Mathematical and numerical modelling of shallow water flow. (English) Zbl 0771.76032

Summary: This paper deals with shallow water equations. We discuss the mathematical model, the admissible boundary conditions, some popular numerical methods in the specialized literature, as well as we propose new approaches based on fractional step and finite element methods.


76M10 Finite element 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


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[1] Agoshkov, V. I.; Quarteroni, A.; Saleri, F. (1992):
[2] Benque, J. P.; Cunge, J. A.; Feuillet, J.; Hauguel, A.; Holly, F. M. (1982): New method for tidal current computation. Journ. of the Waterway, Port, Coastal and Ocean Division, ASCE, 108, 396-417
[3] Benque, J. P.; Haugel, A.; Viollet, P. L. (1982): Numerical methods in environmental fluid mechanics. In: Abbott, M. B.; Cunge, J. A. (eds): Eng. Appl. Comput. Hydraulics, vol II
[4] Casulli, V. (1990): Semi implicit finite difference method the two dimensional shallow water equations. J. Comp. Phys. 86, 56-73 · Zbl 0681.76022
[5] Elvius, T.; Sunstrom, A. (1973): Computationally efficient schemes and boundary conditions for a fine-mesh barotropic model based on the shallow water equations. Tellus, 25/2, 132-156
[6] Galland, J. C.; Goutal, N.; Hervouet, J. M. (1991): TELEMAC: A new numerical model for solving shallow water equations. Adv. Water Resources 1991, 14, 3
[7] Kreiss, H. O. (1970): Initial boundary value problems for hyperbolic equations. Comm. Pure Appl. Math. 23, 277-298 · Zbl 0193.06902
[8] Lax, P. D.; Wendroff, B. (1960): Systems of conservation laws. Comm. Pure Appl. Math. 13, 217-237 · Zbl 0152.44802
[9] Leendertse, J. J. (1967). Aspects of a computational method for long period long wave propagation. RAND Corp. Memorandum RM-5294-PR
[10] Lions, J. L.; Magenes, E. (1972): Nonhomogeneous boundary value problems and applications, I, II. Grund. B. Berlin, Heidelberg, New York: Springer · Zbl 0227.35001
[11] Marchuk, G. (1988): Splitting methods. Nauka, Moscow (in Russian) · Zbl 0653.65065
[12] Oliger, J.; Sundstrom, A. (1978): Theoretical and practical aspects of some initial boundary value problems in fluid dynamics. SIAM J. Appl. Math. 35, 3, 418-446 · Zbl 0397.35067
[13] Peraire, J.; Zienkiewicz, O. C.; Morgan, K. (1986): Shallow water problems: a general explicit formulation Int. J. Num. Math. in Eng. 22, 547-574 · Zbl 0588.76027
[14] Pironneau, O. (1988): methodes des elementes finis pour les fluides, Masson
[15] Quarteroni, A.; Sacchi-Landriani, A.; Valli, A. (1991): Coupling of viscous and inviscid Stokes equations via a domain decomposition method for finite elements. Numer. Math. 59, 831-859 · Zbl 0738.76044
[16] Quarteroni, A. (1990): Domain decomposition method for systems of conservation laws: spectral collocation approximations. SIAM. Sci. Stat. Comput. 11, 6, 1029-1052 · Zbl 0711.65082
[17] Quarteroni, A. (1968): An energy-conserving difference scheme for the storm surge equations. Month. Weather Rev. 96, 3
[18] Stelling, G. S. (1984): On the construction of computational methods for shallow water flow problems. Rijkwaterstaat communications · Zbl 0574.76013
[19] Vreugdenhil, C. B. (1990): Numerical methods for shallow water flow. Von Karman Institute for Fluid Dynamics, Lecture Series 1990-1903
[20] Walters, R. A. (1983): Numerically induced oscillations in finite element approximations to the shallow water equations. Int. J. for Num. Meth. in Fluids, 3, 591-604 · Zbl 0526.76028
[21] Whitham, G. (1974): Linear and non linear waves. New York: Wiley · Zbl 0373.76001
[22] Wilders, P.; Van Stijn, T. L.; Stelling, G. S.; Fokkema, G. A. (1988): A fully implicit splitting method for accurate tidal computations. Int. Journ. for Num. Meth. in Eng. 26, 2707-2721 · Zbl 0666.76044
[23] Yanenko, N. N. (1971): The method of fractional steps. Berlin, Heidelberg, New York: Springer · Zbl 0209.47103
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