c 2006 Society for Industrial and Applied Mathematics

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1 SIAM J. SCI. COMPUT. Vol. 8, No. 5, pp c 6 Socety for Industral and Appled Mathematcs BEHAVIOR OF FINITE VOLUME SCHEMES FOR HYPERBOLIC CONSERVATION LAWS ON ADAPTIVE REDISTRIBUTED SPATIAL GRIDS CH. ARVANITIS AND A. I. DELIS Abstract. In ths work we consder fnte volume schemes combned wth dynamc spatal mesh redstrbuton. We study whether approprate mesh redstrbuton s a satsfactory mechansm for ncreasng the resoluton of numercal solutons for problems of scalar and systems of conservaton laws (CL n one space dmenson, whle beng at the same tme a stablzaton mechansm for selectng the approprate entropy soluton. In order to ncrease the resoluton around shock areas and keep the computatonal cost low, our redstrbuton polcy s to reconstruct spatally the numercal soluton on a new mesh, where the soluton s curvature s almost unformly dstrbuted, whle the node s cardnalty s kept constant. We examne the stablzaton propertes of that redstrbuton process by addng t as a substep on the tme evoluton step of some classcal schemes wth known (unstable characterstcs. Testng the resultng method for several such schemes and on a large number of CL problems that have solutons wth specal characterstcs (shocks, rarefacton areas, steady states and comparng the results wth those produced by schemes wth extra stablzaton mechansms (lke slope/flux lmters, entropy correctons, we conclude that ndeed the proposed redstrbuton adds such stablzaton propertes whle at the same tme ncreasng the resoluton. Key words. fnte volume methods, adaptve grd redstrbuton, hyperbolc conservaton laws AMS subject classfcatons. 35L65, 76M1, 65M1 DOI / Introducton. The applcaton of fnte volume schemes s a very popular choce for computng solutons of systems of conservaton laws (CL n the followng context: fnd u : R d [,T] R M such that (1.1 u(, = u gven d t u + x F (u =. =1 Some classcal schemes, of frst or second order n space, when appled drectly to ths system wll result n computatonal solutons wth dffusve or oscllatory behavor, especally close to shocks. To overcome ths dffculty several modfcatons of such schemes have been proposed n the lterature where the necessary stablty and vscosty mechansms are mposed by hand. Mesh adaptaton s a man current stream to effcently compute numercal solutons of complex systems by ncreasng the resoluton of the essental soluton. Several redstrbuton technques have been ntroduced recently for solvng the problem of proper mesh selecton, startng wth the self-adjustng method of Harten and Hyman [14] to the movng mesh methods of Azarenok et al., Fazo and LeVeque, Stocke et al., Tao Tang, Huazhong Tang, and Receved by the edtors June 1, 5; accepted for publcaton (n revsed form May 15, 6; publshed electroncally November 16, 6. Department of Appled Mathematcs, Unversty of Crete, Heraklon 7149, Crete, Greece (arvas@tem.uoc.gr. The work of ths author was partally supported by the European Unon RTNnetwork HYKE, HPRN-CT--8, and the program Pythagoras of EPEAEK II. Department of Scences, Dvson of Mathematcs, Techncal Unversty of Crete, Unversty Campus, Chana 731, Crete, Greece (adels@scence.tuc.gr. 197

2 198 CH. ARVANITIS AND A. I. DELIS others (see [4], [5], [6], [8], [1], [17], [19], [], [1], [3], [7], [8], [9], [3]. These methods calculate the spatal postons of the nodes of the new mesh, some of them by solvng an Euler Lagrange equaton, others by optmzng proper energy metrcs. All of them have as a common factor that they can be combned wth any numercal scheme (makng approprate modfcatons for ncreasng ts resoluton. In ths work the evolvng mesh s constructed such that ts spatal resoluton s controlled va selectve characterstcs of the computed soluton. Our man am here s to expermentally study the behavor of fnte dfference-volume schemes, wth or wthout stablzaton mechansms, but under the regme of ths adaptvely evolvng mesh. Our choce of classcal schemes ncludes, for example, the frst order Roe scheme, the second order Lax Wendroff and MacCormack schemes, and also some TVD schemes. The adaptve procedure studed n ths work s based on a mesh redstrbuton polcy that evolves wthn every computatonal tme step. The basc prncple of the suggested mesh redstrbuton s to (redstrbute the nodes of the generated partton wth respect to geometrcal characterstcs of the soluton. These characterstcs are defned through a postve functonal of the soluton, the so-called estmator functon [1], [], [3]. Among other estmator functons for evoluton PDEs, lke the arc-length and varance, we choose the curvature of the soluton as such a functon, for ts dffuseless behavor. Experments n [3] wth Galerkn fnte element schemes, whch approxmate CL solutons lke central fnte dfference schemes, have shown that ths redstrbuton procedure has stablzaton propertes of ts own. For example, n ths work we show that schemes lke Lax Wendroff combned wth our redstrbuton polcy provde surprsngly stable solutons free of oscllatons. From the numercal experments produced n the course of ths work we concluded the followng advantages of the use of ths partcular adaptve grd redstrbuton (AGR method. The method when appled to classcal second order schemes, whch produce oscllatng solutons, suppresses the oscllatons, producng TVD-lke approxmatons. Classcal schemes lke the Lax Wendroff or the MacCormack scheme become stable and produce relable solutons. When appled to numercal schemes that do not satsfy entropy condtons, for example, the orgnal Roe scheme, the method approxmates the entropy satsfyng soluton. The method works well for hyperbolc problems wth source terms and produces stable solutons for frst and second order balanced schemes and can converge to correct steady states. The method can automatcally detect, resolve, and track steep wave fronts and dscontnutes, wthout havng to resort to fner grds. The AGR method s of lnear complexty, as we wll see n the next secton (cf. Remark 3, and consequently ts computatonal cost s n favor when compared, for example, to the stablzaton mechansms for hgh resoluton TVD schemes, where we have to solve a number of Remann problems n order to compute the correspondng stablzaton lmters. The mechansm has been proved robust for all the applcatons presented and schemes used. Followng from the above, our man concluson from the present work s that numercal schemes, when combned wth the proposed mesh adaptaton, yeld stable solutons free of oscllatons. The paper s organzed as follows. In the next secton we defne the AGR algorthm. In secton 3 we present fnte volume schemes for scalar CL, adjusted to nonunform grds, whch wll be tested n the regme of the adaptve redstrbuton al-

3 FINITE VOLUMES ON ADAPTIVE REDISTRIBUTED GRIDS 199 gorthm. Secton 4 s devoted to numercal results. In secton 5 numercal schemes for systems of CL, wth or wthout source terms, are detaled, and n secton 6 numercal results for the shallow water system of equatons are presented.. The AGR method n one dmenson. Let X be a partton of a doman [a, b] and {x } N the correspondng nodes n ncreasng order,.e., a = x <x < = 1 <x N 1 <x N = b. Then we ntroduce the followng notaton: Wth σ(x we denote the σ-algebra of X, whch for fnte parttons concdes wth the superset of unons of sets from X. Wth resoluton of X, we denote the measure whch for any measurable set A s defned by resoluton(a = card{k X : K A},.e., the resoluton of the partton X over some set A s the number of partton elements that A contans. If U = {u } N s a vector of values defned on elements of the partton = X, then t shall be notated as U(X. For compatblty wth approxmatons gven from fnte volume schemes, n the followng we shall dentfy any vector of functon values {u } N evaluated on partton nodes {x = } N,.e., U = = U(X, wth the locally constant functon U, defned on [a, b] by (.1 U = N = u χ [x + 1,x+ + u N χ {x + N }, wth χ A denotng the characterstc functon of the set A, and {x + }N = 1 denotng the ponts x + = x, 1 x+ = x + 1, =, 1,...,N 1, x+ = x. N N In the case of evoluton PDEs, where the numercal soluton s constructed lke a sequence of spatal approxmatons of solutons nstances, the redstrbuton process could be appled before every evoluton step. Let Solver denote a (general fnte volume scheme for solvng some evoluton PDE. For n = 1,,... Solver gves sequentally the approxmatons U n of the soluton nstances u(t n, at gven tme moments t <t 1 < <t n R +, startng at t =. The gven ntal data U s defned on the unform partton X of the doman. In the case of unform partton, the evoluton step can be represented for n =1,,... as an equaton on the vector space R (N+1 : (X n,u n (X n =(X n 1, Solver(X n 1,U n 1 (X n 1. On meshes generated by the AGR process, the evoluton step can be represented by the system of vector equatons: (. ( X, Ũ( X = AGR(X n 1,U n 1 (X n 1, (X n,u n (X n = ( X, Solver( X, Ũ( X, where X,Ũ are temporal vectors. Observe that n the case where the AGR process returns the same vectors,.e., the AGR concdes wth the dentty lnear transformaton, then (. reduces to the unform evoluton step. The AGR procedure, for a

4 193 CH. ARVANITIS AND A. I. DELIS gven vector of values U = {u } N defned on the nodes {x = } N of the partton X, =.e., U = U(X, s descrbed by the followng two sequental steps, (.3 X = GMesh(X, U(X, Ũ( X =Rec(X, U(X, X. In what follows we descrbe n detal the two steps GMesh, Rec of our redstrbuton algorthm AGR..1. The GMesh step. At the GMesh step of the AGR procedure (.3, a new partton X of spatal nodes { x } N s formed, wth resoluton controlled by selected = characterstcs of the numercal soluton U. The step s accomplshed n two phases. At the frst phase the selected characterstcs are defned on the doman [a, b] through a strctly postve functonal g of the approxmate soluton, the estmator functon. Snce we are manly nterested n ncreasng the resoluton over areas wth dscontnutes, and takng nto account the results n [3], n ths work we wll also use some power p of proper approxmaton of the soluton s curvature defned by u (x (curv u(x = (1 + u (x 3 as the estmator functon for selectng the resoluton s densty of the new partton. For a gven partton {x } N, the soluton s curvature at the node x = can be approxmated by (curv h u(x = 1+ u(x u(x 1 x +1 x 1 x x u(x +1 u(x 1 x +1 x ( u(x u(x 1 x x 1 1+ ( u(x+1 u(x x +1 x 1+ ( u(x+1, u(x 1 x +1 x 1 whch s the nverse outer radus of the plane ponts A j =(x j,u(x j, j = 1,,+1,.e., (A +1 A (A A 1 (curv h u(x = A +1 A A +1 A 1 A A 1, where denotes the Eucldean norm. Thus, at the dscrete level, the values g of our estmator functon on the node ponts {x } N, are gven by g = = g N = δ p and for =1,...,N 1, ( { (A +1 A (A A 1 } p (.4 g = max δ,, A +1 A A +1 A 1 A A 1 where the power p s left as a free parameter takng values n the range [, 1] and δ s a very small postve number whch ensures that g s strctly postve. The constant δ must be chosen such that, at a computatonal level, g behaves monotoncally as a functon of p even for neglgble values. In our experments, where the calculatons were performed wth ANSI-C computatonal lbrares of double precson, δ was fxed to the value 1 3. In the case of vector soluton u =(u 1,...,u M T, we ntend to use the average of the coordnate estmators as the unform estmator functon for all the coordnates of the soluton. Therefore, n general, nstead of the values g we keep the normalzed verson g / N j= g, =, 1,...,N. j

5 FINITE VOLUMES ON ADAPTIVE REDISTRIBUTED GRIDS 1931 At the second phase, the partton X s actually produced through g h U, the correspondng functon of type (.1 to values (.4. Snce we expect a new mesh X wth resoluton that locally follows the functon g h U, we conclude that the resoluton measure must be dstrbuted n space lke the measure G U (A = (g U dμ A h ntroduced by the postve functon g h U, at least for sets from σ( X. Therefore the new partton X s defned such that A σ( X G U (A =C resoluton(a, wth C = G ([a, b] U (.5. N Applyng sequentally (.5 wth the sets A =[a, x ], =1,,...,N, where ther ndces declare also the resoluton of the partton X over them, we nduce, as t s called n the lterature of movng meshes, the equdstrbuton prncpal: (.6 x = a and for =1,...,N, x s such that x a g h (U(x dx = N b a g h (U(x dx. In the present work, the new mesh X s defned through (.6, but we note that other nvertng algorthms could also be constructed through a more elegant treatment of the generc equdstrbuton prncpal (.5. Let G(x =G U ([a, x] be the dstrbuton functon, whch s an ncreasng local lnear functon defned on the grd X, snce g h U s a postve functon of type (.1. Thus, G s completely defned from the values G = G(x oftheg U measure of the ntervals [a, x ], that s, G =, and for =1,...,N, x G = G U ([a, x ] = (g h U dμ = G 1 + g h (U(x dx [a,x ] x (.7 1 (.1 = G 1 +(g 1 + g (x x 1 /. From (.6 t follows that the x node s gven by nvertng the equaton G( x = N G( x N, so the nodes of the new mesh X are x = a, x N = b, and, for =1,...,N 1, k :=, (.8 G = N G N, k = max k 1 l N {l : G l G }, x = x k + x k+1 x k G k +1 G k ( G G k... The Rec step. At the Rec step of the AGR procedure (.3, a new numercal soluton Ũ = {ũ } N of type (.1 s defned on the new grd X by reconstructng = U from the old grd X to the new one. Snce the schemes used n ths work produce conservatve solutons, the reconstructon step s done so that t s locally conservatve on each nterval [ x +, x+ ] of the new mesh. That s, for =,...,N, we requre 1 x + x + 1 Ũ(x dx = x + x + 1 U(x dx. Snce the functons U, Ũ are of type (.1 defned on the parttons X, X, respectvely, ũ ( x + x + x + = 1 x + 1 U(x dx = x + x + k U(x dx + x + k x + k 1 U(x dx x + 1 x + k 1 U(x dx,

6 193 CH. ARVANITIS AND A. I. DELIS where, usng the real lne orderng of the doman [a, b], x + denotes the larger node k from the old partton not exceedng the node x + of the new partton. Consequently, the new values Ũ are ũ = u and, for =1,...,N, k :=, (.9 k = max {l : x + x + }, l k l N 1 { ũ = ( x + x + k u k +1 + k l=k 1+1 } (x + x+ u l l 1 l ( x+ x+ u 1 k k 1 / ( x + x Intal Data Estmator Integral of Estmator after 1 AGR teraton 1.5 Intal Data Estmator Integral of Estmator after AGR teratons 1.5 Intal Data Estmator Integral of Estmator after AGR teratons Fg. 1. Iteraton of the AGR procedure appled on Remann data. After 1 teraton (left and after teratons for p =3 (mddle and p =.1 (rght. The Gmesh step of the AGR process can be graphcally represented as n Fgure 1. Startng wth some ntal data (sold lne we calculate the estmator functon (dotted lne through (.4 and then the correspondng dstrbuton functon (ncreasng dotted lne usng (.7. The proposed mesh (x poston of the vertcal lnes, under the dstrbuton functon s gven by nvertng a unform mesh (y poston of the horzontal lnes, over the dstrbuton functon through the dstrbuton functon usng (.8. Notce that y-coordnates are vald only for the data whle ether the estmator or ts dstrbuton functon was scaled vertcally to ft on the vewng graph area. Remark 1(the acton of the parameter p. Parameter p controls essentally the maxmum densty of the generated mesh X. Indeed, from relatons (.4, (.6, we conclude that the correspondng nodes x, =1,...,N, satsfy the equaton 1 N b a x curv p (U(x dx = curv p (U(x dx max {curv p (U(x} ( x h h h x 1, x x [ x, x ] 1 1 and therefore ( 1 curvh U p L p [a,b] mn{ x N curv h U x 1 }. L [a,b] Thus, choosng p approprately, the resoluton of the X mesh respects an upper gven bound. The acton of the parameter p n the AGR procedure can also be seen n

7 FINITE VOLUMES ON ADAPTIVE REDISTRIBUTED GRIDS 1933 Fgure 1 (mddle, rght. For p = observe from (.7 that G = x x, so the proposed partton from (.8 concdes wth the unform partton of [a, b]. As the value of p ncreases, the proposed partton becomes more dense on the dscontnuty areas, but over some value of p the process becomes unrelable because the partton lacks nodes over smooth areas. In our experments wth fnte volume schemes an approprate range for the parameter p was [,.1] for frst order and [4,.14] for second order schemes. Remark (smoothng effect of the AGR process. Results n [3] show that the reconstructed functon Ũ of the AGR procedure s more dffusve than the ntal functon U. Indeed ths dffusve behavor can be observed by applyng the AGR procedure teratvely on some ntal data,.e., startng wth some ntal data U on the unform partton X of [a, b], we defne the sequence (X n,u n (X n = AGR(X n 1,U n 1 (X n 1, n =1,,... Experments wth Remann ntal data show that the above sequence attans a lmt par (X, U(X. In Fgure 1 (mddle, rght we present the results after teratons for p =3 (mddle and for p =.1 (rght. In Fgure (mddle, rght we present the trajectores of the nodes for the above teratve process. Notce that after about teratons for p = 3 and about 8 for p =.1 the poston of all the nodes X stablzes. But from (.9 one can observe that reconstructng any functon on the same grd leads to the same functon,.e., U(X = Rec(X, U(X,X, so we conclude that also the reconstructed values U(X after the above teratons became constant. Fgure (left shows the smoothng effect of the AGR process on the ntal data after teratons and for varous values of p Intal Data p=3 p=5 p= Fg.. Focus n the shock area (left after teratons of the AGR procedure and for varous values of parameter p. Trajectores of mesh nodes for p =3 (mddle and for p =.1 (rght. Remark 3(complexty of the AGR process. Observe that both the Gmesh step (.7, (.8 and the Rec step (.9 of the AGR process are of lnear complexty..3. Redstrbuton n the steady state regme. The dffusve behavor of the AGR process mght be one of the reasons that second order schemes, when combned wth the redstrbuton step, produce nonoscllatory approxmatons; however, n the case where the current and the new mesh are almost the same the redstrbuton step should be avoded. In addton, n the case of very senstve CL problems wth steady states we notced that the repeated applcaton of the AGR process prevents the conservaton of the steady states. Takng nto account these observatons, n the numercal experments of ths work we adopt a new verson of the AGR process:

8 1934 CH. ARVANITIS AND A. I. DELIS X = GMesh(X, U, f ( X X >D, Ũ = Rec(X, U, X, else X = X, Ũ = U, where D s a cutoff level for the relatve mean dsplacement X X between the current and the proposed meshes, under whch the Rec step s avoded. The relatve mean dsplacement s measured n the l 1 vector norm, that s, X X = 1 N N+1 = x x b a. Experments show that, for frst order schemes or those wth extra stablzaton mechansms (lke usng slope/flux lmters, the AGR process produces results wth hgher resoluton when the cutoff parameter D s of order 1, whereas, due to the oscllatory behavor of second order schemes lke the classcal Lax Wendroff, the cutoff level can be of order 1 3 or even less. In all cases, when the problem mposes steady states a nonzero value of D s crucal for the solvng process n order to reach the tme nvarant soluton for all schemes. The above observatons become more pronounced for systems of conservaton laws. 3. Numercal schemes on nonunform grds (one-dmensonal scalar equatons. In ths secton we revew some well-known classcal numercal schemes, mplemented at the Solver step, for the dscretzaton of a scalar conservaton law, (3.1 u t + f(u x =, and present them n ther nonunform grd formulaton. We wll also state some well-known dsadvantages that these schemes exhbt. We wll approxmate the soluton u(t, x, x R, t, of (3.1 by the dscrete values u n, Z, n N, and n order to do so we consder a grd of ponts x + 1 and defne the computatonal cells and ther lengths as C =[x 1,x + 1 ], Δx = x + 1 x 1 >. We also denote by x =(x 1 + x + 1 / the centers of the partton cells. The values u n wll be approxmatons of the averages of the exact soluton over the cell u n 1 u(t n,xdx, Δx C wth t n the dscrete tme levels. For consstency, we present the schemes n the usual conservatve formulaton wth explct tme steppng, (3. u n+1 = u n Δt (f n Δx + f n 1, 1 where f n, =1,,...,N 1, s the numercal flux functon that determnes the + 1 scheme. Frst order Roe scheme. The numercal flux of the well-known frst order upwnd Roe scheme s gven by (see [15] (3.3 f ROE + 1 = 1 ( f +1 + f a + 1 Δu + 1,

9 FINITE VOLUMES ON ADAPTIVE REDISTRIBUTED GRIDS 1935 wth f = f(u, Δu + 1 = u +1 u and a + 1 the characterstc speed defned by (3.4 a + 1 = f +1 f u +1 u, u +1 u, a = f u, u +1 = u. A well-known drawback of Roe s scheme s that t may resolve nonphyscal solutons by admttng statonary entropy-volatng expanson shocks. Varous entropy correctons have been proposed; see [18]. For example, we can replace the modul of a n (3.3 by or by the smoother form wth ψ(a = ψ(a = max(δ, a { a, a δ, (a + δ /δ, a <δ, δ + 1 = max{,a + 1 a,a +1 a + 1 }, δ 1 = max{,a 1 a 1,a a 1 }. The Roe numercal flux s then defned as f ROE + = 1 ( (3.5 f f ψ(a + 1 Δu + 1. In our numercal examples to follow we wll see that when the adaptve mechansm s appled, the above entropy correctons are not necessary for the scheme to produce approxmatons of entropy solutons. Ths s a frst order dffusve scheme, but t serves as a key ngredent n developng hgher order methods. The local Lax Fredrchs scheme. A well-known frst order scheme that produces approxmatons of entropy solutons s the local Lax Fredrchs (LF scheme. Its numercal flux functon (see [18] s gven by (3.6 f LF + = 1 ( 1 f +1 + f a max Δu + 1 where a max = max{ f (u,f (u +1 }. Ths scheme produces dffusve numercal solutons and smears dscontnutes and was used for comparson reasons only. The Lax Wendroff scheme. A natural generalzaton of the classcal second order Lax Wendroff (LW scheme for nonunform grds can be gven as a combnaton of the frst order Roe flux plus a correcton flux (see [18]: (3.7 f LW + = f ROE ( Δx (Δx +1 +Δx Δt (Δx +1 +Δx a + 1 a + 1 Δu + 1, wth a + 1 as defned n (3.4. It s easy to see that for a unform grd numercal flux (3.7 s reduced to the classcal LW numercal flux. As for unform grds ths scheme has hghly oscllatory behavor near dscontnutes and wthout entropy correcton cannot produce entropy solutons.,

10 1936 CH. ARVANITIS AND A. I. DELIS The MacCormack scheme. One of the most classcal second order schemes (see [15] s the well-known two-step MacCormack scheme: (3.8 (3.9 (3.1 u (1 u ( u n+1 = 1 = u n Δt ( f n Δx +1 f n, = u n Δt ( f (1 +1 Δx f (1 ( u (1 + u ( Ths smple scheme also exhbts oscllatory behavor near dscontnutes and cannot produce the correct entropy solutons. The MUSCL-TVD scheme. Here we present a second order slope-lmtng scheme based on the MUSCL-TVD nterpolaton formula on nonunform grds; see, for example, []. We defne ( u u 1 u L + 1 u R + 1 = u + h +1 = u +1 h +1 wth h = x x 1 and θ gven by h +., Φ(θ, h ( ( u+ u +1 1 Φ θ = (u +1 u /h +1. (u u 1 /h The Φ s a lmter functon, and there are several optons from whch to choose Φ; see, for example, [5]. Some of the most popular lmters are the MnMod (MM lmter, the Superbee (SB lmter the VanLeer (VL lmter Φ(θ = max(, mn(1,θ, Φ(θ = max(, mn(θ, 1, mn(θ,, Φ(θ = θ + θ 1+ θ, and the monotonzed central (MC lmter Φ(θ = max(, mn((1 + θ/,, θ. The numercal flux for the MUSCL scheme can then be expressed, based on the Roe flux (wthout an entropy fx, as f MUSCL + = 1 ( (3.11 f(u R 1 + +f(u L 1 + a (ur + u L 1 +, 1 θ +1 where a + 1 s evaluated as before, wth u +1 and u replaced by u R + 1, and u L.Ina + 1 smlar way one can defne a MUSCL flux based on the LF flux. Second order TVD schemes reduce to frst order at extrema, and there are also dfferences n the behavor for each lmter; for example, the last three lmters have been shown to exhbt sharper resoluton of dscontnutes, snce they do not reduce the slope as severely as MM near a dscontnuty.

11 FINITE VOLUMES ON ADAPTIVE REDISTRIBUTED GRIDS Numercal examples for scalar conservaton laws. Because all the schemes presented above are explct n tme, stablty requres the tme step Δt to satsfy the CFL condton, as to ensure that the tme step s small enough that waves from neghborng cells do not nteract, wth CFL 1. Based on that we use a varable tme step calculated from { a + CFL=Δt max 1 Δx, a } + 1 (4.1, Δx wth a + = max(,a and a = mn(,a Burgers equaton. We frst consder numercal experments for the Burgers equaton n the nvscd lmt, ( u (4. u t + =. The analytcal solutons to ths problem can be found, for example, n [15]. The frst problem s an academc test case wth ntal condtons, for x [ 5, 5], x (4.3 u(x, = {, x <,, x, and has the followng exact soluton, also known as a transonc rarefacton:, x < t, (4.4 u(x, = x/t, t x t,, x>t. As ponted out n [18], dealng wth transonc rarefactons properly s an mportant component n the development of successful methods. Results for ths problem are presented n Fgures 3 and 4 at t = s. A grd of 61 ponts was used for all schemes wth CFL number equal to.9. All the schemes presented n the prevous secton, wth the excepton of the LF scheme, cannot produce the correct entropy soluton. By the applcaton of the adaptve mechansm all the schemes calculate the correct soluton. The most accurate results were produced wth the MUSCL adaptve scheme usng the SB lmter. We note here that when the adaptve mechansm was appled to the MUSCL scheme, we were able to produce smlar results for all lmters. The second problem was presented n [] and has ntal condtons gven by, x [.,.],, x (., 3.], (4.5 u(x, =, x (3., 4.8], otherwse. The soluton doman s for x [, 5] wth homogeneous Drchlet condtons. Ths problem ncludes two shocks ntally at x = and x = 3, movng to the rght and left, respectvely, and two expanson dscontnutes at x =. and x = 4.8 also expandng to the rght and left, respectvely. The two shocks collde at tme t = 1s and form a sngle shock movng to the left. The numercal results are presented n Fgures 5 and 6 for t = s when the shocks have combned nto a sngle one. A grd of 11 ponts was used for all schemes wth CFL number equal to.9. All the schemes

12 1938 CH. ARVANITIS AND A. I. DELIS LF Roe LF-Adaptve Roe-Adaptve Fg. 3. Transonc rarefacton problem: Numercal soluton (left and grd pont trajectores (rght for the adaptve Roe scheme (p =9, D =. LW-Adaptve MacCormack-Adaptve MUSCL MUSCL-Adaptve Fg. 4. Transonc rarefacton problem: Numercal soluton (left for the LW and MacCormack schemes (p =6, D =and (rght for the MUSCL scheme (p =.1, D =. produce mproved results when the adaptve mechansm s mposed, when compared to those produced n a unform mesh. It s mpressve that even the second order oscllatory LW and MacCormack schemes are now able to produce accurate solutons. The performance of the MUSCL usng the MC lmter scheme s also greatly mproved, snce n the nonadaptve case the scheme fals to produce the entropy correct soluton at expansons. Smlar observatons were made for the other lmters as well. In Fgure 6 one can also see the ablty of the mesh to capture and follow the evoluton of the soluton as demonstrated by the grd pont trajectores. The thrd problem has an ntal profle of a smooth wave gven by u(x, = sn(πx + sn(πx, x [, 1], wth perodc boundary condtons mposed. The soluton propagates to the rght, steepenng untl the sngularty tme t c =64/(19π. Results are presented n Fgures 7 and 8 for the second order schemes for a 61-pont grd and wth a CFL number of at t = 1s, and verfy the observatons made for the second example. Consderably better results are obtaned for the adaptve methods, compared to the ones n unform grd, especally n shock resoluton because of the clusterng of grd ponts near the shock. Agan the grd trajectores show the formaton and followng of the shock layer.

13 FINITE VOLUMES ON ADAPTIVE REDISTRIBUTED GRIDS Roe Roe-Adaptve LW - Adaptve MacCormack - Adaptve Fg. 5. The second Burgers problem: Numercal soluton Roe s scheme (left (p = 79, D =4 and the LW and MacCormack schemes (rght (p =.1, D =.. MUSCL MUSCL-Adaptve Fg. 6. The second Burgers problem: Numercal soluton (left and grd pont trajectores (rght for the adaptve MUSCL scheme (p =9, D = LW LW-Adaptve MacCormack-Adaptve Fg. 7. The thrd Burgers problem: Numercal soluton for the LW and MacCormack schemes (left and grd pont trajectores (rght for the LW scheme (p =6, D =. The fourth problem has ntal condton u(x, =1 x, x [, 1], and has been selected to llustrate convergence to a dscontnuous steady state. The

14 194 CH. ARVANITIS AND A. I. DELIS MUSCL MUSCL-Adaptve Fg. 8. The thrd Burgers problem: Numercal soluton (left and grd pont trajectores (rght for the MUSCL scheme (p =.118, D = LF Roe LF-Adaptve Roe-Adaptve Fg. 9. The fourth Burgers problem: Numercal solutons for the Roe and LF schemes (left and grd pont trajectores (rght for the adaptve Roe scheme (p =.1, D =13. parameters for the calculatons were 61 grd ponts and a CFL number equal to.9. The stablzng tendency of the adaptve method can be clearly seen snce t does not produce oscllatory results, as shown n Fgures 9 and 1 for t = 15s. One can also observe the mprovement n all the calculatons when the adaptve mechansm s mposed. The effect n the grd movement of a zero and a nonzero value for the cutoff parameter D, for Roe s frst order scheme, can be seen n Fgure 11, followng the remarks made n secton Nonconvex conservaton law and the Buckley Leverett equaton. In ths secton we frst apply the adaptve grd method to the scalar conservaton law wth nonlnear nonconvex flux, (4.6 f(u =(u 1(u 4/4 for x [, 1]. The ntal profle s u(x, = sgn(x. As ponted out n [9] solvng ths Remann problem for a scalar conservaton law wth nonconvex flux leads to dffcultes wth some numercal schemes. Numercal solutons for all schemes are shown n Fgure 1 at t =1.s, n a 11-pont grd and for CFL number equal to. Agan by mposng the adaptve mechansm we are able to produce qualty results for all schemes.

15 FINITE VOLUMES ON ADAPTIVE REDISTRIBUTED GRIDS LW MacCormack LW-Adaptve MacCormack-Adaptve MUSCL MUSCL-Adaptve Fg. 1. The fourth Burgers problem: Numercal solutons for the LW and MacCormack schemes (left and MUSCL scheme (rght (p =.13, D =. 1,1 Roe Roe Adaptve (D= Roe Adaptve (D=13 Log ( Relatve L 1 Error,1,1, Tme(sec Fg. 11. The fourth Burgers problem: convergence hstores for the Roe scheme LW MacCormack LW-Adaptve MacCormack-Adaptve. 1.5 MUSCL MUSCL-Adaptve Fg. 1. Nonconvex flux problem: Numercal solutons for the LW and MacCormack schemes (left and MUSCL scheme (rght (p =69, D =.

16 194 CH. ARVANITIS AND A. I. DELIS.9 LW MacCormack LW-Adaptve MacCormack-Adaptve Fg. 13. Buckley Leverett problem: Numercal solutons for the LW and MacCormack schemes (left and grd pont trajectores for the adaptve LW scheme (p =76, D =..9 MUSCL MUSCL-Adaptve Fg. 14. Buckley Leverett problem: Numercal solutons for the MUSCL scheme (p =.1, D =88. Next we consder the scalar Buckley Leverett equaton wth the flux functon f(u = u u +(1 u, wth ntal and boundary condtons for x [, ], u(x, = 1 1+1x, u(,t=1, u(1,t= 1 1. The soluton for the adaptve and unform LW and MacCormack schemes s shown n Fgure 13 n a 11-pont grd. The hghly oscllatory behavor of these schemes s totally suppressed by the adaptve mechansm resolvng and followng the shock layer much more accurately. Improvement n the shock resoluton can also be observed for the MUSCL scheme n Fgure 14. Remark 4(a result on the smoothng effect of the AGR. As a fnal result for scalar CL we present a smple advecton test (see [6] of a contact dscontnuty n order to measure the smoothng effect of the adaptve mechansm. We compute an

17 FINITE VOLUMES ON ADAPTIVE REDISTRIBUTED GRIDS Roe Roe-Adaptve MUSCL MUSCL-Adaptve Fg. 15. Lnear advecton of a contact dscontnuty for the Roe scheme (p =3, D =4 and MUSCL scheme (p =.1, D =. approxmate soluton to the lnear advecton equaton, u t + u x =, wth perodc boundary condtons and ntal condton { sn(x/, x [,π, u(x, = sn(x/, x [π, π, that has a dscontnuty at x = π whch at t = s s advected to x = π +. Results on a 11-pont grd for the Roe and MUSCL schemes, wth CFL =.95, are shown n Fgure 15. The smoothng effect of the adaptve mechansm can be seen for the frst order Roe scheme, but for the second order MUSCL scheme the results have been mproved. 5. Numercal schemes for nonunform grds (one-dmensonal systems of equatons. We consder the general system of conservaton laws wth a source term added,.e., (5.1 U t + F(U x = G(U, U R M ; here F(U s the flux functon and G s the source term. We can numercally approxmate system (5.1 by usng the explct conservatve numercal scheme wth a source term approxmaton wthn the fnte dfference-volume frame, as U n+1 = U n Δt ] (5. [F n Δx + F n 1 + Δt Gn 1 Δx, where agan F n, =1,,...,N 1, s the numercal flux functon, and the approxmaton of the source term G n s consdered as the cell average value (numercal + 1 source ntegral over the computatonal cell satsfyng the relatonshp (5.3 G n = 1 Δx x+ 1 x 1 G(x, U n dx, and after choosng the numercal flux functon t remans to choose an approprate approxmaton (dependng on the scheme used for the numercal source ntegral G n. For the statonary case, U t =, the flux functon and source term balance: F(U x = G(U.

18 1944 CH. ARVANITIS AND A. I. DELIS Therefore, an accurate numercal scheme should also balance the numercal flux wth the source term approxmaton, (5.4 F n + 1 F n 1 = G n. The numercal schemes presented next ncorporate the numercal source ntegral n such a way that (5.4 s satsfed Roe s scheme for systems of conservaton laws wth source terms. The numercal flux for the Roe upwnd scheme can be wrtten n the form (5.5 The matrx J + 1 F ROE + 1 = 1 [ ] F +1 + F J + 1 (U +1 U. s the Roe-type lnearzaton and satsfes (5.6 F +1 F = J + 1 (U +1 U and has real egenvalues (ã k, k =1,...,M, and a complete set of egenvectors + 1 (ẽ k. Representng now U U n terms of the egenvectors of the Roe lnearzaton,.e., (5.7 U +1 U = M k=1 α k + ẽ k 1 +, 1 we obtan from (5.6 (5.8 J + 1 (U +1 U = M k=1 ã k + α k 1 + ẽ k Then the Roe numercal flux can be wrtten as [ ] F ROE + = 1 M (5.9 F F ã k + α k 1 + ẽ k The α + 1 (wave strengths values follow from (5.7. Source term upwndng. The ntegral of G, n relaton (5.3, over the cell [x 1,x + 1 ] s frst splt nto a sum of two ntegrals n the [x 1,x ], [x,x + 1 ]as (5.1 1 Δx x+ 1 x 1 G(x, U n dx = 1 Δx [ x k=1 x 1 G(x, U n dx + x+ 1 x ] G(x, U n dx. The upwnd defnton of the approxmated source terms G n gven by for nonunform grds s (5.11 G n = 1 [ (x x 1 Δx G L (x 1,x, U n 1, U n + (x +1 x ] G R (x,x +1, U n, U n +1,

19 FINITE VOLUMES ON ADAPTIVE REDISTRIBUTED GRIDS 1945 where G L and G R are contnuous functons, gvng the averages of G(x, U n onthe subcells [x 1,x ], [x,x + 1 ], respectvely. These functons gve the upwnd values of the source terms and followng [7] can be defned as [ ] [ ] (5.1 G L = I + J 1 1 J 1 G 1 1, GR + = I J J 1 G The defnton of the J ± 1 and J ± 1 matrces follow from (5.6 for the Roe lnearza- represent approxmatons of ton and are defned n the appendx. The terms G ± 1 G at cells ( 1, and (, + 1, respectvely. Alternatvely, we rewrte (5.1 n decomposed form, (5.13 G L 1 = M k=1 β k ẽ k 1 1 ( 1 + sgn(ã k, GR 1 + = 1 M k=1 β k + ẽ k ( 1 sgn(ã k +, 1 wth the values of β k ± 1 determned from (5.14 M k=1 β k ± ẽ k 1 ± = G 1 ± The LW scheme. The numercal flux for the classcal second order LW scheme for nonlnear systems, on nonunform grds, can be wrtten, followng [18], as ( F LW + = F ROE M Δx (5.15 Δt ã k 1 + ã k 1 + α k 1 + ẽ k 1 +, 1 k=1 Δx + 1 Δx + 1 where Δx + 1 =(Δx +1 +Δx /. A flux-lmted second order TVD verson of the LW scheme s then gven by (5.16 where F LWTVD + 1 = F ROE ( M k=1 Δx Δx + 1 Δt Δx + 1 ã k + 1 Φ(θ k + ã k 1 + α k 1 + ẽ k 1 +, 1 (5.17 θ k + 1 = α k I+ 1 α k + 1 wth I = { 1 f ã k >, f ã k < ; + 1 here Φ can be any of the lmter functons presented n secton 3. The Roe averaged values can be used for the source term approxmatons, and we obtan a flux-lmted second order approxmaton of the source term n the numercal source term (5.1, smlar to the one presented n [16], by settng [ ( ( ] G L 1 G R + 1 = = M k=1 M k=1 β k ẽ k 1 1 β k + ẽ k sgn(ã k 1 1 Φ(θ k 1 [ ( ( 1 sgn(ã k + 1 Φ(θ k Δx Δx 1 Δx Δx + 1 Δt Δx 1 Δt Δx + 1 ã k 1 ã k + 1, ].

20 1946 CH. ARVANITIS AND A. I. DELIS 5.3. The MacCormack scheme. The MacCormack scheme adapted to approxmate systems of conservaton laws wth a source term can be wrtten as (5.18 where U n+1 = 1 ( U (1 U n + U (1 Δt ( F (1 F (1 1 + Δt Δx Δx G(1, 1 = U n Δt ( F n Δx +1 F n Δt + Gn Δx +. 1 Ths scheme s a predctor-corrector one and has the advantage that we do not need to approxmate the egenvalues and egenvectors of the Jacoban matrx. However, spurous oscllatons may occur n the soluton, especally when dscontnutes are present, even when we choose the G dscretzatons n (5.18 n such a way as to satsfy ( The shallow water model. We consder the well-known one-dmensonal shallow water system, wth a geometrcal source term (the bottom topography added, wrtten n dfferental conservaton law form as a sngle vector equaton: (6.1 wth U = [ h hu ] [, F(U = U t + F(U x = G(U, hu hu + g h ] [, G(U = ] ghz. System (6.1 descrbes the flow at tme t atpontx R, where h(x, t sthe total water heght above the bottom, u(x, t s the average horzontal velocty, Z(x s the bottom heght functon, and g the gravtatonal acceleraton. In the followng we wll denote by q = hu the water unt dscharge. An mportant property of system (6.1 s related to the source term: the shallow water system admts nontrval steady states. They are characterzed by (6. (6.3 (hu x =, ( hu + g h = ghz, x.e., (6.4 (6.5 q =(hu = constant, hu + g(h + Z = constant. A partcular case that provdes a benchmark for many approxmatng schemes s the stll water steady state (flow at rest,.e., when u = and h + Z = constant. In [7] the concept of Property C was ntroduced. A gven scheme would satsfy Property C f, n the case of a flow at rest, there s an exact balance between the dscrete components of the flux and a gven source term treatment based on (5.4, approxmatng ths statonary soluton. Thus, when numercally treatng the source terms, for Property C to be satsfed exactly or approxmately (to order O(Δx one must ensure that ths equlbrum soluton would not be perturbed. The source term dscretzatons presented n the prevous sectons along wth those presented n the appendx for the shallow water system satsfy ths property.

21 FINITE VOLUMES ON ADAPTIVE REDISTRIBUTED GRIDS 1947 In the numercal examples presented next we use agan a varable tme step as to satsfy the CFL 1 stablty condton: ã k+ CFL=Δt max 1,k Δx, ã k ( Δx, where for the shallow water model values ã k+ 1 and ã k + 1 are defned n the appendx Idealzed dam-break flow. We frst consder a nonstatonary case, the dam-break problem n a rectangular channel wth flat bottom, Z =. We computed the soluton on a channel of length L = m for tme t = 5s and wth ntal condtons u(x, =, { h(x, = h 1, x 1, h, x > 1, wth h 1 > h. Ths s the correspondng Remann problem for the homogeneous problem. The water depth rato s gven by h /h 1. The dam collapses at t =s and the resultng flow conssts of a shock wave (bore travelng downstream and a rarefacton wave (depresson wave travelng upstream. The upstream depth h 1 was set at 1m. When the depth rato s greater than, the flow throughout the channel remans subcrtcal. For depth ratos smaller than, the flow downstream of the dam poston s supercrtcal whle remanng subcrtcal upstream. For very small values of the rato h /h 1 the flow regme becomes strongly supercrtcal downstream and the shock wave can be dffcult to capture; see [1], for example. Analytcal solutons to ths problem can be found, for example, n [4]. Comparatve results are presented n a 11-pont grd and depth rato 5, for all schemes n Fgures 16 18, usng CFL =.9. All the results produced wth the adaptve method are sgnfcantly better n terms of shock resoluton and nonoscllatory behavor n both components of the soluton. No entropy problems appeared for all adaptve schemes. The grd clusterng and the close followng of the downstream shocks can also be seen. It s mpressve that the LW and MacCormack schemes are able to produce qualty results wth no extra dsspatve mechansm mposed other than the adaptve one. The performance of the TVD scheme, usng the VL lmter, was also mproved. A more quanttatve comparson s afforded n Table 1, where we compare the adaptve and nonadaptve Roe and TVD schemes n terms of computatonal costs, reported as the CPU tmes, the number of tme steps (NT, and the L 1 errors for h and q. It s clear that the adaptve schemes produce accurate results wth fewer grd ponts than the nonadaptve schemes, and even though the total number of tme steps s ncreased, for the adaptve schemes, CPU tmes are substantally smaller (snce n nonadaptve schemes one has to obtan solutons to many more grd ponts. For example, the adaptve grd calculatons wth 11 grd ponts are as accurate as for a fxed grd wth 41 ponts, and requre less CPU tme. The effect of the parameter D can also be seen n Table Flow at rest over topography. We consder system (6.1 wth ntal condtons u(x, = x R, h(x, + Z(x =H x R.

22 1948 CH. ARVANITIS AND A. I. DELIS Roe Roe-Adaptve Roe Roe-Adaptve Fg. 16. Idealzed dam-break flow: results wth Roe s scheme for h (left, q (mddle, and grd pont trajectores (rght for the adaptve Roe scheme (p =518, D = LW LW-Adaptve MacCormack-Adaptve LW LW-Adaptve MacCormack-Adaptve Fg. 17. Idealzed dam-break flow: results wth the LW and MacCormack schemes for h (left, q (mddle, and grd pont trajectores (rght for the adaptve LW scheme (p =.1, D = TVD TVD-Adaptve TVD TVD-Adaptve Fg. 18. Idealzed dam-break flow: results wth the TVD scheme for h (left, q (mddle, and grd pont trajectores (rght for the adaptve TVD scheme (p =.1, D =15. Then, clearly, u(x, t = x R,t, h(x, t+z(x =H x R,t s a soluton to (6.1. We test the schemes wth the adaptve mechansm n order to study ther behavor

23 FINITE VOLUMES ON ADAPTIVE REDISTRIBUTED GRIDS 1949 Table 1 Idealzed dam-break flow: comparatve performance of the Roe and TVD schemes. Descrpton CPU tme (sec NT L 1 error (h L 1 error (q Roe Fxed grd: N = N = N = N = Roe+AGR: N = 51, p =5, D = N = 11, p = 35, D = N = 11, p = 5, D = N = 11, p = 5, D = TVD Fxed grd: N = N = N = N = TVD + AGR: N = 51, p =.1, D = N = 11, p =.1, D = N = 11, p =.1, D = to ths benchmark problem [13, 9, 11] wth Z(x gven by {. 5(x 1, 8 x 1, (6.7 Z(x = otherwse, n a channel of length L = 5m and H = m. Fgures 19 and dsplay the fnal water level and the fnal unt dscharge values, respectvely, for the Roe and MacCormack schemes at tme t = s and n a grd of 11 ponts. Both schemes wth the adaptaton method perfectly preserve ths steady state, up to machne accuracy. Grd ponts are concentrated where most needed, n the topography area. The movement of the grd ponts s stoppng much faster for the MacCormack scheme beng a second order method n tme. Smlar results and observatons were made for the TVD scheme that are not presented here for brevty Steady transcrtcal flow. In ths case, and for the same ntal condtons as n secton 6., we mpose an upstream boundary condton for the dscharge q = 1.53m /s and a downstream boundary condton for the water level H =6m only n the case where the flow s subcrtcal. If the flow becomes supercrtcal downstream, no condton for the water level s mposed. Results at t = 5s n a 51-pont grd are presented n Fgures 1 and for the Roe and TVD schemes (usng the MM lmter and CFL =.9. In Table we quantfy the errors and performance of the schemes for ths test problem. Smlar observatons wth those n secton 6.1 can be deduced. The mportance of usng the parameter D for steady state problems can be clearly seen Steady transcrtcal flow wth shock. The ntal condtons n ths case was taken to be as n the prevous secton, wth H beng the constant water level downstream provded by the boundary condton. In ths test case the upstream boundary condton for the dscharge was q =.18m /s and the downstream boundary condton for the water level was H =.33m.

24 195 CH. ARVANITIS AND A. I. DELIS Topography Roe-Adaptve Roe-Adaptve Fg. 19. Flow at rest over topography: results wth the adaptve Roe scheme for h + Z (left, q (mddle, and grd pont trajectores (rght (p =6, D = Topography MacCormack-Adaptve MacCormack-Adaptve Fg.. Flow at rest over topography: results wth the adaptve MacCormack scheme for h + Z (left, q (mddle, and grd pont trajectores (rght (p =6, D = Roe Roe-Adaptve Roe Roe-Adaptve Fg. 1. Steady transcrtcal flow: results wth Roe s scheme for h + Z (left, dscharge (q (rght (p =75, D =6. Results for a coarse 51-pont grd for the Roe and TVD schemes and also on a 11-pont grd for the TVD scheme wth the MM lmter used are shown n Fgures 3 5. The mprovement n the results compared to the unform grd results s clear. The shock resoluton s very good and the steady state s perfectly preserved for a

25 FINITE VOLUMES ON ADAPTIVE REDISTRIBUTED GRIDS TVD TVD-Adaptve TVD TVD-Adaptve Fg.. Steady transcrtcal flow: results wth the TVD scheme for depth h+z (left, dscharge (q (rght (p =.1, D =5. Table Steady transcrtcal flow: performance of the Roe and TVD schemes. Descrpton CPU tme (sec NT L 1 error (h L 1 error (q Roe Fxed grd: N = N = N = Roe+AGR: N = 51, p =75, D = N = 11, p =75, D = N = 11, p =75, D = TVD Fxed grd: N = N = TVD + AGR: N = 51, p =.1, D = N = 11, p =.1, D = N = 11, p =.1, D = long tme calculaton of t = 8s. The CFL value used n all tests was. Fnally n Fgure 6 the numercal results for the depth are shown for the adaptve MacCormack scheme, where n spte of the formaton of a strong shock the scheme can produce a nonoscllatory steady state soluton, especally when applyng the parameter D Dam-break flow over topography. In ths example we solve the shallow water equatons wth a wavy bottom Z(x, {.3(cos(π(x 1/ 3, x 1 1, (6.8 Z(x = otherwse, and ntal condtons {. Z(x, 1 x<1, (6.9 h(x, =.35 Z(x, 1 x<1, u(x, = { 1, 1 x<1,, 1 x<1,

26 195 CH. ARVANITIS AND A. I. DELIS Roe, Topography Roe-Adaptve Roe Roe-Adaptve Fg. 3. Steady transcrtcal flow wth shock: results wth Roe s scheme for h + Z (left, q (mddle, and grd pont trajectores (rght for the adaptve Roe scheme (p =8, D = TVD TVD-Adaptve.3.3 TVD TVD-Adaptve Fg. 4. Steady transcrtcal flow wth shock: results wth the TVD scheme for h + Z (left, q (mddle, and grd pont trajectores (rght for the adaptve TVD scheme on a 51-pont grd (p =8, D = TVD, Topography TVD-Adaptve TVD TVD-Adaptve Fg. 5. Steady transcrtcal flow wth shock: results wth the TVD scheme for h + Z (left, q (mddle, and grd pont trajectores (rght for the adaptve TVD scheme on a 11-pont grd (p =8, D =75. n order to test the adaptve behavor of the schemes n an unsteady flow over topography; ths test was also presented n [8]. The soluton profles for the water depth and velocty profle at tme t = 1s are shown n Fgures 7 9. A grd of 11 ponts was used n all calculatons. The exact soluton for ths problem s a reference soluton calculated n a unform grd wth 41 grd ponts. The mprovements n the numercal soluton can be observed for all the schemes; even the MacCormack scheme can produce a stable soluton when compared wth the one calculated n a unform grd.

27 FINITE VOLUMES ON ADAPTIVE REDISTRIBUTED GRIDS , Topography MacCormack-Adaptve.35.3, Topography MacCormack-Adaptve Fg. 6. Steady transcrtcal flow wth shock: results for h + Z wth the adaptve MacCormack scheme wth p =.1, D =(left and wth p =.1, D =3 (rght Roe Roe-Adaptve Roe Roe-Adaptve Fg. 7. Dam-break flow over topography: results wth Roe s scheme for h (left, u (mddle, and grd pont trajectores (rght for the adaptve Roe scheme (p =.19, D = TVD TVD-Adaptve TVD TVD-Adaptve Fg. 8. Dam-break flow over topography: results wth the TVD scheme for h (left, u (mddle, and grd pont trajectores (rght for the adaptve TVD scheme (p =.11, D =6. 7. Conclusons. In ths work we have rgorously nvestgated, n a general framework, the numercal behavor of an adaptve grd redstrbuton (AGR mechansm, when combned wth classcal fnte volume numercal methods wth well-known characterstcs, for solvng one-dmensonal hyperbolc conservaton laws. The evolvng mesh s constructed such that ts spatal resoluton s controlled va selectve geometrcal characterstcs of the numercal soluton, by choosng some power of the curvature of the soluton as the estmator functon. A conservatve reconstructon procedure for the numercal soluton s also appled at each evoluton step.