A family of stable numerical solvers for the shallow water equations with source terms. (English) Zbl 1083.76557

Summary: We introduce a multiparametric family of stable and accurate numerical schemes for 1D shallow water equations. These schemes are based upon the splitting of the discretization of the source term into centered and decentered parts. These schemes are specifically designed to fulfill the enhanced consistency condition of Bermúdez and Vázquez, necessary to obtain accurate solutions when source terms arise. Our general family of schemes contains as particular cases the extensions already known of Roe and Van Leer schemes, and as new contributions, extensions of Steger-Warming, Vijayasundaram, Lax-Friedrichs and Lax-Wendroff schemes with and without flux-limiters. We include some meaningful numerical tests, which show the good stability and consistency properties of several of the new methods proposed. We also include a linear stability analysis that sets natural sufficient conditions of stability for our general methods.


76M12 Finite volume methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction


Full Text: DOI


[1] Bermudez, A.; Dervieux, A.; Desideri, J.A.; Vázquez Cendón, M.E., Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes, Comput. meth. appl. mech. engrg., 155, 49, (1998) · Zbl 0961.76047
[2] Bermúdez, A.; Vázquez Cendón, M.E., Upwind methods for hyperbolic conservation laws with source terms, Comput. fluids, 23, 8, 1049-1071, (1994) · Zbl 0816.76052
[3] P. Brufau. Simulación bidimensional de flujos hidrodinámicos transitorios en gemotrı́as irregulares, Ph.D. thesis, Universidad de Zaragoza, 2000
[4] Burguete, J.; Garcı́a-Navarro, P., Efficient construction of high-resolution TVD conservative schemes for equations with source terms, application to shallow water flows, Int. J. numer. meth. fluids, 37, 209-248, (2001) · Zbl 1003.76059
[5] T. Chacón Rebollo, E.D. Ferńandez Nieto, M. Gómez Mármol, A flux-splitting solver for shallow water equations with source terms, Int. J. Numer. Meth. Fluids, in press
[6] E. Godlewski, P.A. Raviart, Hyperbolic systems of conservation laws, Mathematiques et Applications, Ellipses, Paris, 1991 · Zbl 0768.35059
[7] Godlewski, E.; Raviart, P.A., Numerical approximation of hyperbolic systems of conservation laws, (1996), Springer-Verlag · Zbl 0860.65075
[8] N. Goutal, F. Maurel, in: Proceedings of the 2nd Workshop on Dam-Break Wave Simulation, Technical Report HE-43/97/016/A, Electricité de France, Département Laboratoire National d’Hydraulique, Groude Hydraulique Fluviale, 1997
[9] LeVeque; Yee, H.C., A study of numerical methods for hyperbolic conservation laws with stiff source terms, J. comput. phys., 86, 187-210, (1990) · Zbl 0682.76053
[10] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1997), Springer · Zbl 0888.76001
[11] M.E. Vázquez Cendon, Estudio de esquemas descentrados para su aplicacion a las leyes de conservación hiperbólicas con términos fuente, Ph.D. thesis, Universidad de Santiago de Compostela, 1994
[12] Vázquez Cendón, M.E., Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry, J. comput. phys., 148, 497-526, (1999) · Zbl 0931.76055
[13] Zhou, J.G.; Causon, D.M.; Mingham, C.G.; Ingram, D.M., The surface gradient method for the treatment of source terms in the shallow-water equations, J. comput. phys., 168, 1-25, (2001) · Zbl 1074.86500
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.