×

zbMATH — the first resource for mathematics

Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. (English) Zbl 0931.76055
Summary: This paper deals with the numerical solution of the shallow water equations in channels with irregular geometry but with a locally rectangular cross-section. This type of channel leads to the presence of source terms involving the gradient of the depth and the breadth of the channel. Extensions of the \(Q\)-scheme of van Leer and Roe are proposed which generate natural upwind discretizations of the source terms. We analyze the consistency of the proposed schemes. A stationary solution that emphasizes the source terms considered is obtained which is used to test the proposed extensions in terms of a “conservation” property. We also obtain a low-order asymptotic unsteady analytical solution for small Froude numbers. The numerical results confirm the improved properties of the proposed schemes for a transient test problem. \(\copyright\) Academic Press.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
76M20 Finite difference methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
86A05 Hydrology, hydrography, oceanography
Software:
HE-E1GODF; HLLE
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alcrudo, F; Garcı́a-Navarro, P, A high-resolution Godunov-type scheme in finite volumes for the 2D shallow-water equations, Int. J. numer. methods fluids, 16, 489, (1993) · Zbl 0766.76067
[2] Bermúdez, A; Dervieux, A; Desideri, J.A; Vázquez, M.E, Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes, Comput. methods appl. mech. eng., 155, 49, (1998) · Zbl 0961.76047
[3] Bermúdez, A; Vázquez, M.E, Upwind methods for hyperbolic conservation laws with source terms, Comput. & fluids, 23, 1049, (1994) · Zbl 0816.76052
[4] Garcı́a-Navarro, P; Alcrudo, F; Savirón, J.M, 1D open channel flow simulation using TVD maccormack scheme, J. hydraulic eng., 118, 1359, (1992)
[5] P. Garcı́a-Navarro, M. E. Vázquez-Cendón, Some Considerations and Improvements on the Performance of Roe’s Scheme for 1D Irregular Geometries, Departamento de Matemática Aplicada, Universidad de Santiago de Compostela, 1997, Comput. & Fluids
[6] Glaister, P, Approximate Riemann solutions of the shallow water equations, J. hydraulic res, 26, 293, (1988)
[7] Glaister, P, Prediction of supercritical flow in open channels, Comput. & math. appl., 24, 69, (1992) · Zbl 0763.76049
[8] Godlewski, E; Raviart, P.A, Numerical approximation of hyperbolic systems of conservation laws, 118, (1996) · Zbl 1155.76374
[9] Goutal, N; Maurel, F, Proceedings of the 2nd workshop on dam-break wave simulation, (1997)
[10] Greenberg, J.M; LeRoux, A, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. numer. anal., 33, 1, (1996) · Zbl 0876.65064
[11] Harten, A, On a class of high resolution total-variation-stable finite-difference schemes, SIAM J. numer. anal., 21, 1, (1984) · Zbl 0547.65062
[12] Harten, A; Lax, P; van Leer, A, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM rev., 25, 35, (1983) · Zbl 0565.65051
[13] LeVeque, R, Numerical methods for conservation laws, (1990) · Zbl 0723.65067
[14] LeVeque, R; Yee, H.C, A study of numerical methods for hyperbolic conservation laws with stiff source terms, J. comput. phys., 86, 187, (1990) · Zbl 0682.76053
[15] Liu, T.-P, Quasilinear hyperbolic systems, Commun. math. phys., 68, 141, (1979) · Zbl 0435.35054
[16] MacDonald, I, Tests problems with analytic solutions for steady open channel flow, (1994)
[17] Priestley, A, Roe-type schemes for super-critical flows in rivers, (1989)
[18] Roe, P.L, Approximate Riemann solvers, parameter vectors, and difference schemes, J. comput. phys., 43, 357, (1981) · Zbl 0474.65066
[19] Roe, P.L, Upwind differenced schemes for hyperbolic conservation laws with source terms, Proceedings of the conference on hyperbolic problems, 41, (1986)
[20] Toro, E.F, Riemann solvers and numerical methods for fluid dynamics. A practical introduction, (1997) · Zbl 0888.76001
[21] van Leer, B, On the relation between the upwind-differencing schemes of Godunov, engquist-osher and roe, SIAM J. sci. statist. comput., 5, 1, (1984) · Zbl 0547.65065
[22] M. E. Vázquez-Cendón, Estudio de esquemas descentrados para su aplicación a las leyes de conservación hiperbólicas con términos fuente, Tesis doctoral, Departamento de Matemática Aplicada, Universidad de Santiago de Compostela, Spain, 1994
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.