ENTROPY CONSERVATIVE SCHEMES AND ADAPTIVE MESH SELECTION FOR HYPERBOLIC CONSERVATION LAWS

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1 Journal of Hyperbolc Dfferental Equatons Vol. 7, No. 3 () c World Scentfc Publshng Company DOI:.4/S ENTROPY CONSERVATIVE SCHEMES AND ADAPTIVE MESH SELECTION FOR HYPERBOLIC CONSERVATION LAWS CHRISTOS ARVANITIS and CHARALAMBOS MAKRIDAKIS, Department of Appled Mathematcs Unversty of Crete, 749 Heraklon-Crete, Greece Insttute of Appled and Computatonal Mathematcs FO.R.T.H. Heraklon, 7, Greece NIKOLAOS I. SFAKIANAKIS Department of Mathematcs Unversty of Venna, 9 Venna, Austra Receved Aug. 9 Accepted 5 Jun. Communcated by P. G. LeFloch Abstract. We consder numercal schemes whch combne non-unform, adaptvely redefned spatal meshes wth entropy conservatve schemes for the evoluton step for shock computatons. We observe that the resultng adaptve schemes yeld approxmatons free of oscllatons n contrast to known fully dscrete entropy conservatve schemes on unform meshes. We conclude that entropy conservatve schemes are transformed to entropy dmnshng schemes when combned wth the proposed geometrcally drven mesh adaptvty. Keywords: Adaptve mesh refnement; entropy conservatve schemes; non-classcal shocks. Mathematcs Subject Classfcaton : 65M5. Introducton We consder scalar conservaton laws n one space dmenson, u t + f(u) x =, x [a, b], t [,T], (.) wth ntal data u havng compact support n a much smaller nterval. We are nterested n studyng numercal approxmatons of the soluton u of (.) overnon-unform, adaptvely redefned spatal meshes. Adaptvty s a man theme n modern scentfc computng of complex physcal phenomena. It s therefore mportant to nvestgate the behavor of adaptve schemes for hyperbolc 383

2 384 C. Arvants, C. Makrdaks & N. I. Sfakanaks problems, such as (.), whch exhbt several nterestng and not trval characterstcs. In ths work, we nvestgate the behavor of certan geometrcally drven adaptve algorthms when combned wth the mportant class of entropy conservatve schemes of Tadmor. The adaptve schemes we wll consder have the followng structure: n every tme step t = t n the mesh that we consder s: M n x = {a = x n < <x n N = b} wth varable space step szes h n = x n + xn, =,...,N. We note moreover, that the mesh s reconstructed n every tme step. We consder also numercal approxmatons U n of the soluton u over the mesh Mx n at tme t = t n wth U n = {U n,...,un N }. The creaton and manpulaton of such non-unform meshes and the tme evoluton of the approxmate solutons s dctated by the Man Adaptve Scheme (MAS) whch n short s descrbed by the followng procedures In every tme step, construct new mesh/space; Update numercal approxmatons over the new space; Evolve n tme wth the numercal scheme. In the next secton, we shall dscuss n detal the man adaptve scheme, we just note here that we use a fxed number of spatal nodes and that the mesh reconstructon procedure s based on nformaton attaned by the geometry of the numercal approxmaton tself. Applcatons of the man adaptve scheme on several problems, pont out a strong stablzaton property emanatng from the mesh reconstructon [, 3, 4]. Ths stablzaton property led us to combne MAS wth the margnal class of entropy conservatve schemes. Entropy conservatve schemes were frst ntroduced by Tadmor [5] and further studed by LeFloch and Rhode [], Tadmor [6], Lukáčová-Medvďová and Tadmor [3]. They are sem-dscrete numercal schemes whch satsfy an entropy equalty. On one hand these schemes are nterestng on ther own rght, for they appear n the context of zero dsperson lmts, complete ntegrable systems and computaton of non-classcal shocks. On the other hand, they are mportant as buldng blocks for the constructon of entropy stable schemes [3, 5, 6]. Out of ther propertes we shall focus on the followng: explct n tme dscretzaton of the entropy conservatve schemes leads to entropy producton and numercal nstabltes occur n the form of oscllatons [6]. The am of ths work s to combne the man adaptve scheme wth entropy conservatve schemes for the evoluton step. We note n advance that the stablzaton propertes of MAS are able to tame out the oscllatons produced by the fully dscrete entropy conservatve schemes; hence we conclude that the entropy dsspaton due to proper mesh reconstructon counterbalances the entropy producton due to explct n tme dscretzaton. In other words the man concluson of ths work s

3 Adaptve Mesh Selecton 385 that entropy conservatve schemes are transformed to entropy dmnshng schemes when combned wth the geometrcally drven mesh redstrbuton of Sec.. Ths work s dvded n the followng sectons, n Sec., we present the MAS and gve some detals regardng the steps and the mplementaton of ths scheme. In Sec. 3, we are followng [5, 6] and we brefly overvew the noton of entropy varables and the constructon of entropy conservatve sem-dscrete schemes. In Sec. 4, we nvestgate the computatonal behavor of the schemes by consderng several numercal tests/problems suggested n [, 6]. In each problem, we construct the respectve sem-dscrete schemes and dscuss on ther fully dscrete versons. We shall wtness, as mentoned above, that the combnaton of MAS wth fully dscrete, explct n tme entropy conservatve schemes, tames out oscllatons. In addton, the applcaton of adaptvty to entropy conservatve schemes desgned to capture non-classcal behavor lead to approxmatons of smlar type to entropy dmnshng schemes. Thus, adaptve entropy conservatve schemes fal on resolvng non-classcal shocks.. The Man Adaptve Scheme We focus n ths secton on the study of the Man Adaptve Scheme (MAS) and ts propertes. To start wth, we note that n the unform mesh case the evoluton of the numercal approxmatons s dctated solely by the numercal scheme under consderaton. In the contrary, n the non-unform case, the scheme conssts of three ntermedate procedures n each tme step. These comprse the Man Adaptve Scheme (MAS). Defnton. (MAS). Gven mesh Mx n = {a = x n < < x n N = b} and approxmatons U n = {U n,...,un n }, Step. (Mesh reconstructon) Construct new mesh Mx n+ = {a = x n+ < <x n+ N = b}. Step. (Soluton update) Gven Mx n,un and Mx n+ a. construct a functon V n (x) such that V n (x n )=U n; b. compute/update approxmatons Û n = {Û n,...,û N n n+ } over Mx. Step 3. (Tme evoluton) Gven Mx n+, Û n march n tme to compute U n+ = {U n+,...,u n+ N }. A frst comment s that n the unform mesh case the mesh does not change wth tme so there s no need for mesh reconstructon Mx n+ (no Step ); hence there s no need for soluton update (no Step ). In ths case the MAS (Defnton.) reduces to tme evoluton step (Step 3) whch s just the the usual numercal scheme over a unform mesh. The extra steps of the non-unform MAS on one hand change sgnfcantly both the computaton and the analyss of the numercal approxmatons U n and on the other hand, under condtons, are responsble for stablzaton propertes of the MAS.

4 386 C. Arvants, C. Makrdaks & N. I. Sfakanaks The use of non-unform adaptvely redefned meshes, n the context of fnte dfferences, has been studed n the past; we menton for nstance the work of Harten and Hyman [9], Dorf and Drury [6] and Tang and Tang [7], among others. See also []. In these works however standard entropy stable schemes were consdered. The approach that we shall follow, for the mesh reconstructon step of MAS (Step ) was frst ntroduced by Arvants, Katsaouns and Makrdaks [] and by Arvants [4]. Ths approach s dfferent n the sense that t utlzes geometrc nformaton attaned from the numercal soluton and redstrbutes a fxed number of nodes accordng to an equ-dstrbuton prncple. Propertes and varatons of MAS procedure have been studed later n [, 3, 5]. We start the dscusson of MAS wth the mesh reconstructon procedure (Step ). Ths procedure s a way of relocatng the nodes of the mesh accordng to the geometrc nformaton contaned n the dscrete numercal soluton. The basc dea s smple and geometrc: n areas where the numercal soluton s smoother/flatter the densty of the nodes s low, n the contrary n areas where the numercal soluton s less smooth/flat the densty of the nodes should be hgher In more detal, at a gven tme step, the basc prncples of the mesh reconstructon procedure (Step of MAS) are: (a) Locate the regons of space where ncreased accuracy s demanded, through a postve functonal g Uh of the approxmate soluton U h. (b) Fnd a partton {x n } N n= of the space Ω wth predefned cardnalty N, and densty that follows the estmator functon g Uh,.e. the new mesh satsfes the equdstrbuton prncple: g Uh = g [x,x +] N Uh, =,,...,N. Ω Examples of geometrc estmator functons that have been used are the arclength estmator, the gradent estmator and the curvature estmator. For reasons deployed n [4, 5], we use the curvature of the approxmate soluton as estmator functon whch, for the case of smooth functon U s defned as follows: U (x) g U (x) =. (.) ( + (U (x)) ) 3 At the dscrete level, gven a partton {x } N =, we approxmate the curvature g Uh (x ) of the dscrete soluton {U h, } N =,wthg U h, gven by g Uh, = R(A,A,A + )

5 Adaptve Mesh Selecton 387 where R(A,A,A + ) s the radus of the crcumcrcle that passes through the three ponts A j =(x j,u h,j ),j =,,+,thats (A + A ) (A A ) g Uh, =, =,...,N (.) A + A A + A A A For equdstant ponts A,A,A +,relaton(.) s a second order approxmaton of the curvature (.). Closng wth the mesh reconstructon step, we note that one can easly prove that the computatonal cost of constructng a new mesh wth ths procedure s O(N), where N s the fxed number of spatal nodes. We now move to the soluton update procedure (Step of MAS). We have an old mesh Mx n n+ from the ntaton of the MAS and a new mesh Mx from the mesh reconstructon procedure (Step of MAS). The approxmate soluton U n s defned over the old mesh Mx n, and we want to redefne the approxmate soluton over the new mesh Mx n+. To ths end we consder vertex centered grds. That s, we consder a partton of the doman n cells C n defned as: C n =[x n /, xn +/ ), xn +/ = xn + xn +. Over the old set of cells C n we construct the pecewse constant functon V n (x) that satsfes V n (x n )=U n. The set of cells C n s defned through the old mesh Mx n. Smlarly we consder a new set of cells C n+ defned through the new mesh Mx n+ as follows: C n+ =[x n+ /, xn+ +/ ), +/ = xn + xn+ +. xn+ Now we can compute the updated values Û n = {Û n,...,û N n } n a mass conservatve way []. Defnton. (Conservatve constructon of Û n ). We take nto account the confguraton of the new nodes x n+ wth respect to the old cells C n.sowehave the followng cases: If the confguraton of the new nodes x n+, xn+, x n+ + s such that x n+ / Cn k and x n+ +/ Cn l, wth k<l, we defne û n = x n+ (x n k+/ xn+ / )un k + l j=k+ x n j un j +(xn+ +/ xn l / )un l If the confguraton of the new nodes x n+, xn+, x n+ + s such that x n+ /,xn+ +/ Cn k,.

6 388 C. Arvants, C. Makrdaks & N. I. Sfakanaks we defne û n = un k. Ths constructon s conservatve, respects the maxmum prncple and s of lnear complexty wth respect to the fxed number of nodes N. Proposton.3. The constructon defned n Defnton. s conservatve. Proof. The area defned by the new approxmaton û n û n xn+ =(x n k+/ xn+ / )un k + We wrte the followng term û n + xn+ + =(xn l+/ xn+ +/ )un l + }{{} B l j=k+ m j=l+, over the new cell Cn+,s x n j un j +(xn+ +/ xn l / )un l. }{{} A x n j un j +(xn+ +3/ xn m / )un m and sum wth the prevous one û n xn+.wewenotcethatthetermsa and B merge wth the exstng sums and we end up wth the followng rght-hand sde (x n k+/ xn+ / )un k + whch s suffcent for conservaton. m j=k+ x n j u n j +(x n+ +3/ xn m / )un m, We can also prove that ths constructon of Û n respects the maxmum prncple. Proposton.4 (Maxmum prncple). The values of the new pont approxmatons û n are bounded by the maxmum of the old pont approxmatons u n, max û n max u n. Proof. In the case where the confguraton of the new nodes x n+, xn+, x n+ + s such that x n+ / Cn k and x n+ +/ Cn l, wth k<l, the new approxmaton û n s defned as û n = (x n x n+ k+/ xn+ / )un k + l j=k+ x n j un j +(xn+ +/ xn l / )un l.

7 Adaptve Mesh Selecton 389 Snce the ntervals n the enumerator of the rght-hand sde sum up to x n+ we mmedately bound as follows û n max{ u n j,j = k,...,l}. In the case where the confguraton of the new nodes x n+, xn+, x n+ + s such that x n+ /, xn+ +/ Cn k, the new approxmaton û n s defned as û n = un k, so agan the bound û n max{ un j,j = k,...,l} s vald. Fnally, for the tme evoluton procedure (Step 3 of MAS), we use any numercal scheme vald for non-unform meshes. The numercal schemes that we consder n ths work are explct (n tme) dscretzatons of entropy conservatve schemes. 3. Entropy Conservatve Schemes In ths paragraph, we present a short overvew of the constructon of entropy conservatve schemes followng [5, 6]. We wll restrct our attenton to the constructon of 3-pont second order accurate schemes. The constructon starts by assumng that the one dmensonal conservaton law u t + f(u) x = wthf a smooth, convex flux functon s equpped wth a convex entropy functon U(u) along wth an entropy flux functon F, whch satsfes F (u) =U (u)f (u). Remark 3.. The entropy functon U s not to be confused wth the numercal approxmatons U n snce for the later we shall always use a superscrpt. The entropy functon U provdes the new varables the entropy varables [7, 8], v(u) =U (u), and due to the convexty of U the mappng u(v) s -; hence t serves as a change of varables u = u(v). Wth respect to the new varables v, the ntal conservaton

8 39 C. Arvants, C. Makrdaks & N. I. Sfakanaks law yelds u(v)+ g(v) =, g(v) =f(u(v)). t x The potental functons that follow play an essental role n the constructon and analyss of the entropy conservatve schemes, namely the entropy potental, φ(v) s defned as u(v) = d φ(v),.e. φ(v) =vu(v) U(u(v)), (3.) dv and the entropy flux potental ψ(v) s defned as g(v) = d ψ(v) or ψ(v) =vg(v) F (u(v)). (3.) dv We can now ntroduce the entropy conservatve numercal flux g + = g(v + ξ(v + v ))dξ = ψ(v +) ψ(v ), ξ= v + v whch provdes the entropy conservatve centered sem-dscrete numercal scheme: d dt u (t) = (g x + g ). (3.3) It s proven n [5, Theorem 4.] that the sem-dscrete scheme (3.3) snfact entropy conservatve. Remark 3.. As descrbed by Tadmor n [5], conservatve schemes wth more numercal vscosty than the entropy conservatve ones are entropy stable (Theorem 5.). Furthermore, conservatve schemes contanng more vscosty than an entropy stable scheme are also entropy stable (Theorem 5.3). So there s a type of orderng n the class of entropy conservatve/stable schemes wth the entropy conservatve beng a margnal class. The entropy conservatve scheme (3.3) are sem-dscrete schemes, for the fully dscrete numercal scheme, we perform explct n tme dscretzaton, through whch the scheme (3.3) recasts nto u n+ = u n t (g x + g ). (3.4) To justfy our choce of explct tme dscretzaton, we refer to [6] tonotethat mplct tme dscretzaton enforces entropy stablty, on the contrary explct tme dscretzaton leads to entropy producton. The entropy conservatve schemes, though, do not lead to entropy dsspaton; hence there s no counterbalance for the entropy producton due to explct tme dscretzaton. Ths leads to nstabltes whch numercally are presented n the form of oscllatons. Hence by combnng the MAS wth entropy conservatve schemes for the tme evoluton (Step 3), we can test the entropy dsspaton propertes of the mesh reconstructon (Step ) and the soluton update (Step ) of the MAS.

9 Adaptve Mesh Selecton Adaptvty and Numercal Dsspaton We shall dscuss four test cases where the entropy dsspaton of the MAS s exhbted. The frst three are 3-pont entropy conservatve schemes. We follow the work of Tadmor [5, 6] up to the pont of constructng a sem-dscrete entropy conservatve scheme. Then we use an explct verson of the scheme as the evoluton step n our adaptve algorthm. The fnal example s a 5-pont entropy conservatve sem-dscrete scheme emanatng from the work of LeFloch and Rhode [] desgned to approxmate nonclasscal shocks. The tme dscretzaton n ths example s a 4th order Runge Kutta scheme. In all numercal experments we studed, we notced that the adaptve mesh case elmnates the oscllatons wth optmal CFL condton. In the more nterestng examples, we dsplay the results of both unform and adaptve mesh selecton. For the rest we just present the adaptve mesh case, snce t s well known that the relevant unform mesh schemes produce oscllatons of large magntude rather fast. 4.. Problem Frst example s the nvscd Burgers equaton: ( ) u t + u = x equpped wth entropy functon U(u) = ln u. We follow the steps stated n the ntroducton up to the pont of constructng the respectve sem-dscrete entropy conservatve scheme. We then select the temporal dscretzaton and create the fully-dscrete entropy conservatve scheme. In ths example, we have U(u) = ln u and F (u) = u. By the convexty of the entropy functon U(u), we perform the followng change of varables v(u) =U (u) v(u) = u u(v) = v, the entropy varable flux reads g(v) =f(u(v)) or g(v) = v, and the model equaton recasts nto u(v)+ g(v) =. t x For the spatal dscretzaton, we shall make use of the entropy flux potental ψ(v), ψ(v) =vg(v) F (u(v)),.e. ψ(v) = v,

10 39 C. Arvants, C. Makrdaks & N. I. Sfakanaks ada5 ada5 ada ada5 ada5 ada ada5 ada5 ada Fg.. (Problem.) Burgers equaton wth box ntal condtons. The par entropy/entropy flux consdered s (U(u),F(u)) = ( ln u, u). We present the numercal solutons for tmes t =., t =.5 and t =.375 only for the non-unform adaptve case snce the unform exhbts large scale oscllatons. In each graph, we present the numercal solutons for N = 5, 5, cells. We note the lack of oscllatons n all three tme steps despte the ncreasng number N of cells. and the entropy conservatve flux shall be g + = ψ(v +) ψ(v ) or g v + v + = = v + v u +u. We employ the last result to wrte the entropy conservatve centered sem-dscrete numercal scheme, d dt u (t) = x (g + g ),.e. d dt u (t) = u (t) u +(t) u (t) x + x.

11 Adaptve Mesh Selecton 393 Fnally as we explaned earler we dscretze explctly n tme and the fully dscrete scheme s: u n+ = u n t x + x u n (u n + u n ). (4.) We refer to Fg. for an ntal presentaton of the adaptve mesh case. The numercal scheme used s (4.) and the ntal condton s a box. The test s performed for a sequence of ncreasng number of cells. We also refer to Fgs. and 3 for more elaborate tests, where we agan use the scheme (4.) but ths tme wth dfferent ntal condtons. We test the adaptve mesh case versus the unform mesh.5 un5 ada5.5 un5 ada un5 ada5.5 un5 ada Fg.. (Problem.) Burgers equaton wth slow slope ntal condton. The par entropy/entropy flux consdered s (U(u),F(u)) = ( ln u, u). We exhbt tmes t =,t =3,t =6,t =9for the soluton, the unform and the adaptve mesh case. We note that the unform mesh case exhbts oscllatons that rapdly spread to both drectons. The adaptve case s clean of oscllatons and captures very accurately the soluton. In ths test we utlzed N = 5 cells and the CFL was set to.45.

12 394 C. Arvants, C. Makrdaks & N. I. Sfakanaks.5. un5 un5 un5.5. un5 un5 un ada5 ada5 ada5.5. ada5 ada5 ada ada5 ada5 ada Fg. 3. (Problem.) Burgers equaton wth slow slope ntal condton. The par entropy/entropy flux consdered s (U(u),F(u)) = ( ln u, u). We exhbt tmes t =3andt = 9 wth ncreasng number of cells N = 5, 5 and 5, separately for the unform and the adaptve case versus the soluton of the problem. In the frst lne, we present the unform case (left t =3,rght t = 9) and we note that t exhbts oscllatons whch grow wth respect both to tme and to the number of cells N used for the computaton. In the second lne, we present the adaptve case for thesamedata(leftt =3,rghtt = 9) and we note that the numercal approxmatons are clean of oscllatons and moreover they are ndstngushable from the soluton n ths plottng scale. The graph n the thrd lne s agan the adaptve case for t =9wherethstmewehavefocused n the regon of the jump, we note here that the lmtng profle of the results (as the number of cells N ncreases) s the soluton of the problem. For ths test the CFL s.45.

13 Adaptve Mesh Selecton 395 case and the soluton of the respectve problem. The outcome of these tests s that n the adaptve mesh case oscllatons are suppressed and at the same tme the soluton of the problem s resolved accurately. 4.. Problem For the second example we use the same method, 3-pont entropy conservatve scheme wth adaptve mesh selecton and explct tme dscretzaton. The convex entropy n ths example s U(u) =e u. In ths example the model equaton s, u t +(e u ) x =. The entropy that we want conserved s U(u) =e u wth entropy flux F (u) =U (u)f (u),.e. F (u) = eu. Due to the convexty of the entropy U we perform the followng change of varables, Hence the entropy varables flux reads v(u) =U (u) v(u) =e u or u(v) =lnv. g(v) =f(u(v)),.e. g(v) =v, whch provdes us wth the new model equaton u(v)+ g(v) =. t x We defne the entropy flux potental ψ(v) va, ψ(v) =vg(v) F (u(v)) or ψ(v) = v. So the entropy conservatve flux s wrtten g + = ψ(v +) ψ(v ),.e. g v + v + = (eu+ + e u ), and the entropy conservatve centered sem-dscrete numercal scheme reads, d dt u (t) = (g x + g ) hence d dt u (t) = eu+(t) e u (t). x + x Fnally we dscretze explctly n tme to get the fully dscrete scheme u n+ = u n t x + x (e un + e u n ). (4.) We refer to Fg. 4 for a graphcal presentaton of a test performed for ths scheme over an ntal condton that leads to shocks and rarefactons. We compare the

14 396 C. Arvants, C. Makrdaks & N. I. Sfakanaks.5 un ada.5 un ada un ada.5 un ada Fg. 4. (Problem.) u t +(e u ) x = wth mult shock ntal condtons. Entropy/entropy flux (U(u),F(u)) = (e u, eu ). Exhbtng tme steps t =., t =.3, t =.6, t =.9 for the soluton, the unform and the adaptve mesh case. We note the large ampltude oscllatons of the unform mesh case. The adaptve mesh case s clean and resolves accurately the soluton. In ths example, we utlzed a mesh of 5 cells and the CFL was set to.3. adaptve mesh case versus the unform one versus the soluton of the problem. As n the prevous case the adaptve mesh case s clean of oscllatons and resolves the soluton accurately whereas the unform one exhbts strong oscllatons Problem 3 Ths test s agan Burgers equaton but ths tme the entropy s U(u) = u f(s)ds. Assumng that u s postve u, the entropy U s convex. The choce of the specfc entropy functon leads to entropy varables flux g(v) = v. Theworkctedn[5, 6] s agan used as was prevously descrbed. The model we use s the nvscd Burgers equaton: ( ) u t + u =. x

15 Adaptve Mesh Selecton 397 The entropy we want to conserve n ths case s u u U(u) = f(s)ds = s ds = 6 u3, wth entropy flux defned va F (u) =U (u)f (u),.e. F (u) = 8 u4. The convexty of U (u ) provdes us wth the new varables v, The entropy varables flux s gven by v(u) =U (u) v(u) = u u(v) = v. g(v) =f(u(v)) or g(v) =v, and the new model equaton s wrtten as, u(v)+ g(v) =. t x We move on to the entropy flux potental ψ(v) ψ(v) =vg(v) F (u(v)),.e. ψ(v) = v, whch provdes us wth the entropy conservatve flux g + = ψ(v +) ψ(v ) or g v + v + = 4 (u + + u ). Hence the entropy conservatve centered sem-dscrete numercal scheme reads d dt u (t) = (g x + g ) hence d dt u (t) = u + u. 4 x + x To fully dscretze the prevous equaton we once agan use forward Euler tme dscretzaton and the fully dscrete numercal schemes assumes the form = u n t.5 (u + u )(u + + u ). (4.3) x + x u n+ We refer to Fgs. 5 and 6 for a graphcal presentatons of tests performed for ths scheme. More specfcally, n Fg. 5, we exhbt a comparson of the adaptve mesh case versus the unform case where the ntal condton s consdered to be u (x) =.5sn(π(x +.5)) +.5. We wtness appearance of oscllatons on behalf of the unform case as soon as the smoothness of the profle s lost, whle at the same tme the adaptve mesh case remans clean of oscllatons. In Fg. 6,weperformtests only for the adaptve mesh case where ths tme the number of cells N vares. The numercal solutons converge to a sharp lmtng profle as the number of cells N ncreases.

16 398 C. Arvants, C. Makrdaks & N. I. Sfakanaks.4 adaptve unform.4 adaptve unform adaptve unform Fg. 5. (Problem 3.) Burgers equaton wth u (x) =.5sn(π(x +.5)) +.5 ntal condtons. The par entropy/entropy flux consdered s (U(u),F(u)) = ( 6 u3, 4 u4 ). Exhbtng tmes t =.3, t =.66, t =.85 for the unform and the adaptve mesh case. We note agan the the adaptve mesh case exhbts no oscllatons where as n the unform mesh case they appear as soon as the smoothness of the soluton s lost. In ths test we used 3 cells and the CFL was set to Problem 4 We now consder a more nvolved case, namely a 5-pont entropy conservatve scheme ntroduced n [] for approxmatng non-classcal shocks. We consder the one-dmensonal model: wth entropy functon U(u) = u t +(u 3 ) x = u f(s)ds U(u) = 4 u4.

17 Adaptve Mesh Selecton 399 ada5 ada5 ada ada5 ada5 ada ada5 ada5 ada ada5 ada5 ada Fg. 6. (Problem 3.) We agan consder Burgers equaton wth u (x) =.5sn(π(x +.5)) +.5 ntal condtons. The par entropy/entropy flux consdered s (U(u),F(u)) = ( 6 u3, 4 u4 ). In ths test we present tmes t =.66 (left) and t =.85 (rght) only for the adaptve case snce the unform case has already produced spurous oscllatons. We perform the test over dfferent cell numbers N = 5, 5,. In the frst lne, we note that the numercal solutons are clean of oscllatons even n hgh spatal resolutons. In the second lne, we see the same graphs as n the frst, ths tme focused on the area of the dscontnuty, where we note that the ncrease of the spatal resoluton leads to a sharp lmtng profle. In ths test the CFL was set to.4. The entropy flux n ths case s gven by the relaton: F (u) =U (u)f (u) F (u) = u6. Due to the convexty of the entropy functon, we defne the new varables, entropy varables v as v(u) =U (u) v(u) =u 3 u(v) = 3 v.

18 4 C. Arvants, C. Makrdaks & N. I. Sfakanaks Applyng ths change of varables we conclude to the new entropy varables: g(v) =f(u(v)) g(v) =v. The man step now s the constructon of the entropy conservatve flux g+/. LeFloch and Rohde (Theorem 3.) state that the numercal scheme d dt u (t) = x (g +/ g / ), g +/ = g (v,v,v +,v + ), 4 un4 ada4 4 un4 ada un4 ada Fg. 7. (Problem 4.) Burgers equaton wth a jump u l =4,u r = 5 atx = 8 for ntal condtons. The par entropy/entropy flux consdered s (U(u),F(u)) = ( 4 u4, u6 ). We exhbt tmes t =. (left), t =. (rght), t =.3 (bottom) for the unform and the adaptve mesh case. We frst note that n the unform case oscllaton appear both at the top and the bottom of the jump. We note that especally n the top the oscllatons seem to propagate wth a pattern (see Fg. 8), these oscllatory parts are resolved by the adaptve case wthout any oscllatons. Next we note that n the unform case a plateau created at level y 3.5 whch the adaptve case does not resolve (see Fg. 8). Ths test was performed usng N = 4 cells and 4th order Runge Kutta temporal dscretzaton of the scheme (4.4).

19 Adaptve Mesh Selecton 4 wth entropy conservatve flux g (v,v,v +,v + ) defned by g (v,v,v +,v + )= g(v + s(v + v ))ds ((v + v + ) B (v,v +,v + ) (v v ) B (v,v,v + )), 4 un8 un4 un 4 un8 un4 un ada ada4 ada8 4 ada ada4 ada Fg. 8. (Problem 4.) Burgers equaton wth a jump u l =4,u r = 5 atx = 8 for ntal condtons. The par entropy/entropy flux consdered s (U(u),F(u)) = ( 4 u4, u6 ). We exhbt tmes t =. (left) t =.3 (rght) separately for the unform and the adaptve mesh case over dfferent mesh szes N =, 4, 8. In the frst lne, we present the numercal solutons over unform meshes and we note that the pattern of oscllatons and ther frequency vares wth N (most notably at the top) whle at the same tme they keep ther speed of propagaton. We also note that the creaton of the plateau at level y 3.5 s persstent wth the ncrease of N revealng that ths schemes resolves non-classcal shock on unform meshes. In the second lne, we present the area of the jump n the adaptve case. We notce that the results do not exhbt any oscllatons and at the same tme they converge (as N ncreases) to a lmtng profle that lacks the plateau exhbted n the unform case. Ths reveals the falure of the adaptve case to resolve non-classcal shocks. These tests where performed after 4th order Runge Kutta temporal dscretzaton of the scheme (4.4).

20 4 C. Arvants, C. Makrdaks & N. I. Sfakanaks s entropy conservatve and thrd order accurate provded, B (v, v, v) =B(v) =Dg(v). The choces B =orb = B, whereb(v) =Dg(v), produce classcal shocks, whle other choces such as B =5B produce non-classcal shocks. We refer to [] for a thorough dscusson on non-classcal shocks n conservaton laws. We just menton here that these shocks can occur only when the flux functon f s not convex, they satsfy a sngle nstead of the whole set entropy nequalty and they are under-compressve n the sense that characterstcs pass through t rather than focus on t. We fnsh wth the constructon of the entropy conservatve sem-dscrete numercal scheme by selectng B =5B, hence the scheme yelds: d dt u (t) = (g(v + ) g(v )) x 5 ( g(v + )+g(v + ) g(v )+g(v )). (4.4) x The numercal tests are performed wth a Runge Kutta 4th order temporal dscretzaton and a jump u l = 4, u r = 5 atx = 8 ntal condton. We refer to Fgs. 7 and 8 for graphcal representaton of the comparson tests of the adaptve case versus the unform case. Out of the test performed, t s evdent that the schemes of [] when combned wth the adaptve mesh selecton algorthm of Sec. are transformed to entropy dmnshng schemes and they no longer approxmate the non-classcal shock, but rather the entropy soluton of the conservaton law. 5. Conclusons In ths work we consdered a mesh reconstructon procedure as descrbed n Sec. and we combned t wth the entropy conservatve schemes of Tadmor for the evoluton step of the MAS. The entropy conservatve schemes of Tadmor are desgned to serve as buldng block to construct entropy dsspatve schemes for scalar and systems of conservaton laws [6]. These schemes consttute an mportant margnal class for the numercal study of hyperbolc problems. The man conclusons regardng the use of the MAS along wth entropy conservatve schemes for the tme evoluton step are: () The entropy conservatve schemes, when combned wth adaptve mesh selecton converge to the entropy soluton wthout oscllatons. () Optmal CFL condton can be accomplshed even wth explct tme dscretzaton. (3) These schemes are used as a bass for the constructon of numercal schemes for non-classcal shock computaton. Our results show that even n ths case these

21 Adaptve Mesh Selecton 43 schemes combned wth approprate mesh selecton converge to the entropy soluton rather than to the non-classcal shock. It s mportant therefore to observe the strong stablzaton mechansm of mesh selecton for ths class of schemes. On the other hand, such schemes should be used wth care f one wants to compute non-classcal shock behavor n non-unform meshes. Acknowledgments Research partally supported by the European Unon grant No. MEST-CT-5- (Dfferental Equatons wth Applcatons n Scence and Engneerng DEASE). References [] Ch. Arvants and A. Dels, Behavor of fnte volume schemes for hyperbolc conservaton laws on adaptve redstrbuted spatal grds, SIAM J. Sc. Comput. 8 (6) [] Ch. Arvants, Th. Katsaouns and Ch. Makrdaks, Adaptve fnte element relaxaton schemes for hyperbolc conservaton laws, Math. Model. Anal. Numer. 35 () [3] Ch. Arvants, Ch. Makrdaks and A. Tzavaras, Stablty and convergence of a class of fnte element schemes for hyperbolc systems of conservaton laws, SIAM J. Numer. Anal. 4 (4) [4] Ch. Arvants, Fnte elements for hyperbolc conservaton laws: New methods and computatonal technques, Ph.D. thess, Unversty of Crete (). [5] Ch. Arvants, Mesh redstrbuton strateges and fnte element method schemes for hyperbolc conservaton laws, J. Sc. Computng 34 (8) 5. [6] E. Dorf and L. Drury, Smple adaptve grds for d ntal value problems, J. Comput. Phys. 69 (987) [7] K. O. Fredrchs and P. D. Lax, Systems of conservaton laws wth convex extenson, Proc. Nat. Acad. Sc. USA 68 (97) [8] E. Godlewsk and P. A. Ravart, Hyperbolc Systems of Conservaton Laws (Ellpses, 99). [9] A. Harten and J. Hyman, Self adjustng grd methods for one-dmensonal hyperbolc conservaton laws, J. Comput. Phys. 5 (983) [] B. T. Hayes and P. LeFloch, Non-classcal shocks and knetc relatons: Scalar conservaton laws, Arch. J. Math. Anal. 39 (997) 56. [] A. Kurganov, G. Petrova and B. Popov, Adaptve semdscrete central-upwnd schemes for nonconvex hyperbolc conservaton laws, SIAM J. Sc. Comput. 9(6) (7) 38 4 (n electronc). [] P. G. LeFloch and Ch. Rohde, Hgh-order schemes, entropy nequaltes and nonclasscal shocks, J. Numer. Anal. SIAM () 3 6. [3] M. Lukáčová-Medvďová and E. Tadmor, On the entropy stablty of Roe-type fnte volume methods, to appear n Proceedngs HYP 8. [4] N. Sfakanaks, Adaptve mesh reconstructon, total varaton bound, preprnt (8). [5] E. Tadmor, The numercal vscosty of entropy stable schemes for systems of conservaton laws, Math. Comput. (987) 9 3.

22 44 C. Arvants, C. Makrdaks & N. I. Sfakanaks [6] E. Tadmor, Entropy stablty theory for dfference approxmatons of nonlnear conservaton laws and related tme dependent problems, Acta Numer. (3) [7] H. Tang and T. Tang, Adaptve mesh methods for one- and two-dmensonal hyperbolc conservaton laws, SIAM J. Numer. Anal. 4 (3)