A new two-dimensional shallow water model including pressure effects and slow varying bottom topography
ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 2, pp. 211-234.

The motion of an incompressible fluid confined to a shallow basin with a slightly varying bottom topography is considered. Coriolis force, surface wind and pressure stresses, together with bottom and lateral friction stresses are taken into account. We introduce appropriate scalings into a three-dimensional anisotropic eddy viscosity model; after averaging on the vertical direction and considering some asymptotic assumptions, we obtain a two-dimensional model, which approximates the three-dimensional model at the second order with respect to the ratio between the vertical scale and the longitudinal scale. The derived model is shown to be symmetrizable through a suitable change of variables. Finally, we propose some numerical tests with the aim to validate the proposed model.

DOI : 10.1051/m2an:2004010
Classification : 35L65, 65M60
Mots clés : Navier-Stokes equations, Saint Venant equations, free surface flows
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     title = {A new two-dimensional shallow water model including pressure effects and slow varying bottom topography},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {211--234},
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Ferrari, Stefania; Saleri, Fausto. A new two-dimensional shallow water model including pressure effects and slow varying bottom topography. ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 2, pp. 211-234. doi : 10.1051/m2an:2004010. http://www.numdam.org/articles/10.1051/m2an:2004010/

[1] V.I. Agoshkov, D. Ambrosi, V. Pennati, A. Quarteroni and F. Saleri, Mathematical and numerical modelling of shallow water flow. Comput. Mech. 11 (1993) 280-299. | Zbl

[2] V.I. Agoshkov, A. Quarteroni and F. Saleri, Recent developments in the numerical simulation of shallow water equations. Boundary conditions. Appl. Numer. Math. 15 (1994) 175-200. | Zbl

[3] J.P. Benque, J.A. Cunge, J. Feuillet, A. Hauguel and F.M. Holly, New method for tidal current computation. J. Waterway, Port, Coastal and Ocean Division, ASCE 108 (1982) 396-417.

[4] J.P. Benque, A. Haugel and P.L. Viollet, Numerical methods in environmental fluid mechanics. M.B. Abbot and J.A. Cunge Eds., Eng. Appl. Comput. Hydraulics II (1982) 1-10.

[5] S. Ferrari, A new two-dimensional Shallow Water model: physical, mathematical and numerical aspects Ph.D. Thesis, a.a. 2002/2003, Dottorato M.A.C.R.O., Università degli Studi di Milano.

[6] S. Ferrari, Convergence analysis of a space-time approximation to a two-dimensional system of Shallow Water equations. Internat. J. Appl. Analysis (to appear). | MR | Zbl

[7] J.F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Contin. Dyn. Syst. Ser. B 1 (2001) 89-102. | Zbl

[8] R.H. Goodman, A.J. Majda and D.W. Mclaughlin, Modulations in the leading edges of midlatitude storm tracks. SIAM J. Appl. Math. 62 (2002) 746-776. | Zbl

[9] E. Grenier, Boundary layers for parabolic regularizations of totally characteristic quasilinear parabolic equations. J. Math. Pures Appl. 76 (1997) 965-990. | Zbl

[10] E. Grenier and O. Guès, Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems. J. Differential Equations 143 (1998) 110-146. | Zbl

[11] O. Guès, Perturbations visqueuses de problèmes mixtes hyperboliques et couches limites. Grenoble Ann. Inst. Fourier 45 (1995) 973-1006. | Numdam | Zbl

[12] M.E. Gurtin, An introduction to continuum mechanics. Academic Press, New York (1981). | MR | Zbl

[13] F. Hecht and O. Pironneau, FreeFem++:Manual version 1.23, 13-05-2002. FreeFem++ is a free software available at: http://www-rocq.inria.fr/Frederic.Hecht/freefem++.htm

[14] J.M. Hervouet and A. Watrin, Code TELEMAC (système ULYSSE) : Résolution et mise en œuvre des équations de Saint-Venant bidimensionnelles, Théorie et mise en œuvre informatique, Rapport EDF HE43/87.37 (1987).

[15] S. Jin, A steady-state capturing method for hyperbolic systems with geometrical source terms. ESAIM: M2AN 35 (2001) 631-645. | Numdam | Zbl

[16] A. Kurganov and L. Doron, Central-upwind schemes for the Saint-Venant system. ESAIM: M2AN 36 (2002) 397-425. | Numdam | Zbl

[17] O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Ural'Ceva, Linear and quasilinear equations of parabolic type. Providence, Rhode Island. Amer. Math. Soc. (1968).

[18] D. Levermore and M. Sammartino, A shallow water model with eddy viscosity for basins with varying bottom topography. Nonlinearity 14 (2001) 1493-1515. | Zbl

[19] E. Miglio, A. Quarteroni and F. Saleri, Finite element approximation of a quasi-3D shallow water equation. Comput. Methods Appl. Mech. Engrg. 174 (1999) 355-369. | Zbl

[20] J. Rauch and F. Massey, Differentiability of solutions to hyperbolic initial-boundary value problems. Trans. Amer. Math. Soc. 189 (1974) 303-318. | Zbl

[21] M. Sammartino and R.E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equations on a half-space. I. Existence for Euler and Prandtl Equations; II. Construction of the Navier-Stokes solution. Comm. Math. Physics 192 (1998) 433-461 and 463-491. | Zbl

[22] D. Serre, Sytems of conservation laws. I and II, Cambridge University Press, Cambridge (1996). | Zbl

[23] G.B. Whitham, Linear and nonlinear waves. John Wiley & Sons, New York (1974). | MR | Zbl

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