A well-balanced gas-kinetic scheme for the shallow-water equations with source terms. (English) Zbl 1017.76071

Summary: We describe an extension of gas-kinetic BGK scheme to shallow-water equations with source terms. In the current study, the particle velocity change due to the gravitational force and variable river bottom is implemented explicitly in the flux evaluation. The current scheme is a well-balanced method, which presents accurate and robust results in both steady and unsteady flow simulations.


76M28 Particle methods and lattice-gas methods
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Full Text: DOI


[1] Alcrudo, F.; Benkhaldoun, F., Exact solutions to the Riemann problem of the shallow water equations with a bottom step, Comput. fluids, 30, 643, (2001) · Zbl 1048.76008
[2] E. Audusse, M. O. Bristeau, and, B. Perthame, Kinetic Schemes for Saint-Venant Equations with Source Terms on Unstructured Grids, INRIA Report No. 3989, 2000.
[3] Bermudez, A.; Vazquez, M.E., Upwind methods for hyperbolic conservation laws with source terms, Comput. fluids, 23, 1049, (1994) · Zbl 0816.76052
[4] Bhatnagar, P.L.; Gross, E.P.; Krook, M., A model for collision processes in gases. I: small amplitude processes in charged and neutral one-component systems, Phys. rev., 94, 511, (1954) · Zbl 0055.23609
[5] Chow, V.T., open-channel hydraulics, (1959), McGraw-Hill New York
[6] Ghidaoui, M.S.; Deng, J.Q.; Gray, W.G.; Xu, K., A Boltzmann based model for open channel flows, Int. J. numer. methods fluids, 35, 449, (2001) · Zbl 1013.76053
[7] Glaister, P., Approximate Riemann solutions of the shallow water equations, J. hydraul. res., 26, 293, (1988)
[8] Greenberg, J.M.; Lerroux, A.Y., A well-balanced schemes for the numerical processing of source terms in hyperbolic equations, SIAM J. numer. anal., 33, 1, (1996)
[9] W. H. Hui, and, C. H. Pan, The water surface formulation for two-dimensional shallow water flow with bottom topography, preprint, 2001.
[10] Hubbard, M.E.; Garcia-Navarro, P., Flux difference splitting and the balancing of source terms and flux gradients, J. comput. phys., 165, 89, (2000) · Zbl 0972.65056
[11] Jenny, P.; Muller, B., Rankine – hugoniot – riemann solver considering source terms and multidimensional effects, J. comput. phys., 145, 575, (1998) · Zbl 0926.76079
[12] Jin, S., A steady-state capturing method for hyperbolic systems with geometrical source terms, Math. model numer. anal., 35, 631, (2001) · Zbl 1001.35083
[13] Jin, S.; Kim, Y.J., On the computation of roll waves, Math. model numer. anal., 35, 463, (2001) · Zbl 1001.35084
[14] LeVeque, R.J., Balancing source terms and flux gradients in high-resolution Godunov methods: the quasisteady wave-propagation algorithm, J. comput. phys., 146, 346, (1998) · Zbl 0931.76059
[15] Noussair, A., Riemann problem with nonlinear resonance effects and well-balanced Godunov scheme for shallow water fluid flow past an obstacle, SIAM J. numer. anal., 39, 52, (2001) · Zbl 1001.35085
[16] B. Perthame, and, C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term, CALCOLO, in press. · Zbl 1008.65066
[17] Slyz, A.; Prendergast, K.H., Time-independent gravitational fields in the BGK scheme for hydrodynamics, Astron. astrophys. suppl. ser., 139, 199, (1999)
[18] Toro, E.F., shock-capturing methods for free-surface shallow flows, (2001), Wiley New York · Zbl 0996.76003
[19] Vazquez-Cendon, 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, (1999) · Zbl 0931.76055
[20] Xu, K., A gas-kinetic scheme for the Euler equations with heat transfer, SIAM J. sci. comput., 20, 1317, (1999) · Zbl 0959.76064
[21] Xu, K., A gas-kinetic BGK scheme for the navier – stokes equations and its connection with artificial dissipation and Godunov method, J. comput. phys., 171, 289, (2001) · Zbl 1058.76056
[22] 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, (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.