Internatonal Symposum on Computers & Informatcs (ISCI 05) Hgh resoluton entropy stable scheme for shallow water equatons Xaohan Cheng,a, Yufeng Ne,b, Department of Appled Mathematcs, Northwestern Polytechncal Unversty, X an 7007, Shaanx, Chna a chengxh68@63.com, b yfne@nwpu.edu.cn Abstract: A hgh resoluton entropy stable scheme s proposed for solvng shallow water equatons n ths paper. The scheme contans two parts: entropy conservatve flux and numercal dffuson operator. To acheve hgh resoluton, a lmter s employed to guarantee the numercal dffuson term beng added around the dscontnutes automatcally. Numercal experments are presented to demonstrate the proposed scheme s capacty. Keywords: entropy stable scheme, lmter, shallow water equatons, hgh resoluton Introducton Shallow water equatons, also known as the Sant-Venant system, are wdely used to model flows n rvers, lakes and near-shore oceans. In one dmensonal case the equatons can be wrtten as a system of balance laws U + f( U) = s( xu, ), wth U = [ h, hu] T beng the conservatve vector, T f = [ hu, hu + gh ] beng the flux vector, and s = [0, ghb ] T x beng the source term. Here, bx ( ) denotes the bottom topography, hxt (,) s the water t x () 05. The authors - Publshed by Atlants Press 078
heght above the bottom and uxt (,) s the depth-averaged water velocty. The constant g s the gravty acceleraton. In Eq. (), only the geometrcal source term s consdered. Other effects, such as wnd forces and frcton on the surface, are neglected. The most strkng feature of balance laws, as well as conservaton laws, s that solutons wth dscontnutes may appear n a fnte tme even for suffcent smooth ntal data []. Thus, solutons are n general sought n the weak sense. Due to the non-unqueness of weak solutons, an addtonal crteron, termed entropy condton, must be mposed to select the physcal soluton of our nterest. Numercal schemes whch respect a dscrete verson of the entropy dsspaton statement were called entropy stable schemes. Entropy stable scheme was frst presented by Tadmor [] and has attracted much attenton n recent years [3,4,5]. Most of them were based on entropy conservatve flux and sutable numercal dffuson operator. Entropy conservatve flux preserves the entropy exactly and behaves well n smooth regons. However, spurous oscllaton wll be produced around the dscontnutes, such as shocks. Thus, some dsspatve mechansm should be contaned to acheve entropy stablty. Compared wth the popular used Weghted Essentally Non-Oscllatory (WENO), Dscontnuous Galerkn (DG) methods [6,7], entropy stable schemes satsfy the system s addtonal condton,.e. entropy nequalty, and can avod some unphyscal phenomena. These methods are promsng when solvng realstc engneerng problems. In ths paper, a hgh resoluton entropy stable method s developed to solve shallow water equatons. The scheme conssts of a second order entropy conservatve flux and a Roe-type numercal dsspaton term. Unlke the approach n [8], numercal dsspaton term s actvated on the whole the whole doman and lowers the accuracy n smooth areas. Although a constructon of second order accurate s performed to avod ths, the resultng scheme cannot be entropy stable. Here, a lmter, whch sgnals the smooth degree of solutons, s employed to make the numercal dsspaton work around the dscontnutes automatcally. By ths way, the obtaned scheme stll guarantees entropy stablty. 079
When handng the source term, the well balance property s also taken nto account. Fnally, numercal examples are presented to verfy how well the scheme performs n practce. Numercal method For smplcty, unform grds of sze sem-dscrete fnte volume scheme s gven by x are consdered. Then, a standard Here, U s the cell average on I x / x+ / d U = ( F + / F / ) s. dt x () = [, ], F + / s the numercal flux consstent wth the flux f and s s a dscretzed source term. Tme ntegraton s carred out wth a second order strong stablty preservng (SSP) Runge-Kutta method [9]: gven a soluton n U at tme step t n, the soluton at the next tme level n U + s advanced by * n n n U = U + t LU ( ), n+ n * n * U = ( U + U + t LU ( ), ) (3) where L s the rght-hand sde of (). Shallow water system possesses an entropy functon E( U ) = ( hu + gh + ghb) and the entropy varables are defned E u by V= = [ gh ( + b), u] U functons are gven by T. Then, the correspondng entropy flux = + and 3 Q( U ) hu gh u 080
J ( x, U ) ghbu, as = respectvely. Defne the entropy potental T ψ = V f Q = guh. Accordng to the lemma n [8], a numercal flux F + s entropy conservatve f EC / V F g b h u T EC + / + / = ψ + + / + /, + / + / wth the notaton: a = a a + +, + / + / (4) a = ( a + a ). Then, EC analogous to the case wth flat bottom topography, F + / s determned by h+ /u+ / EC F + / =. g h + / + h + /( u + / ) To acheve the well balanced property, the source term s approxmated by 0 s = g. ( h+ / b + h / b + / / ) x Entropy conservatve flux preserves the total entropy and performs well n smooth regons. But t wll produce spurous oscllatons n the vcnty of dscontnutes. Ths problem s handled by usng a sutable numercal operator. Consder the followng numercal flux, F = F R F Λ R V ES EC T + / + / + / + / + / + / + / (5) wth, 08
R + / + / =, g u + / gh+ / u+ / + gh+ / ( + / + / + / + / ) Λ = dag u gh, u + gh, Φ = dag( φ, φ ). + / + / + / The parameter φ + / s computed by the Combned Superbee lmter[0]. We want to stress here that the above defned numercal flux s entropy stable. The ntroduced Φ possesses a property of beng close to at shocks and vanshng away from shocks so that the numercal dffuson works around dscontnutes automatcally. Numercal experments In all smulatons below, the gravtatonal constant s fxed to be g = and Neumann boundary condtons are mposed. The reference solutons are computed from a well balanced second order central upwnd scheme from [] on a mesh of 000 ponts. Frst, small perturbaton of a steady state soluton s consdered. The bottom contans a hump, 0.5(cos(0 π ( x 0.5)) + ), x 0.5 < 0. bx ( ) = 0, otherwse. The ntal data s set to be xt, = 0 ( h+ b) = + ε and u, = 0 = 0. The perturbaton constant ε equals to 0.0 for 0. < x < 0.. Ths example s smulated on a mesh of 00 grds and the water surface level h+ bat tme t = 0.7 s presented n Fg.. It s seen that the scheme resolves the wave accurately wthout numercal oscllatons. xt 08
Fg.. Small perturbaton of a steady state soluton. (a) Intal water surface level h+ b(sold lne) and bottom topography (dotted lne); (b) Water surface level h+ bat t = 0.7 (square: numercal soluton; sold lne: reference soluton) Fg.. A statonary shock n the transcrtcal case. (a) Intal water surface level h+ b(sold lne) and bottom topography (dotted lne); (b) Water surface level h+ bat t =.8 (square: numercal soluton; sold lne: reference soluton) 083
Next, a transcrtcal flow s smulated. The bottom topography s the same as the prevous example. The ntal water surface level s set to be and the ntal water velocty s taken to be 0.3. In ths example, a steady-state appears on the surface. The computaton s performed on a mesh of 00 grds up to tme t =.8 and Fg. dsplays the water surface level. Clearly, the shock s fully captured. Conclusons In ths paper, a hgh resoluton entropy stable method has been developed for smulatng shallow water flows. Its prncpal advantage, and a major dfference from other exstng methods, s that t acheves entropy stablty. The scheme s accuracy s mproved by usng a lmter. Numercal smulatons demonstrate the proposed scheme s effcent and hgh resoluton. Acknowledgements Ths research was fnancally supported by the Natonal Natural Scence Foundaton of Chna (7043, 476) and the Doctorate Foundaton of Northwestern Polytechncal Unversty (CX046). References []. RJ LeVeque, Fnte volume methods for hyperbolc problems. Vol. 3. 00: Cambrdge unversty press. []. E Tadmor. Math. Comp., 987, 49(79):9-03. [3]. E Tadmor. Acta Numer., 003, ():45-5. [4]. US Fjordholm, S Mshra, and E Tadmor. SIAM J. Numer. Anal., 0, 50():544-573. [5]. A Hltebrand and S Mshra. Numer. Math., 04, 6():03-5. [6]. C-W Shu. SIAM Rev., 009, 5():8-6. [7]. B Cockburn and C-W Shu. J. Sc. Comput., 00, 6(3):73-6. [8]. US Fjordholm, S Mshra, and E Tadmor. J. Comput. Phys., 0, 084
30(4):5587-5609. [9]. S Gottleb, DI Ketcheson, and C-W Shu. J. Sc. Comput., 009, 38(3):5-89. [0]. PK Sweby. SIAM J. Numer. Anal., 984, (5):995-0. []. A Kurganov and D Levy. ESAIM: Math. Modell. Numer. Anal., 00, 36(3):397-45. 085