Hostname: page-component-797576ffbb-6mkhv Total loading time: 0 Render date: 2023-12-09T02:20:28.735Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "useRatesEcommerce": true } hasContentIssue false

A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation*

Published online by Cambridge University Press:  24 June 2010

Emmanuel Audusse
Univ. Paris 13, Institut Galilée, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France.
Marie-Odile Bristeau
INRIA Paris-Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France.
Benoît Perthame
INRIA Paris-Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France. Lab. J.-L. Lions, Univ. P. et M. Curie, BC187, 4 place Jussieu, 75252 Paris Cedex 05, France.
Jacques Sainte-Marie
INRIA Paris-Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France. Saint-Venant Laboratory, 6 quai Watier, 78400 Chatou, France.
Get access


The standard multilayer Saint-Venant system consists in introducing fluid layers that are advected by the interfacial velocities. As a consequence there is no mass exchanges between these layers and each layer is described by its height and its average velocity. Here we introduce another multilayer system with mass exchanges between the neighboring layers where the unknowns are a total height of water and an average velocity per layer. We derive it from Navier-Stokes system with an hydrostatic pressure and prove energy and hyperbolicity properties of the model. We also give a kinetic interpretation leading to effective numerical schemes with positivity and energy properties. Numerical tests show the versatility of the approach and its ability to compute recirculation cases with wind forcing.

Research Article
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Audusse, E., A multilayer Saint-Venant system: Derivation and numerical validation. Discrete Contin. Dyn. Syst. Ser. B 5 (2005) 189214. CrossRef
Audusse, E. and Bristeau, M.O., Transport of pollutant in shallow water flows: A two time steps kinetic method. ESAIM: M2AN 37 (2003) 389416. CrossRef
Audusse, E. and Bristeau, M.O., A well-balanced positivity preserving second-order scheme for shallow water flows on unstructured meshes. J. Comput. Phys. 206 (2005) 311333. CrossRef
Audusse, E. and Bristeau, M.O., Finite-volume solvers for a multilayer Saint-Venant system. Int. J. Appl. Math. Comput. Sci. 17 (2007) 311319. CrossRef
Audusse, E., Bouchut, F., Bristeau, M.O., Klein, R. and Perthame, B., A fast and stable well-balanced scheme with hydrostatic reconstruction for Shallow Water flows. SIAM J. Sci. Comput. 25 (2004) 20502065. CrossRef
Audusse, E., Bristeau, M.O. and Decoene, A., Numerical simulations of 3d free surface flows by a multilayer Saint-Venant model. Int. J. Numer. Methods Fluids 56 (2008) 331350. CrossRef
Barré de, A.J.C. Saint-Venant, Théorie du mouvement non permanent des eaux avec applications aux crues des rivières et à l'introduction des marées dans leur lit. C. R. Acad. Sci. Paris 73 (1871) 147154.
Bouchut, F., An introduction to finite volume methods for hyperbolic conservation laws. ESAIM: Proc. 15 (2004) 107127.
Bouchut, F. and Morales de, T. Luna, An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment. ESAIM: M2AN 42 (2008) 683698. CrossRef
Bouchut, F. and Westdickenberg, M., Gravity driven shallow water models for arbitrary topography. Commun. Math. Sci. 2 (2004) 359389. CrossRef
M.O. Bristeau and J. Sainte-Marie, Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems. Discrete Contin. Dyn. Syst. Ser. B 10 (2008) 733–759.
Castro, M.J., García-Rodríguez, J.A., González-Vida, J.M., Macías, J., Parés, C. and Vázquez-Cendón, M.E., Numerical simulation of two-layer shallow water flows through channels with irregular geometry. J. Comput. Phys. 195 (2004) 202235. CrossRef
Castro, M.J., Macías, J. and Parés, C., A q-scheme for a class of systems of coupled conservation laws with source term. application to a two-layer 1-D shallow water system. ESAIM: M2AN 35 (2001) 107127. CrossRef
Decoene, A. and Gerbeau, J.-F., Sigma transformation and ALE formulation for three-dimensional free surface flows. Int. J. Numer. Methods Fluids 59 (2009) 357386. CrossRef
Decoene, A., Bonaventura, L., Miglio, E. and Saleri, F., Asymptotic derivation of the section-averaged shallow water equations for river hydraulics. M3AS 19 (2009) 387417.
Ferrari, S. and Saleri, F., A new two-dimensional Shallow Water model including pressure effects and slow varying bottom topography. ESAIM: M2AN 38 (2004) 211234. CrossRef
FreeFem++ home page, (2009).
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.
P.L. Lions, Mathematical Topics in Fluid Mechanics, Incompressible models, Vol. 1. Oxford University Press, UK (1996).
F. Marche, Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects. Eur. J. Mech. B, Fluids 26 (2007) 49–63.
Mohammadi, B., Pironneau, O. and Valentin, F., Rough boundaries and wall laws. Int. J. Numer. Methods Fluids 27 (1998) 169177. 3.0.CO;2-4>CrossRef
Nwogu, O., Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterw. Port Coast. Ocean Eng. ASCE 119 (1993) 618638. CrossRef
Peregrine, D.H., Long waves on a beach. J. Fluid Mech. 27 (1967) 815827. CrossRef
B. Perthame, Kinetic formulation of conservation laws. Oxford University Press, UK (2002).
Perthame, B. and Simeoni, C., A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201231. CrossRef
Salençon, M.J. and Thébault, J.M., Simulation model of a mesotrophic reservoir (Lac de Pareloup, France): Melodia, an ecosystem reservoir management model. Ecol. model. 84 (1996) 163187. CrossRef
Shankar, N.J., Cheong, H.F. and Sankaranarayanan, S., Multilevel finite-difference model for three-dimensional hydrodynamic circulation. Ocean Eng. 24 (1997) 785816. CrossRef
Ursell, F., The long wave paradox in the theory of gravity waves. Proc. Cambridge Phil. Soc. 49 (1953) 685694. CrossRef
M.A. Walkley, A numerical Method for Extended Boussinesq Shallow-Water Wave Equations. Ph.D. Thesis, University of Leeds, UK (1999).