that all the characterstcs emaatg from p respectvely p +1/ ], [p 1,p+1/ ] respectvely [p +1/ ], [p+1/ [p,p+1/ sp [ M,M] 1 f 1,p ,p +1 7 do ot tersect the le segmets ] Ths mples the followg codto 31

5 Admrth et al / J Math Aal Appl Fg 31 Notato for the two step mootozato scheme Wth the above otato ad assmpto, we ca ow derve the two step mootozato scheme Assme that { } Z for 0, are gve wth [ M,M] As Secto let w R, +1,x +3/ whch eed ot be a etropy solto be ay solto ad defe wx,t = w +1/ x, t for x, t P The from codto 31, w s a well-defed wea solto ad let +1/ = 1 h x +1 x w x, + 1/ t dx Mass coservato the cells gves 0 = wt + fw x dxdt = P +1/ P +1/ wνt + fwν x ds Evalatg the tegral o the bodary of P +1/ ad sg the CFL codto 31 characterstcs do ot tersect the le segmets, see Fg 31 we obta +1/ = = [ = t p +1/ +1 p +1 [ f fw 1 w dt 1 t [ f +1 f ] p +1/ p f 1 fw 1 ] w dt ]

6 43 Admrth et al / J Math Aal Appl Now repeatg the argmet wth data { +1/ } at t +1/ ad +1 = h 1 x+3/ x +1/ wx, + 1 t dx we have +1 = +1/ 1 [ f +1/ = +1/ / +1/ [ f +1/ Ths the two step mootozato scheme reads +1/ = + +1 [ ] f +1 f, +1 = +1/ / [ f +1/ f +1/ 1 f +1/ 1 f +1/ 1 ] +1/ ] 1 ] 3 Observe that there s a bacward shft evalatg +1 ad that t eeds oly pot evalatos of the flx fcto f The two step scheme 3 was already trodced ad aalyzed [5,6] ad t ca be wrtte a compact form as follows For a,b R let Ha,b= a f b f a = a + b fb fa 33 The scheme 3 ca be rewrtte as or +1/ = [ f +1 f ], +1 = +1/ 1 [ f +1/ +1/ = H, +1, +1 +1/] f 1, 34 = H +1/ 1, +1/ so we formlate the two step mootozato scheme the compact form = H H 1,,H, Covergece of the two step mootozato scheme Cocerg covergece we have the ma reslt Theorem 31 Let 0 BVR ad M The der the CFL codto 31 the two step fte volme scheme { } gve 35 coverges to the qe etropy solto Proof Let a,b [ M,M], the from 31 we have H a a, b = f a 0,, H b a, b = 1 1 f b 0 36 ad hece H s a odecreasg fcto each of ts argmet Therefore from 35 the scheme s a three pot mootoe scheme Let gx,y,z = HHX,Y,HY,Z, the gx, X, X = H HX,X,HX,X = HX,X= X 37 The scheme s L -stable, sce, whe assmg [ M,M] for all Z, from 36 we ca wrte M = g M, M, M g 1,, +1 = +1 gm,m,m = M

7 Admrth et al / J Math Aal Appl Fally let s wrte the scheme coservatve form Expadg 35 we obta +1 f H, +1 f H 1, = H 1, = 1 [ f + f H, +1 f 1 f H 1, ] = [ f + f H, +1 + f 1 f H 1, ] 1 Therefore +1 = [ F, +1 F 1, ] 38 where the mercal flx F a, b s gve by F a, b = 1 [ ] a f a + f Ha,b + = 1 [ fa+ f Ha,b Ha,b + a ] = 1 [ fa+ f Ha,b b a + 1 ] fb fa, or F a, b = 1 [ fa+ fb+ f Ha,b + a b ] 39 4 From 37 F a, a = fa so the flx s cosstet ad coseqetly the solto of the two step mootozato scheme 3 whch ca be wrtte alteratvely as 33, 35 or 38, 39 coverges to the qe etropy solto of problem 11, 1 Ths proves the theorem 33 Comparso wth the Lax Fredrchs LF ad the two step Lax Wedroff Rchtzer LWR scheme O oe had the two step mootozato scheme 38, 39 gves +1 = [ f f H, +1 f 1 f H 1, ] 310 O the other had the two step LWR scheme pt α = β = 1/ Eq 19 of [3] s gve by w +1 = f H +1, f H 1,, whle the LF scheme reads v +1 = f +1 f 1 It follows that +1 = w+1 + v +1 Hece the solto gve by scheme 38, 39 s the average of that gve by the LF ad LWR schemes Ths remar was already made by Toro [5] Therefore eve thogh the LWR scheme s ot L -stable, by tag ts average wth the L -stable LF scheme we obta a L -stable coverget scheme I terms of mercal vscosty, the mercal vscosty coeffcet Q +1/ of the two step mootozato scheme 38, 39 s determed by Hece F a, b = 1 fa+ fb Q +1/ b a

8 434 Admrth et al / J Math Aal Appl Fg 3 Comparso of the Godov, Lax Fredrch ad the two step mootozato scheme for f= / wth tal codto x, 0 = f x<0ad1fx>0 The solto s show at t = 1 ad was calclated wth h = /100, t = /300 Therefore b a Q +1/ = Q +1/ = fa+ fb F a, b = fa+ fb 1 fa+ fb+ f Ha,b + a b = 1 a b fa+ fb f Ha,b = 1 fa+ fb f a f b f a a b = 1 fa+ fb fa+ f ξ f b f a a b = 1 fb fa + f ξ fb fa b a f ξ + b a = f ξ fb fa f ξ b a 1 + f ξ fb fa b a ad QLF +1/ Let f= ad deote by Q G +1/ Fredrchs schemes respectvely The we have Q G +1/ = 1 + = Q +1/ 1 = Q LF +1/ f ξ the mercal vscosty coeffcet of the Godov ad the Lax Ths shows that the performace of scheme 38, 39 s better tha the Lax Fredrchs scheme terms of mercal vscosty as ca be observed o Fg 3 4 Geeralzed two step mootozato schemes There are may ways to geeralze the two step mootozato preseted the prevos secto I ths secto we geeralze t to a famly of two step schemes Let γ 1,γ [0, 1] satsfyg γ 1 + γ = 1 ad β 1,β [ 1, 1] Gve the dscretzato steps h, t of space ad tme, we frther dscretze tme by dvdg the tme step to two sbsteps γ 1 t, γ t, ad we move the space dscretzato pot p to p +1/ by legth β 1 h at tme t + γ 1 t ad frther move p +1/ by legth β h bac to oe of the dscretzato pots at tme t + γ 1 + γ t = t +1 See Fg 41 for γ 1 = 1/3, γ = /3, β 1 = 1/, β = 1/ ad Fg 31 for γ 1 = γ = 1/, β 1 = β = 1/ I ths way we bld le segmets whch, addto of the les t = t, t = γ 1 t, t = t +1 wll form the bodares of the cotrol volmes for the two step fte volme sheme

9 Admrth et al / J Math Aal Appl Fg 41 Cotrol volmes for a geeralzed two step mootozato scheme wth γ 1 = 1/3, γ = /3, β 1 = 1/, β = 1/ γ l,β l, l = 1,, are chose also order to satsfy the CFL codto γ l sp f m β l, 1 β l, l = 1,, 41 [ M,M] order to esre that, for l = 1, the characterstcs leavg p, ad for l =, that leavg p+1/ do ot tersect the le segmets For l = 1,, let δ l = γ l β l δ 1 s the slope of the segmet [p,p+1/ ] ad δ s the slope of the segmet coectg p +1/ to oe of the dscretzato pots at tme t +1 For a,b R defe { a γl f δl b f δl a f δ l > 0, H δl a, b = l = 1, 4 b γ l f δl b f δl a f δ l < 0, We ca geerate for dfferet famles of coverget schemes, depedg o the sgs of the β s Scheme 1 β 1 0, β 0, β 1 + β = 1 If M, from the CFL codto 41, Scheme 1 reads +1/ = H δ1, +1 = γ 1 f δ1 +1 fδ1, +1/ = H δ 1, +1/ +1/ = 1 γ +1/ +1/ f δ fδ 1 +1 Note that the case β l = γ l = 1/, l = 1,, correspods to the two step scheme preseted the prevos secto Scheme β 1 > 0, β < 0 wth β 1 = β

10 436 Admrth et al / J Math Aal Appl If M, der the CFL codto 41, Scheme reads +1/ = H δ1, +1 = γ 1 f δ1 +1 fδ1, +1/ = H δ 1, +1/ +1/ = γ +1/ f δ +1 fδ +1/ 1 Ths case correspods to the stato show Fg 41 The other two cases are β 1 0, β 0, β 1 + β = 1 ad β 1 < 0, β > 0, β = β 1 ad they ca be dealt exactly as above 5 Geeralzed -step mootozato schemes We ow geeralze the method to steps Let 1 be a teger ad for l = 1,,,let0 γ l 1 satsfyg l=1 γ l = 1 We trodce sbtervals of t,t +1 deoted by [t + l,t + l+1 ], l = 0,, 1, wth t + l = t + l l=1 γ l t, l = 1,, Let X 1 <X <X 3 be ay three cosectve space dscretzato pots Ths they satsfy X 3 X = X X 1 = h We ow trodce admssble crves ρ :[t,t +1 ] R s sad to be a admssble crve f 1 ρ s cotos ad ρt = X, ρ [t + l,t + l+1 ] s a le segmet for 0 l 1, 3 ρt +1 {X 1,X,X 3 } Examples of admssble crves are show Fgs 31, 41, 51, 5 Deote ΓX 1,X,X 3,γ 1,,γ,= { ρ : [t,t +1, ] [X 1,X 3 ]; ρ s admssble }, Γ + X 1,X,X 3,γ 1,,γ,= { ρ ΓX 1,X,X 3,γ 1,,γ,; ρt +1 = X 3 }, Γ 0 X 1,X,X 3,γ 1,,γ,= { ρ ΓX 1,X,X 3,γ 1,,γ,; ρt +1 = X }, Γ X 1,X,X 3,γ 1,,γ,= { ρ ΓX 1,X,X 3,γ 1,,γ,; ρt +1 = X 1 } For ρ ΓX 1,X,X 3,γ 1,,γ,we deote by δ l,l= 1,, the slopes of the le segmets of ρ o the terval [t + l,t + l+1 ], l = 0,, 1, ad the assocated β l are defed by β l = γ l δ l For each Z let ρ Γx 1/,x +1/,x +3/,γ 1,,γ,satsfyg j {, 0, +} sch that ρ Γ j x 1/,x +1/,x +3/,γ 1,,γ, Z The slopes of the ρ s are the same for all s ad are deoted by δ 1,,δ The ρ s wll be the lateral bodares of the cotrol volmes sed to defe the fte volme scheme We assme that {ρ } Z satsfy the CFL codto sp f m β l,1 β l for 1 l 51 γ l [ M,M] Wth the otatos as 4 we ca ow defe the -step scheme as follows Gve { } wth M, defe dctvely for 1 l 1, l 1 l 1 + l + + H δl,+1 f δ l > 0, = l = 1,, 5 l 1 l H δl, f δ l < 0, 1

11 Admrth et al / J Math Aal Appl Fg 51 Cotrol volmes for a 3-step mootozato scheme, γ 1 = γ = γ 3 = 1 3, β 1 = 1 3, β = 5 6, β 3 = 1 ρ Γ x 1/,x +1/,x +3/,γ 1,γ,γ 3, Fg 5 Cotrol volmes for a 3-step mootozato scheme, γ 1 = γ = γ 3 = 1 3, β 1 = 1, β = 1 3, β 3 = 5 6 ρ Γ + x 1/,x +1/,x +3/,γ 1,γ,γ 3,

12 438 Admrth et al / J Math Aal Appl The +1 = H δ H δ H δ H δ , f ρ t +1 = x +3/, +,+1 f ρ t +1 = x 1/, +, f ρ t +1 = x +1/ ad δ < 0, +,+1 f ρ t +1 = x +1/ ad δ > 0 As Theorem 31, der the CFL codto 51, t follows easly that H δl a, b s mootoe each of ts varable ad H δl a, a = a Hece the scheme 5 ad 53 coverges to a qe etropy solto of problem 11, 1 f 0 M ad 0 BVR Example 1 If δ l 0, l = 1,,, the scheme 5, 53 ca be wrtte as follows Defe for l 3, H X 1,X,X 3 = H δ Hδ1 X 1,X, H δ1 X,X 3, H l X 1,X,,X l+1 = H δl H l 1 X 1,,X l, H l 1 X,,X l+1, ad FX 1,,X = γ 1 f δ1 X 1 + γ f δ H1 X 1,X X 1 + γ l+1 f δl+1 H l X 1,,X l+1 54 l= The +1 = F, +1 +, F 1, + 55 If =, β l = γ l = 1 the scheme 55 cocdes wth scheme 38, 39 If β l = γ l = 1 for 1 l, the the CFL codto 51 gves sp f 1 m [ M,M], 1 1 = 1 Hece the CFL codto reads ow sp [ M,M] f 1 If =, δ l > 0, l = 1,, the scheme 55 ca be wrtte as +1/ = H δ1, +1 = γ 1 f δ1 +1 fδ1 = 1 β 1 + β 1 +1 γ 1 f +1 f, +1 = 1 β +1/ 1 + β +1/ γ f +1/ +1/ f 1 Frthermore f we let β 1 = β = 1, γ 1 = 3 4, γ = 1 4, the the CFL codto 51 becomes sp f /3 [ M,M] We cosder ow the case whe the slopes δ l s chage sg Let = 3, γ 1 + γ + γ 3 = 1, β 1 = β + β 3 wth β 0, β 3 0, the Let +1/3 +/3 = γ 1 f δ1 +1 fδ1, = +1/3 γ +1/3 +1/3 f δ fδ 1, = +/3 γ 3 +/3 +/3 f δ3 fδ FX 1,X,X 3 = γ 1 f δ1 X 3 + γ f δ Hδ1 X,X 3 + γ 3 f δ3 Hδ Hδ1 X 1,X, H δ1 X,X 3, the the scheme reads +1 = F 1, +1, F, 1, 53

13 Admrth et al / J Math Aal Appl Exteso to systems Cosder a hyperbolc system of coservato laws U t + FU x = 0, x R, t>0, Ux,0 = U 0, x R, where U s a -vector ad F : R R a C 1 -map For α R let X = x + αt, τ = t, VX,τ= Ux,t The V satsfes 61 6 V τ + F V αv X = 0, X R, τ>0, 63 VX,0 = U 0 X, X R 64 If U s a egevale of F U, the U α s a egevale of F U αi Hece f the egevales of F U are boded, the we ca choose α large eogh sch that all the egevales correspodg to 63 are postve Therefore, f we have a L -bod for a solto of 61, 6 the we ca covert t to a solto of 63, 64 wth all egevales postve Frthermore f we defe F α U = FU U α for α 0, the we ca defe the scheme 34 for the system 61, 6 provded that all the waves are trapped as before I the same way -step schemes ca be defed for systems Ther advatage s that they are pot evalato schemes Oe ca expect a better accracy by gog to -step schemes ad choosg proper γ l s ad δ l s Ths may also exted to the mltdmesoal case ad wth a dffso term o the rght-had sde 7 Coclso Usg the techqe of mootozato we showed how to costrct a famly of mltstep schemes wth oly pot vale evalatos of the flx fcto Ths famly cldes the Force ad proposes a alteratve for the Msta schemes For all these schemes we proved covergece of the approxmate solto to the etropy solto of the cotos problem We also gave hts o how to exted them to systems ad hgh resolto schemes I forthcomg papers we wll exted these schemes to hgher resolto schemes ad to the dscotos flx case We wll also gve a example of applcato to a system of coservato laws represetg a problem of polymer floodg Acowledgmet The athors acowledge fdg from Ido-Frech Cetre for Promoto of Advaced Research der Project Refereces [1] Admrth, S Mshra, GD Veerappa Gowda, Optmal etropy soltos for coservato laws wth dscotos flx-fctos, J Hyperbolc Dffer Eq [] LC Evas, Partal Dfferetal Eqatos, Grad Std Math, vol 19, Amerca Mathematcal Socety, 00 [3] E Godlews, P-A Ravart, Hyperbolc Systems of Coservato Laws, Trmestvell, 1991 [4] SN Krzov, Frst order qaslear eqatos several depedet varables, Math USSR Sb [5] EF Toro, Rema Solvers ad Nmercal Methods for Fld Dyamcs, Sprger-Verlag, 1999 [6] EF Toro, SJ Bllet, Cetered TVD schemes for hyperbolc coservato laws, IMA J Nmer Aal [7] EF Toro, MUSTA: A mlt-stage mercal flx, Appl Nmer Math