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Bermúdez, Alfredo; Vázquez, M. Elena

Upwind methods for hyperbolic conservation laws with source terms. (English) Zbl 0816.76052

Comput. Fluids 23, No. 8, 1049-1071 (1994).
From the summary: The paper deals with the extension of some upwind schemes to hyperbolic systems of conservation laws with source term. More precisely, we give methods to get natural upwind discretizations of the source term when the flux is approximated by using flux-difference or flux-splitting techniques. In particular, the \(Q\)-schemes of Roe and van Leer and the flux-splitting techniques of Steger-Warming and Vijayasundaram are considered. Numerical results for a scalar advection equation with nonlinear source and for the one-dimensional shallow water equations are presented.

Cited in 5 Reviews
Cited in 366 Documents

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35L65 Hyperbolic conservation laws

Keywords:

\(Q\)-schemes; flux-difference techniques; flux-splitting techniques; scalar advection equation; one-dimensional shallow water equations

Software:

HLLE
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\textit{A. Bermúdez} and \textit{M. E. Vázquez}, Comput. Fluids 23, No. 8, 1049--1071 (1994; Zbl 0816.76052)
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References:

[1] Harten, A.; Lax, P.; van Leer, A., On upstream differencing and Godunov-type scheme for hyperbolic conservation laws, SIAM Rev., 25, 35 (1983) · Zbl 0565.65051
[2] LeVeque, R., Numerical Methods for Conservation Laws (1990), Birkhäuser: Birkhäuser Basel · Zbl 0723.65067
[3] Godlewski, E.; Raviart, P. A., Hyperbolic systems of conservation laws, (Mathematiques & Applications, N. 3/4 (1991), Ellipses-Edition Marketing) · Zbl 1155.76374
[4] van Leer, B., (Proc. of the 13 Int. Conf. on Numerical Methods in Fluid Dynamics. Proc. of the 13 Int. Conf. on Numerical Methods in Fluid Dynamics, Rome 6-10 July (1992)) · Zbl 0467.76075
[5] Roe, P. L., Upwind diffenced schemes for hyperbolic conservation laws with source terms, (Carasso; Raviart; Serre, Proc. Conf. Hyperbolic Problems (1986), Springer), 41-51
[6] Glaister, P., Approximate Riemann solutions of the shallow water equations, J. Hydraulic Res., 26, 293 (1988)
[7] LeVeque, R.; Yee, H. C., A study of numerical methods for hyperbolic conservation laws with stiff source terms, J. Comp. Phys., 86, 187 (1990) · Zbl 0682.76053
[8] Bermúdez, A.; Vázquez, M. E., Flux-vector and flux-difference splitting methods for the shallow water equations in a domain with variable depth, (Partridge, P. W., Computer Modelling of Seas and Coastal Regions (1992), Computational Mechanics Publications Elsevier Applied Science: Computational Mechanics Publications Elsevier Applied Science London), 256-267
[9] Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43, 357 (1981) · Zbl 0474.65066
[10] Steger, J.; Warming, R. F., Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods, J. Comput. Phys., 40, 263 (1981) · Zbl 0468.76066
[11] Vijayasundaram, G., Transonic flow simulations using an upstream centered scheme of Godunov in finite elements, J. Comput. Phys., 63, 416 (1986) · Zbl 0592.76081
[12] Taylor, C.; Davies, J., Tidal and long wave propagation, (A finite element approach. A finite element approach, Computers Fluids, 3 (1975)), 125 · Zbl 0345.76009
[13] Kawahara, M.; Takeuchi, N.; Yoshida, T., Two step explicit finite element method for tsunami wave propagation analysis, Int. J. Num Meth. Eng., 12, 331 (1978) · Zbl 0375.76003
[14] Lynch, P. R.; Cray, W. C., A wave equation model for finite tidal computations, Computers Fluids, 7, 207 (1979)
[15] Zienkiewicz, O. C.; Heinrich, J. C., A unified treatment of the steady-state shallow water and two-dimensional Navier-Stokes equations, (Finite element and penalty function approach. Finite element and penalty function approach, Comp. Meth. Appl. Mech. Eng., 17/18 (1979)), 673 · Zbl 0414.76014
[16] Goussebaile, J.; Hecht, F.; Labadie, G.; Reinhart, L., Finite element solution of the shallow water equations by a quasi-direct decomposition procedure, Int. J. Num. Meth. Fluids, 4, 1117 (1984) · Zbl 0554.76021
[17] Peraire, J.; Zienkiewicz, O. C.; Morgan, K., Shallow water problems: a general explicit formulation, Int. J. Num. Meth. Eng., 22, 547 (1986) · Zbl 0588.76027
[18] Bermúdez, A.; Rodriguez, C.; Vilar, M. A., Solving shallow water equations by a mixed implicit finite element method, IMA J. Num. Anal., 11, 79 (1991) · Zbl 0713.76069
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