Comptes Rendus
no. 11
Numerical analysis
Numerical scheme for multilayer shallow-water model in the low-Froude number regime
[Schéma numérique pour les modèles de Saint-Venant multi-couche à faible nombre de Froude]

Présenté par : Philippe G. Ciarlet

Martin Parisot1 ; Jean-Paul Vila2
1 Institut de mathématiques de Toulouse, UMR CNRS 5219, UPS IMT, 31062 Toulouse cedex 9, France
2 Institut de mathématiques de Toulouse, UMR CNRS 5219, INSA, 31077 Toulouse, France
Comptes Rendus. Mathématique, Volume 352 (2014) no. 11, pp. 953-957.

Le but de cette note est de présenter un schéma numérique multi-dimensionnel rapprochant les solutions du modèle de Saint-Venant multi-couche en régime de faible nombre de Froude. La stratégie proposée est basée sur un modèle régularisé où la vitesse de transport est modifiée par un gradient de pression dans les équations de la masse et de la quantité de mouvement. La solution numérique satisfait la dissipation d'énergie, jouant le rôle de l'entropie du point de vue mathématique, et les principales propriétés physiques nécessaires aux simulations dans le cadre des écoulements océaniques.

The aim of this note is to present a multi-dimensional numerical scheme approximating the solutions to the multilayer shallow-water model in the low-Froude-number regime. The proposed strategy is based on a regularized model where the advection velocity is modified with a pressure gradient in both mass and momentum equations. The numerical solution satisfies the dissipation of energy, which acts for mathematical entropy, and the main physical properties required for simulations within oceanic flows.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.09.020

Martin Parisot 1 ; Jean-Paul Vila 2

1 Institut de mathématiques de Toulouse, UMR CNRS 5219, UPS IMT, 31062 Toulouse cedex 9, France
2 Institut de mathématiques de Toulouse, UMR CNRS 5219, INSA, 31077 Toulouse, France 
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Martin Parisot; Jean-Paul Vila. Numerical scheme for multilayer shallow-water model in the low-Froude number regime. Comptes Rendus. Mathématique, Volume 352 (2014) no. 11, pp. 953-957. doi : 10.1016/j.crma.2014.09.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.09.020/

[1] R. Abgrall; S. Karni Two-layer shallow water system: a relaxation approach, SIAM J. Sci. Comput., Volume 31 (2009), pp. 1603-1627

[2] E. Audusse A multilayer Saint–Venant model: derivation and numerical validation, Discrete Contin. Dyn. Syst., Ser. B, Volume 5 (2005), pp. 189-214

[3] F. Bouchut; T. Morales de Luna An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment, ESAIM: Math. Model. Numer. Anal., Volume 42 (2008), pp. 683-698

[4] F. Bouchut; V. Zeitlin A robust well-balanced scheme for multi-layer shallow water equations, Discrete Contin. Dyn. Syst., Ser. B, Volume 13 (2010), pp. 739-758

[5] M. Castro; J. Macías; C. Parés 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: Math. Model. Numer. Anal., Volume 35 (2001), pp. 107-127

[6] S. Dellacherie Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number, J. Comput. Phys., Volume 229 (2010), pp. 978-1016

[7] V. Duchêne Asymptotic shallow water models for internal waves in a two-fluid system with a free surface, SIAM J. Math. Anal., Volume 42 (2010), pp. 2229-2260

[8] N. Grenier; J.-P. Vila; P. Villedieu An accurate low-mach scheme for a compressible two-fluid model applied to free-surface flows, J. Comput. Phys., Volume 252 (2013), pp. 1-19

[9] J. Gula; V. Zeitlin; F. Bouchut Instabilities of buoyancy driven coastal currents and their nonlinear evolution in the two-layer rotating shallow water model. Part II. Active lower layer, J. Fluid Mech., Volume 665 (2010), pp. 209-237

[10] R.R. Long Long waves in a two-fluid system, J. Meteorol., Volume 13 (1956), pp. 70-74

[11] R. Monjarret, Local well-posedness of the multi-layer shallow water model with free surface, in press.

[12] M. Parisot, J.-P. Vila, Centered-potential regularization of advection upstream splitting method: application to the multilayer shallow water model in the low-Froude number regime, in press.

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☆ This work was supported by the French Naval Hydrographic and Oceanographic Service (grant number N11CR0001).

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