×

zbMATH — the first resource for mathematics

Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. (English) Zbl 1142.65070
Summary: We propose a new class of finite volume schemes of arbitrary accuracy in space and time for systems of hyperbolic balance laws with stiff source terms. The new class of schemes is based on a three stage procedure. First a high-order weighted essentially non-oscillatory (WENO) reconstruction procedure is applied to the cell averages at the current time level. Second, the temporal evolution of the reconstruction polynomials is computed locally inside each cell using the governing equations. In the original ENO scheme of A. Harten, B. Engquist, S.Osher, and S. Chakravarthy [ibid. 71, 231–303 (1987; Zbl 0652.65067)] and in the ADER schemes of V. A. Titarev and E. F. Toro [ibid. 204, No. 2, 715–736 (2005; Zbl 1060.65641)], this time evolution is achieved via a Taylor series expansion where the time derivatives are computed by repeated differentiation of the governing partial differential equation with respect to space and time, i.e. by applying the so-called Cauchy-Kovalewski procedure.
However, this approach is not able to handle stiff source terms. Therefore, we present a new strategy that only replaces the Cauchy-Kovalewski procedure compared to the previously mentioned schemes. For the time-evolution part of the algorithm, we introduce a local space-time discontinuous Galerkin (DG) finite element scheme that is able to handle also stiff source terms. This step is the only part of the algorithm which is locally implicit. The third and last step of the proposed ADER finite volume schemes consists of the standard explicit space-time integration over each control volume, using the local space-time DG solutions at the Gaussian integration points for the intercell fluxes and for the space-time integral over the source term.
We show numerical convergence studies for nonlinear systems in one space dimension with both non-stiff and with very stiff source terms up to sixth order of accuracy in space and time. The application of the new method to a large set of different test cases is shown, in particular the stiff scalar model problem of R. J. LeVeque and H. C. Yee [ibid. 86, No. 1, 187–210 (1990; Zbl 0682.76053)], the relaxation system of S. Jin and Z. Xin, [Commun. Pure Appl. Math. 48, No. 3, 235–276 (1995; Zbl 0826.65078)] and the full compressible Euler equations with stiff friction source terms.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
Software:
HE-E1GODF; HLLE
PDF BibTeX Cite
Full Text: DOI
References:
[1] Bao, W.; Jin, S., Error estimates on the random projection methods for hyperbolic conservation laws with stiff reaction terms, Applied numerical mathematics, 43, 315-333, (2002) · Zbl 1063.65094
[2] Béreux, F., Zero-relaxation limit versus operator splitting for two-phase fluid flow computations, Computer methods in applied mechanics and engineering, 133, 93-124, (1996) · Zbl 0887.76079
[3] Bermudez, A.; Vazquez, M.E., Upwind methods for hyperbolic conservation laws with source terms, Computers and fluids, 23, 8, 1049-1071, (1994) · Zbl 0816.76052
[4] Botchorishvili, R.; Perthame, B.; Vasseur, A., Equilibrium schemes for scalar conservation laws with stiff sources, Mathematics of computation, 72, 241, 131-157, (2001) · Zbl 1017.65070
[5] Botta, N.; Klein, R.; Langenberg, S.; Lützenkirchen, S., Well balanced finite volume methods for nearly hydrostatic flows, Journal of computational physics, 196, 539-565, (2004) · Zbl 1109.86304
[6] Bouchut, F.; Ounaissa, H.; Perthame, B., Upwinding of source term at interfaces for Euler equations with high friction, Computers and mathematics with applications, 53, 361-375, (2007) · Zbl 1213.76123
[7] Bourlioux, A.; Majda, A.J.; Roytburd, V., One-dimensional front tracking based on high resolution wave propagation methods, SIAM journal on applied mathematics, 51, 303-343, (1991) · Zbl 0731.76076
[8] Buet, C.; Després, B., Asymptotic preserving and positive schemes for radiation hydrodynamics, Journal of computational physics, 215, 2, 717-740, (2006) · Zbl 1090.76046
[9] Burman, E.; Sainsaulieu, L., Numerical analysis of two operator splitting methods for an hyperbolic system of conservation laws with stiff relaxation, Computer methods in applied mechanics and engineering, 128, 291-314, (1995) · Zbl 0867.76055
[10] Caflish, R.E.; Jin, S.; Russo, G., Uniformly accurate schemes for hyperbolic systems with relaxation, SIAM journal on numerical analysis, 34, 246-281, (1997) · Zbl 0868.35070
[11] Castro, M.; Gallardo, J.M.; Pares, C., High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products, applications to shallow-water systems, Mathematics of computation, 75, 1103-1134, (2006) · Zbl 1096.65082
[12] Chen, G.Q.; Levermore, C.D.; Liu, T.P., Hyperbolic conservation laws with stiff relaxation terms and entropy, Communications on pure and applied mathematics, 47, 6, 787-830, (1994) · Zbl 0806.35112
[13] Dumbser, M., Arbitrary high order schemes for the solution of hyperbolic conservation laws in complex domains, (2005), Shaker Verlag Aachen
[14] Dumbser, M.; Käser, M., Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems, Journal of computational physics, 221, 693-723, (2007) · Zbl 1110.65077
[15] Dumbser, M.; Käser, M.; Titarev, V.A.; Toro, E.F., Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems, Journal of computational physics, 226, 204-243, (2007) · Zbl 1124.65074
[16] Dumbser, M.; Munz, C.D., Building blocks for arbitrary high order discontinuous Galerkin schemes, Journal of scientific computing, 27, 215-230, (2006) · Zbl 1115.65100
[17] Dumbser, M.; Schwartzkopff, T.; Munz, C.D., Arbitrary high order finite volume schemes for linear wave propagation, (), 129-144
[18] Einfeldt, B., On Godunov-type methods for gas dynamics, SIAM journal on numerical analysis, 25, 294-318, (1988) · Zbl 0642.76088
[19] Einfeldt, B.; Munz, C.D.; Roe, P.L.; Sjgreen, B., On Godunov-type methods near low densities, Journal of computational physics, 92, 273-295, (1991) · Zbl 0709.76102
[20] Engquist, B.; Osher, S., One sided difference approximations for nonlinear conservation laws, Mathematics of computation, 36, 321-351, (1981) · Zbl 0469.65067
[21] Glimm, J.; Marshall, G.; Plohr, B., A generalized Riemann problem for quasi-one-dimensional gas flows, Advances in applied mathematics, 5, 1-30, (1984) · Zbl 0566.76056
[22] Godunov, S.K., Finite difference methods for the computation of discontinuous solutions of the equations of fluid dynamics, Mathematics of the USSR. sbornik, 47, 271-306, (1959) · Zbl 0171.46204
[23] Golse, F.; Jin, S.; Levermore, C.D., The convergence of numerical transfer schemes in diffusive regimes. I: the discrete-ordinate method, SIAM journal on numerical analysis, 36, 1333-1369, (1999) · Zbl 1053.82030
[24] Gosse, L., A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms, Computers and mathematics with applications, 39, 135-159, (2000) · Zbl 0963.65090
[25] Gosse, L.; Toscani, G., Asymptotic-preserving and well-balanced schemes for radiative transfer and the rosseland approximation, Numerische Mathematik, 98, 2, 223-250, (2004) · Zbl 1120.65343
[26] Greenberg, J.M.; LeRoux, A.Y., A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM journal on numerical analysis, 33, 1, 1-16, (1996) · Zbl 0876.65064
[27] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformly high order essentially non-oscillatory schemes, III, Journal of computational physics, 71, 231-303, (1987) · Zbl 0652.65067
[28] Harten, A.; Lax, P.D.; van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM review, 25, 1, 35-61, (1983) · Zbl 0565.65051
[29] Helzel, C.; LeVeque, R.J.; Warnecke, G., Modified fractional step method for the accurate approximation of detonation waves, SIAM journal on scientific computing, 22, 1489-1510, (2000) · Zbl 0983.65105
[30] Hsiao, L.; Liu, T.P., Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Communications in mathematical physics, 143, 599-605, (1992) · Zbl 0763.35058
[31] Huang, F.; Marcati, P.; Pan, R., Convergence to the Barenblatt solution for the compressible Euler equations with damping and vacuum, Archive for rational mechanics and analysis, 176, 1-24, (2005) · Zbl 1064.76090
[32] Jiang, G.-S.; Shu, C.W., Efficient implementation of weighted ENO schemes, Journal of computational physics, 202-228, (1996) · Zbl 0877.65065
[33] Jin, S., Runge – kutta methods for hyperbolic conservation laws with stiff relaxation terms, Journal of computational physics, 122, 1, 51-67, (1995) · Zbl 0840.65098
[34] Jin, S., Efficient asymptotic-preserving (ap) schemes for some multiscale kinetic equations, SIAM journal on scientific computing, 21, 441-454, (1999) · Zbl 0947.82008
[35] Jin, S., A steady-state capturing method for hyperbolic systems with geometrical source terms, Mathematical modelling and numerical analysis, 35, 4, 631-645, (2001) · Zbl 1001.35083
[36] Jin, S.; Levermore, C.D., Numerical schemes for hyperbolic conservation laws with stiff relaxation terms, Journal of computational physics, 126, 2, 449-467, (1996) · Zbl 0860.65089
[37] Jin, S.; Xin, Z., The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Communications on pure and applied mathematics, 48, 235-277, (1995) · Zbl 0826.65078
[38] LeFloch, P.; Raviart, P.A., An asymptotic expansion for the solution of the generalized Riemann problem, part I: general theory, Annales de l’institut Henri Poincaré, 5, 2, 179-207, (1988) · Zbl 0679.35064
[39] A.Y. LeRoux, Riemann solvers for some hyperbolic problems with a source term, in: ESAIM: Proceedings CANUM, vol. 6, 1998, pp. 75-90.
[40] LeVeque, R.J.; Shyue, K.-M., One-dimensional front tracking based on high resolution wave propagation methods, SIAM journal on scientific computing, 16, 348-377, (1995) · Zbl 0824.65094
[41] LeVeque, R.J.; Yee, H.C., A study of numerical methods for hyperbolic conservation laws with stiff source terms, Journal of computational physics, 86, 1, 187-210, (1990) · Zbl 0682.76053
[42] Liu, X.D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, Journal of computational physics, 115, 200-212, (1994) · Zbl 0811.65076
[43] Lowrie, R.B.; Morel, J.E., Methods for hyperbolic systems with stiff relaxation, International journal for numerical methods in fluids, 40, 413-423, (2002) · Zbl 1019.76029
[44] Miniati, F.; Colella, P., A modified higher order godunov’s scheme for stiff source conservative hydrodynamics, Journal of computational physics, 224, 2, 519-538, (2007) · Zbl 1117.76039
[45] Naldi, G.; Pareschi, L., Numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation, SIAM journal on numerical analysis, 37, 4, 1246-1270, (2000) · Zbl 0954.35109
[46] Osher, S.; Solomon, F., Upwind difference schemes for hyperbolic conservation laws, Mathematics of computation, 38, 339-374, (1982) · Zbl 0483.65055
[47] Pareschi, L.; Russo, G., Implicit-explicit runge – kutta schemes for stiff systems of differential equations, Advances in the theory of computational mathematics, 3, 269-288, (2000) · Zbl 1018.65093
[48] Pember, R.B., Numerical methods for hyperbolic conservation laws with stiff relaxation, I. spurious solutions, SIAM journal on applied mathematics, 53, 5, 1293-1330, (1993) · Zbl 0787.65062
[49] Pember, R.B., Numerical methods for hyperbolic conservation laws with stiff relaxation. II. higher order Godunov methods, SIAM journal on scientific computing, 14, 4, 824-859, (1993) · Zbl 0812.65083
[50] Perthame, B.; Simeoni, C., Convergence of the upwind interface source method for hyperbolic conservation laws, () · Zbl 1008.65066
[51] Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P., Numerical recipes in Fortran 77, vol. 1, (1996), Cambridge University Press · Zbl 0878.68049
[52] Qiu, J.; Dumbser, M.; Shu, C.W., The discontinuous Galerkin method with lax – wendroff type time discretizations, Computer methods in applied mechanics and engineering, 194, 4528-4543, (2005) · Zbl 1093.76038
[53] Qiu, J.; Shu, C.W., Finite difference WENO schemes with lax – wendroff type time discretization, SIAM journal on scientific computing, 24, 6, 2185-2198, (2003) · Zbl 1034.65073
[54] Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes, Journal of computational physics, 43, 357-372, (1981) · Zbl 0474.65066
[55] Schwartzkopff, T.; Dumbser, M.; Munz, C.D., Fast high order ADER schemes for linear hyperbolic equations, Journal of computational physics, 197, 532-539, (2004) · Zbl 1052.65078
[56] Shu, C.W.; Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes II, Journal of computational physics, 83, 32-78, (1989) · Zbl 0674.65061
[57] Strang, G., On the construction and comparison of difference schemes, SIAM journal on numerical analysis, 5, 3, 506-517, (1968) · Zbl 0184.38503
[58] Stroud, A.H., Approximate calculation of multiple integrals, (1971), Prentice-Hall Inc. Englewood Cliffs, New Jersey · Zbl 0379.65013
[59] Titarev, V.A.; Toro, E.F., ADER: arbitrary high order Godunov approach, Journal of scientific computing, 17, 1-4, 609-618, (2002), December · Zbl 1024.76028
[60] Titarev, V.A.; Toro, E.F., ADER schemes for three-dimensional nonlinear hyperbolic systems, Journal of computational physics, 204, 715-736, (2005) · Zbl 1060.65641
[61] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1999), Springer · Zbl 0923.76004
[62] Toro, E.F.; Millington, R.C.; Nejad, L.A.M., Towards very high order Godunov schemes, (), 905-938 · Zbl 0989.65094
[63] Toro, E.F.; Spruce, M.; Speares, W., Restoration of the contact surface in the harten – lax – van leer Riemann solver, Journal of shock waves, 4, 25-34, (1994) · Zbl 0811.76053
[64] Toro, E.F.; Titarev, V.A., Solution of the generalized Riemann problem for advection – reaction equations, Proceedings of the royal society of London series A, 458, 271-281, (2002) · Zbl 1019.35061
[65] Toro, E.F.; Titarev, V.A., ADER schemes for scalar hyperbolic conservation laws with source terms in three space dimensions, Journal of computational physics, 202, 196-215, (2005) · Zbl 1061.65103
[66] van der Vegt, J.; van der Ven, H., Space – time discontinuous Galerkin finite element method with dynamics grid motion for inviscid compressible flows, part I. general formulation, Journal of computational physics, 182, 546-585, (2002) · Zbl 1057.76553
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.