zbMATH — the first resource for mathematics

Some approximate Godunov schemes to compute shallow-water equations with topography. (English) Zbl 1084.76540
Summary: We study here the computation of shallow-water equations with topography by Finite Volume methods, in a one-dimensional framework (though all methods introduced may be naturally extended in two dimensions). All methods are based on a discretisation of the topography by a piecewise function constant on each cell of the mesh, from an original idea of Le Roux et al. Whereas the Well-Balanced scheme of Le Roux is based on the exact resolution of each Riemann problem, we consider here approximate Riemann solvers. Several single step methods are derived from this formalism, and numerical results are compared to a fractional step method. Some test cases are presented: convergence towards steady states in subcritical and supercritical configurations, occurrence of dry area by a drain over a bump and occurrence of vacuum by a double rarefaction wave over a step. Numerical schemes, combined with an appropriate high-order extension, provide accurate and convergent approximations.

76M12 Finite volume methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Full Text: DOI
[1] Bon C. Modélisation et simulations numériques d’écoulements hydrauliques et de ruissellement en topographic quelconque, PhD thesis, Université Bordeaux I, France, 1997
[2] Botchorishvili R, Perthame B, Vasseur A. Schémas d’équilibre pour les lois de conservation scalaires avec des termes sources raides, INRIA Report RR-3891, 2000. To appear in Mathematical Methods and Numerical Analysis
[3] Buffard, T.; Gallouët, T.; Hérard, J.M., A naive Godunov scheme to compute a non-conservative hyperbolic system, Int. ser. numer. math., 129, 129-138, (1998)
[4] Buffard, T.; Gallouët, T.; Hérard, J.M., A naive Godunov scheme to solve shallow-water equations, C. R. acad. sci. Paris, I-326, 885-890, (1998)
[5] Buffard, T.; Gallouët, T.; Hérard, J.M., A sequel to a rough Godunov scheme. application to real gas flows, Comput. fluids, 29, 7, 813-847, (2000) · Zbl 0961.76048
[6] Chinnayya A, Le Roux AY. A new general Riemann solver for the shallow-water equations with friction and topography, 1999. Available from the conservation law preprint server: Available from: http://www.math.ntnu.no/conservation/
[7] Colombeau, J.F., Multiplication of distributions, (1992), Springer Berlin · Zbl 0731.46023
[8] Dal Maso, G.; Le Floch, P.G.; Murat, F., Definition and weak stability of nonconservative products, J. math. pures appl., 74, 483-548, (1995) · Zbl 0853.35068
[9] Eymard, R.; Gallouët, T.; Herbin, R., Finite volume methods, (), 729-1020 · Zbl 0981.65095
[10] Gallouët T, Hérard JM, Seguin N. Some recent Finite Volume schemes to compute Euler equations using real gas EOS. LATP Report 00-021, Université de Provence, France, 2000. To appear in International Journal for Numerical Methods in Fluids
[11] Gallouët T, Hérard JM, Seguin N. On the use of some symetrizing variables to deal with vacuum, 2001 [submitted in revised form]
[12] Garcia-Navarro, P.; Vazquez-Cendon, M.E., On numerical treatment of the source terms in the shallow water equations, Comput. fluids, 29, 8, 951-979, (2000) · Zbl 0986.76051
[13] Godlewski, E.; Raviart, P.A., Numerical approximation of hyperbolic systems of conservation laws, (1996), Springer Berlin · Zbl 0860.65075
[14] Godunov, S.K., A difference method for numerical calculation of discontinuous equations of hydrodynamics, Mat. sb., 271-300, (1959), [in Russian] · Zbl 0171.46204
[15] Gosse, L.; Le Roux, A.Y., A well balanced scheme designed for inhomogeneous scalar conservation laws, C. R. acad. sci. Paris, 1, 323, 543-546, (1996) · Zbl 0858.65091
[16] Goutal N, Maurel F. In: Proceedings of the 2nd Workshop on Dam-break Wave Simulation, EDF-DER Report HE-43/97/016/B, 1997
[17] Greenberg, J.M.; Le Roux, A.Y., A well balanced scheme for the numerical processing of source terms in hyperbolic equation, SIAM. J. numer. anal., 33, 1, 1-16, (1996) · Zbl 0876.65064
[18] Greenberg, J.M.; Le Roux, A.Y.; Baraille, R.; Noussair, A., Analysis and approximation of conservation laws with source terms, SIAM J. numer. anal., 34, 5, 1980-2007, (1997) · Zbl 0888.65100
[19] Le Roux AY. Discrétisation des termes sources raides dans les problèmes hyperboliques. In: Systèmes hyperboliques: Nouveaux schémas et nouvelles applications. Écoles CEA-EDF-INRIA ‘problèmes non linéaires appliqués’, INRIA Rocquencourt (France), March 1998 [in French]. Available from: http://www-gm3.univ-mrs.fr/∼leroux/publications/ay.le_roux.html
[20] LeVeque, R.J., Numerical methods for conservation laws, (1990), Birkhäusër Basel · Zbl 0682.76053
[21] LeVeque, R.J., Balancing source terms and flux gradients in high-resolution Godunov methods, J. comput. phys., 146, 346-365, (1998) · Zbl 0931.76059
[22] Lions, P.L.; Mercier, B., Splitting algorithms for the sum of two nonlinear operators, SIAM J. numer. anal., 16, 6, 964-979, (1979) · Zbl 0426.65050
[23] Masella, J.M.; Faille, I.; Gallouët, T., On an approximate Godunov scheme, Int. J. comp. fluid dyn., 12, 133-149, (1999) · Zbl 0944.76041
[24] Roe, P.L., Approximate Riemann solvers, parameter vectors and difference schemes, J. comput. phys., 43, 357-372, (1981) · Zbl 0474.65066
[25] Seguin N. Génération et validation de Rozavel, un code équilibre en hydraulique 2D.–Mémoire de D.E.A., Université de Bordeaux I, 1999. Available from: http://www-gm3.univ-mrs.fr/∼leroux/publications/n.seguin.html
[26] Seguin N, Vovelle J. Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients LATP report, 2002, centre de Mathematique et d’Informatique, 39 rue Joliot Curie, 13453 Marseille cedex 13, France · Zbl 1059.65514
[27] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1997), Springer Berlin · Zbl 0888.76001
[28] Van Leer, B., Toward the ultimate conservative difference scheme V. A second order sequel to godunov’s method, J. comp. phys., 32, 101-136, (1979) · Zbl 1364.65223
[29] Yanenko NN. Méthode à pas fractionnaires. Résolution de problèmes polydimensionnels de physique mathématique, Librairie Armand Colin, 1968. Traduit du russe par P.A. Nepomiastchy · Zbl 0185.41803
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.