Stability properties of a family of chock capturing methods for hyperbolic conservation laws

Transcription

1 Proceedings of te 3rd IASME/WSEAS Int. Conf. on FLUID DYNAMICS & AERODYNAMICS, Corfu, Greece, August 0-, 005 (pp48-5) Stability properties of a family of cock capturing metods for yperbolic conservation laws VIRGINIA ALARCÓN, SERGIO AMAT AND SONIA BUSQUIER Departamento de Matemática Aplicada y Estadística. Universidad Politécnica de Cartagena. Paseo Alfonso XIII, Cartagena(Murcia). Spain. sergio.amat@upct.es, sonia.busquier@upct.es Abstract: In tis paper a family of local sock capturing metods for yperbolic conservation laws is presented. We analyze some stability properties in order to obtain te convergence of te scemes. Key Words: Hyperbolic conservation laws, local sock capturing metods, stability, convergence. Introduction We consider numerical approximations to weak solutions of nonlinear yperbolic conservation laws: u t + f(u) x = 0, () u(x, 0) = u 0 (x). () te initial data u 0 (x) are supposed to be piecewise smoot functions tat eiter periodic or of compact support. Let be u n j = u (x j, t n ) denote a numerical approximation to te exact solution u(x j, t n ) of ()-() defined on a computational grid x j = j, t n = n t in conservation form: u n+ j = u n j λ( ˆf n j+ ˆf n j ), (3) were λ = t and te numerical flux is a function of k variables ˆf n j+ = ˆf(u n j k+,..., u n j+k), (4) wic is consistent wit (), i.e. ˆf(u,..., u) = f(u). (5) Te importance of te following lemma (see [5]) is because it implies tat approximating te numerical flux to a ig order accuracy it is enoug to reconstruct g(x j+ ) (see equation (6)) up to te same order. Lemma (Su and Oser) If a function g(x) satisfies f(u(x)) = x+ x g(ξ)dξ (6)

2 Proceedings of te 3rd IASME/WSEAS Int. Conf. on FLUID DYNAMICS & AERODYNAMICS, Corfu, Greece, August 0-, 005 (pp48-5) ten f(u(x)) x = g(x + ) g(x ). First order metods give poor accuracy in smoot regions of te flow. Socks tend to be eavily smeared and poorly resolved on te grid. Tese effects are due to te large amount of numerical dissipation in tese scemes. Marquina [4] introduced a new local tird order accurate sock capturing metod (PHM), te main advantage of tis metod lies on te property tat it is localer tan ENO and TVD upwind scemes of te same order, (and, tus, giving better resolution of corners), because numerical fluxes depend only on four variables. In [] and [] we present some modifications of te PHM. Tey are based on different reconstructions. From te numerical experiments, te metods become efficient since tey are low cost and tey are not very sensitive to te Courant-Friedrics-Lewy (CFL) number. To complete te scemes, Su and Oser developed a special family of Runge- Kutta time integration scemes tat ave a TVD property [5], [6]. Te TVD property prevents te time stepping sceme from introducing spurious spatial oscillations into upwind-biased spatial discretization. Te aim of tis paper is to analyze some stability properties in order to obtain te convergence of tese type of scemes. Te paper is organized as follows: section contains te reconstruction step. In section 3 te complete algoritm is presented. Finally, some stability properties and a convergence analysis are studied in 4. Piecewise reconstructions We define a computational grid x j = j, j integer, > 0, were te cells are C j = {x : x j were x j+ = x j +. Our grid data are: (d j+ v j = d j+ x x j+ }, (7) xj+ g(ξ)dξ, (8) x j = v j+ v j, (9) = g (x j+ ) + O( )). We required te following conditions for every j: d i+ v j = xj+ r j (ξ)dξ, (0) x j = r j(x i+ ), i = j, j. () Taylor series expansions sow tat conditions (0) and () imply tird order accuracy of te reconstruction r j (x).. Local Total Variation Bounded reconstructions Te reconstruction procedure is repeated at every time step, tus te cange in total variation of te reconstruction must be controlled. Te local total variation of te reconstruction {r j } is defined by LT V j = T V (r j ), were T V (r j ) means te total variation of te function r j (x) in te cell C j. Te size of LT V j determines locally te increasing of te total variation of te reconstruction. In [4], te following definition is given:

3 Proceedings of te 3rd IASME/WSEAS Int. Conf. on FLUID DYNAMICS & AERODYNAMICS, Corfu, Greece, August 0-, 005 (pp48-5) Definition A metod of reconstruction is local total variation bounded (LTVB) if tere exists a constant M > 0, independing of (depending only on te function g(x) to be reconstructed), suc tat LT V j M, for all j. Te metod of reconstruction wit r j (x) is not LTVB. For nonlinear fluxes, te metod becomes unstable. As PHM sceme, we will assign a different value to te central derivative. Since our reconstruction is local, we restrict our discussion to te cell C 0 and te grid data of te cell: v 0, d, and d. Te algoritm defines te modified reconstruction r 0 (x) suc tat its derivative interpolates d 0 at x 0 and te lateral grid derivative wit smallest absolute value. Were, if C 0 is a nontransition (d d > 0) cell, ten we define d 0 suc tat d 0 r (x 0 ) = O( ) in ), r 0 (x )) = smoot regions and max( r 0 (x O(). In transition cells we consider d 0 = 0. Teorem Te above metod of reconstruction of te function g(x) is a local preprocessed reconstruction procedure tat is LTVB. 3 Piecewise Metods Te final algoritm is based on te first order Roe sceme, wit te entropy-fix correction due to Su and Oser [6], for local piecewise reconstruction. Te numerical fluxes are reconstructed from te upwind side, except tat if te wind canges direction at te cell, ten a local Lax- Friedrics flux decomposition is performed. ALGORITHM For every j do If f (u) does not canges of sign between u n j and un j+, ten Upwindness Pase (Roe-speed) If ā j+ else ā j+ 0 ten = v j+ v j u j+ u j () = r j (x j+ ) (3) = r j+ (x (j+) ) (4) else Flux decomposition pase = max u n j <u<u n f (u) j+ M j+ v k + = (v k + M j+ Computation of f +, as (3) vk = (v k M j+ Computation of f, as (4) end = f + + f u n k ) k = j, j, j + u n k ) k = j, j +, j + 4 Stability and Convergence Analysis Our metod is consistent and in conservation form, ten te Lax-Wendroff teorem ([3] Capter ) says tat if a sequence of te approximations converges ten te limit is a weak solution.

4 Proceedings of te 3rd IASME/WSEAS Int. Conf. on FLUID DYNAMICS & AERODYNAMICS, Corfu, Greece, August 0-, 005 (pp48-5) Definition Te Total Variation of a discrete solution is defined by T V (u) = j u n j+ u n j To obtain te convergence, we use te following two teorems (Teorem 5. and Teorem 5. in [3]). Teorem Consider a conservative metod wit a Lipscitz continuous numerical flux and suppose tat for eac initial data u 0 tere exists some k 0, R > 0 suc tat T V (u n ) R n, k wit k < k 0, nk T. Ten te metod is TV-stable. Teorem 3 A conservative sceme in conservation form, wit Lipscitz continuous numerical flux and TV-stable is convergent. Tus, it is enoug to guarantee tat te numerical flux is Lipscitz continuous and tat te T V (u n ) is uniformly bounded for all n, k wit k < k 0, nk T. We assume tat te numerical flux associated to te considered reconstruction is Lipscitz continuous. Finally, we ave to find a positive constant C suc tat T V (u n ) < C for all n, k wit k < k 0, nk T. In practice, a Runge-Kutta TVD for te time discretization is used. Tus, we do not ave stability problems wit te time. In order to simplify te notation we consider te simplest Euler sceme. From u n+ j = u n j λ( ˆf n j+ ˆf n j ), u n+ j = u n j λ( ˆf n j ˆf n j 3 ). we obtain (u n+ j u n+ j ) = (un j u n j ) λ( ˆf j + ˆf j 3 ). (5) Remark We are interesting to analyze te reconstruction step only. Te oters parts of te algoritm is classical and common for tis type of scemes. Tus, in order to present a more clear proof, we do not consider te upwind pase. We summarize te basic ingredients to obtain te desired stability. ) Let us coose a number > 0, suc tat tere is at least two cells between two jumps of g(x). Since g(x) is a piecewise smoot function, tere exist two constant M, M > 0 depending only on derivatives of g in smoot regions, suc tat: a) For all j, except for a finite number of insolated j s (for wic d j+ = O( )), d j+ < M. In particular, d c and d l are lees tan M always. b) d j+ r (x j ) M ( u n j+ u n j + u n j u n j ). c) (r (x j ) r (x j )) = f (β n j )(u n j u n j ). ) If let us consider vj n = un j, j. Since f is smoot, f(u n j ) f(vn j ) = f (θj n)(un j vn j ). 3) Let be te CFL condition 0 αj n := λ(f (θ n j ) + f (β n j ) ) for all j. Using tis properties and relation (5) we arrive to u n+ j u n+ j ( αn j ) u n j u n j

5 Proceedings of te 3rd IASME/WSEAS Int. Conf. on FLUID DYNAMICS & AERODYNAMICS, Corfu, Greece, August 0-, 005 (pp48-5) and ten + α n j u n j u n j + tm ( u n j+ u n j + u n j u n j + u n j u n j ), T V (u n+ ) ( + αk)t V (u n ), were 0 α 3 M is independent of u n. By induction T V (u n+ ) ( + αk) n+ T V (u 0 ). Tus, for (n + )k T, [4] Marquina A. (994), Local piecewise yperbolic reconstruction of numerical fluxes for nonlinear scalar conservation laws, SIAM J.Sci.Comput., 5, 4, pp [5] Su C.W. and Oser S.J. (987), Efficient implementation of essential non- Oscillatory sock capturing scemes, J.Comput.Pys., 77, pp [6] Su C.W. and Oser S.J. (989), Efficient implementation of essential non- Oscillatory sock capturing scemes II, J.Comput.Pys., 83, pp T V (u n+ ) e αt T V (u 0 ), and ence te metod is total variation stable. Acknowledgments Researc supported in part by te Spanis grants MTM and 00675/PI/04. References [] Amat S., Busquier S. and Candela V.F. (003), A polynomial approac to PHM metod, Preprint U.P. Cartagena, Int.J.Comput.Fluid.D., 7(3), pp [] Amat S., Busquier S. and Candela V.F. (003), Local Total Variation Bounded metods for yperbolic conservation laws, J. of Comp. Metods in Sciences and Engineering, 3(3), pp [3] Leveque R.J. (990). Numerical Metods for Conservation Laws. Birkäuser Verlag (Lectures in Matematics).