E COLE NORMALE SUPERIEURE A moded structured central scheme for 2D hyperbolc conservaton laws Theodoros KATSAOUNIS Doron LEVY LMENS - 98-30 Département de Mathématques et Informatque CNRS URA 762
A moded structured central scheme for 2D hyperbolc conservaton laws Theodoros KATSAOUNIS Doron LEVY LMENS - 98-30 June 1998 Laboratore de Mathematques de l'ecole Normale Supereure 45 rue d'ulm 75230 PARIS Cedex 05 Tel : (33)(1) 01 44 32 30 00 Adresse electronque : katsaoun@dm.ens.fr, dlevy@dm.ens.fr
A Moded Structured Central Scheme for 2D Hyperbolc Conservaton Laws Theodoros Katsaouns y Doron Levy y Abstract We present a new central scheme for approxmatng solutons of two-dmensonal systems of hyperbolc conservaton laws. Ths method s based on a modcaton of the staggered grd proposed n [5] whch prevents the crossngs of dscontnutes n the normal drecton, whle retanng the smplcty of the central framework. Our method satses a local maxmum prncple whch s based on a more compact stencl. Unlke the prevous method, t enables a natural extenson to adaptve methods on structured grds. Key words. Hyperbolc conservaton laws, central derence schemes, non-oscllatory schemes, adaptve methods. 1 Introducton In [5] Jang and Tadmor presented a second-order two-dmensonal central scheme for approxmatng solutons of systems of hyperbolc conservaton laws whch extends the one-dmensonal Nessyahu-Tadmor (NT) scheme [7]. Smlar approach was taken by Armnjon, et al. n [1], [3]. Followng the central framework whose prototype s the Lax and Fredrchs scheme [4], a Godunov-type scheme was constructed. Frst, a pecewse-lnear MUSCL-type [6] nterpolant was reconstructed from the gven cell-averages. Spurous oscllatons n the reconstructon were avoded by mplementng a non-lnear lmtng mechansm [8]. Ths nterpolant was then evolved exactly n tme and nally projected on ts staggered cell-averages. Due to the staggerng, there was no need to solve two-dmensonal Remann problems. Unfortunately, the staggerng was not sucent to elmnate the dscontnutes from the problem, and one was actually left wth one-dmensonal Remann problems n the normal drecton. In the method proposed n [5], nstead of explctly solvng these 1D Remann problems, the values around the dscontnutes were averaged. The dsspatve treatment of those dscontnutes n the normal drecton resulted n several numercal consequences such as smearng of the dscontnutes as evdent n the numercal results presented n [5]. In ths work we present a new central scheme whch was desgned to avod the crossngs of dscontnutes n the normal drecton. Our goal s obtaned by exchangng the orgnal staggered mesh wth an alternatve rotated and stretched mesh. All that follows, s a drect mplementaton of the prevously desgned methods wth our new meshes. Ths new structured mesh can be vewed as a degenerate verson of the unstructured mesh used n [2]. Along wth the man advantage of our new method, several byproducts are n hand. Frst, the method n [5] when vewed n every two tme steps conssted of a 9-ponts stencl. Our method, however, s based on a more compact 5-ponts stencl. Moreover, unlke the method n [5], our y Departement de Mathematques et d'informatque, Ecole Normale Supereure, 45 rue d'ulm, 75230 Pars Cedex 05, France; dlevy@dm.ens.fr, katsaoun@dm.ens.fr 1
new method can be easly extended to adaptve central schemes whch are based on structured staggered grds. Such an extenson seems to be hghly non-trval n the prevous framework. Ths paper s organzed as follows: we start n x2 by presentng our new central method. The smplcty compared wth an adaptve unstructured framework s emphaszed by the explct formulaton of the method outlned below. We then proceed n x3 to formulate and prove a maxmum prncple on the scheme. We end n x4 wth several numercal examples. Acknowledgment: Research was supported by Hyperbolc systems of conservaton laws TMR grant #ERBFMRXCT960033. We would lke to thank Prof. E. Tadmor for hs remarks. 2 The 2D Method We consder the two-dmensonal system of conservaton laws v t + f(v) x + g(v) y = 0; (2.1) augmented wth the ntal data, v 0 (x; y) = v(x; y; t = 0). To approxmate solutons of (2.1), we rst ntroduce a unform rectangular mesh n the (x; y) plane, wth spacngs taken as x; y. On top of ths mesh, we then buld a staggered mesh, whose cells are of the shape of damonds, consult Fgure 2.1 (a). x y (a) (b) Fgure 2.1: (a) The Staggered Mesh. Sold lnes - rectangular grd. Dotted lnes - damond-shaped grd. (b) A Structured Adaptve Grd. By and ~ we denote the rectangular S cells and the damond cells, respectvely. The resultng meshes are therefore abbrevated as = and ~ S = ~. For smplcty of notatons we use a xed tme-step t, and denote the dscrete tme by t n = nt. By u n we denote an approxmaton to the cell-average v n n cell and at tme t n. In the rst phase of the staggerng we assume that u n ~ are gven and we wsh to compute u n+1. In the second phase, the roles of ~ and are exchanged. The reconstructon of our Godunov-type second-order method starts by reconstructng a pecewselnear nterpolant from the gven cell-averages u(x; y; t n ) = X ~ P (x; y; t n ) ~ : (2.2) Here, ~ denoted the characterstc functon of the cell ~ whle P s a lnear polynomal ; P (x; y; t n ) = u n ~ + u 0 ~ x? xc x + u 8 ~ y? yc y 2
where the center of ~ s denoted by (x c ; y c ), and u 0 ~, u 8 ~, are the dscrete slopes n the x- and y-drectons respectvely, u 0 ~ x u x (x c ; y c ; t n ) + O(x) 2 ; u 8 ~ y u y (x c ; y c ; t n ) + O(y) 2. The reconstructon of the slopes utlzes nonlnear lmters descrbed n the remarks below. N N W E NW NE (a) S (b) W SW SE E (c) S Fgure 2.2: Reconstructon: (a) squares from damonds, (b)-(c) damonds from squares. An exact evoluton of (2.2) followed by a projecton on ts staggered cell-averages results wth (consult Fgure 2.2 (a)) { u n+1 = u n? 1 j j Z t n+1 t n Z Z f x + The rst term of the RHS of (2.3), I 1, equals I 1 = u n = un N + u n E + u n S + u n W 4 + 1 24 Z Z g y dxdyd := I 1 + I 2 : (2.3) x(u 0 W? u0 E) + y(u 8 S? u8 N) The ntegrals over the uxes n the second term of the RHS of (2.3), I 2, are approxmated by a second-order md-pont quadrature rule, an approxmaton whch s vald as long as t s done n a smooth regon. The smoothness requrement results wth a CFL condton on the stablty of the method whch due to geometrc consderaton equals 1=4 (compared wth 1=2 n [5]). Ths quadrature can be then explctly wrtten as I 2 =? f(u n+1=2 E )? f(u n+1=2 W )? g(u n+1=2 N )? g(u n+1=2 S ) : ; (2.4) where = t=x and = t=y are the usual xed mesh ratos. The values at tme t n+1=2 requred n (2.4) are predcted usng a rst-order Taylor expanson (whch s sucent for overall second-order accuracy due to the predctor-corrector structure of the method). Hence, for example, = u n E? f(u 2 E) 0? g(u 2 E) 8, and analogously for the other ntermedate values. In the second phase of the staggerng we assume that the values, u n are known and we wsh to compute u n+1 ~. In order to smplfy the notatons we agan advance n tme from tme to tn t n+1. The computaton s analogous to the rst phase, only ths tme due to the lack of symmetry between the two phases of the staggerng, the resultng formulas are derent. Here, there are two possbltes whch are schematcally drawn P n Fgure 2.2 (b)-(c). For both cases an exact evoluton n tme of our reconstructon, u(x; y; t n ) = P (x; y; t n ), yelds (compare wth (2.3)), u n+1=2 E u n+1 ~ = u n ~? 1 j~ j Z t n+1 t n Z Z ~ f x + Z Z ~ g y dxdyd := I 1 + I 2 : (2.5) 3
We start wth the case descrbed n Fgure (b). Here, I 1 = u n ~ = un W +un E + x 2 6 (u0 W? u0 E), and X I 2 =?p 2 + 2 f(u n+1=2 @ c ~ ) ~n x + g(u n+1=2 j @ c ~ ) ~n y : (2.6) j u n+1=2 @ c j ~ = u n @ c j ~? t 2 @ j ~ The md-values, u n+1=2 @ c ~, requred n (2.6) are once agan predcted by Taylor expanson j hf u (u n@cj ) u 0 + g ~ @cj ~ u (u n@cj ) u 8@cj ; ~ ~ where @ c ~ j represents the center of the edges of the damond cells, j = fne,se,nw,swg (see Fgure 2.2 (b)). Here, Du NE = Du SE = Du E and Du NW = Du SW = Du W, wth D denotng the dscrete dervatve ether n the x- or n the y-drecton. The pont-values, u n @ c ~, are also j computed by a Taylor expanson, e.g., u n NE = u n E? x 4 u0 E + y 4 E. u8 Analogous computatons hold for the last case descrbed n Fgure 2.2 (c). For the sake of brevty we lst only the rst term on the RHS of (2.5), I 1, whch n ths case equals I 1 = u n ~ = u n N +un S 2 + y 6 (u8 S? u8 N). Remarks: 1. Reconstructon of the Dervatves. A reconstructon of the dervatves wthout creatng spurous oscllatons requres non-lnear bult-n lmters. One can use, for example, for the x-dervatve n a rectangular cell (j; k), a lmtng on the rght/centered/left dervatves u 0 = MM (u n? j;k j+1;k j;k); 1 un 2 (un? j+1;k j?1;k); (u un n? j;k j?1;k) ; (2.7) un wth 1 2, and MMfu1; u2; : : :g = 8 < : mn fu g; f v > 0; 8; max fu g; f v < 0; 8; 0; otherwse: The choce of = 1 agrees wth the classcal Mn-Mod lmter (see [8] and [5] for more detals). Analogous expresson holds for the y-dervatve. The same routne s repeated for the dervatves n the damond cells. Only ths tme, snce the cells are not algned wth the axes, one has to lmt the dervatves after projectng them on the x and y drectons. For systems, the dervatves are computed component-wse,.e., f 0 j;k are gven by (2.7), consult [7, 5]. = f u(u j;k )u 0 j;k, where u0 j;k 2. Adaptve Mesh. A possble extenson of the method to adaptve method on unstructured meshes s demonstrated n Fgure 2.1 (b). An equvalent extenson wth the prevous method n [5] seems to be mpossble, at least wthout dealng wth complcated cases at the boundares (corners, etc...) and wth uneven dvsons of the cells. Our method formulated on the new mesh requres no specal corner treatment. We consder ths to be the great advantage of our method over the other avalable structured 2D methods. Moreover, snce there s no upwndng nvolved, none of the reected-waves problems whch are typcal to upwndng methods on adaptve structured meshes should appear. 4
3. Ecent Implementaton. We note that the smplcty of the scheme can be drectly projected onto ts mplementaton. It s unnecessary to use the standard methods of the unstructured framework n order to mplement our method. The smplest data structure to store the values n the damonds would be to dvde them nto trangles and to store the values n the trangles n a two-dmensonal array, such that every pont n the array corresponds to the rectangle whch these trangles belong to. In fact, the values of only two trangles out of four n each rectangle should be stored as the values of the other two can be retreved from the neghborng cells. 3 A Maxmum Prncple for Scalar Approxmatons An equvalent maxmum prncple to Theorem 1 n [5] mples to our new scheme. Snce our scheme s non-symmetrc between the two phases of the staggerng, t s natural to formulate the theorem n a non-staggered verson by consderng two jont tme-steps. Ths results wth a local bound on the cell-averages based on values taken from a 5-pont stencl. An equvalent two tme-steps formulaton of Theorem 1 n [5] would be based on a 9-pont stencl. Theorem 3.1 Consder the two-dmensonal scalar scheme (2.3),(2.5). Assume that the dscrete slopes satsfy the lmter property (2.7). Then for any 1 < 2 there exsts a sucently small CFL number C, such that f the followng CFL condton s fullled max( max u jf u (u)j; max jg u (u)j) C ; u then the followng local maxmum prncple holds mnfu n j?1;k; u n j;k; u n j+1;k; u n j;k?1; u n j;k+1 g un+2 j;k maxfun j?1;k; u n j;k; u n j+1;k; u n j;k?1; u n j;k+1 g: The proof of Theorem 3.1 s analogous to the proof of [5, Theorem 1] and we omt t for brevty. The key observaton for the proof s that every new staggered cell average can be wrtten as a convex combnaton of sums and derences of the cell-averages n the supportng cells. 4 Numercal Examples In Table 4.1 we present the L 1 and L 1 errors and convergence rate estmates for the lnear oblque advecton v t +v x +v y = 0 subject to v 0 = sn((x+y)). Equal spacngs were used, x = y = 1=N. The CFL was taken as 0:2 and the tme T = 0:5. We used the MM lmter wth = 1. These results are ndeed comparable wth those presented n [5, Table 4.1]. N L 1 error L 1 order L 1 error L 1 order 20 0.334716-0.120738-40 0.020296 4.04 0.023076 2.39 80 0.005177 1.97 0.009852 1.23 160 0.001661 1.64 0.004155 1.25 320 0.000698 1.25 0.001740 1.26 Table 4.1: Lnear Oblque Advecton. L 1 and L 1 errors and convergence rates. 5
y y y 0 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9 (a) 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x (b) 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (c) 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Fgure 4.1: Perodc 2D-Burgers. N = 160; (a) - JT, CFL=0.4; (b) - JT, CFL=0.2; (c) - The new method, CFL=0.2. We end by presentng an example demonstratng the non-oscllatory behavor of our method. In Fgure 4.1 we show the results obtaned for the perodc two-dmensonal Burgers' equaton, v t + vv x + vv y = 0 n [?1; 1] [?1; 1], for tme T = 0:5, subject to the ntal condtons, v 0 (x; y) =?1:0; x < 0; y < 0;?0:2; x > 0; y < 0; v 0 (x; y) = 0:8; x < 0; y > 0; 0:5; x > 0; y > 0: The label JT refers to the method of [5]. The contour plots n Fgure 4.1 are zoomed nto [0; 1] [?1; 0]. Clearly, our method handles better 'dagonal' waves compared wth the method of [5], whle less smearng the dscontnutes (compare the results of both methods for the same CFL). References [1] Armnjon P., Stanescu D., Vallon M.-C., A Two-Dmensonal Fnte Volume Extenson of the Lax-Fredrchs and Nessyahu-Tadmor Schemes for Compressble Flow, Proc. 6th. Int. Symp. on CFD, Lake Tahoe, 1995, M. Hafez and K. Oshma, edtors, Vol. IV, pp.7-14. [2] Armnjon P., Vallon M.-C. and Madrane A., A Fnte Volume Extenson of the Lax-Fredrchs and Nessyahu-Tadmor Schemes for Conservaton Laws on Unstructured Grds, IJCFD, 9, (1997), pp.1-22. 6
[3] Armnjon P., Vallon M.-C., Generalsaton du Schema de Nessyahu-Tadmor pour Une Equaton Hyperbolque a Deux Dmensons D'espace, C.R. Acad. Sc. Pars, t. 320, sere I. (1995), pp.85-88. [4] Fredrchs K. O., Lax P. D., Systems of Conservaton Equatons wth a Convex Extenson, Proc. Nat. Acad. Sc., 68, (1971), pp.1686-1688. [5] Jang G.-S., Tadmor E., Nonoscllatory Central Schemes for Multdmensonal Hyperbolc Conservaton Laws, SIAM J. Sc. Comp., to appear. [6] van Leer B., Towards the Ultmate Conservatve Derence Scheme, V. A Second-Order Sequel to Godunov's Method, JCP, 32, (1979), pp.101-136. [7] Nessyahu H., Tadmor E., Non-oscllatory Central Derencng for Hyperbolc Conservaton Laws, JCP, 87, no. 2 (1990), pp.408-463. [8] Sweby P. K., Hgh Resoluton Schemes Usng Flux Lmters for Hyperbolc Conservaton Laws, SINUM, 21, no. 5 (1984), pp.995-1011. 7