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Journal of Computational Mathematics

A two-dimensional method for a family of dispersive shallow water models
Nora Aïssiouene1; Marie-Odile Bristeau2; Edwige Godlewski2; Anne Mangeney3; Carlos Parés Madroñal4; Jacques Sainte-Marie2
1 Sorbonne Université, Institut Carnot Smiles, 4 Place Jussieu, F-75252 Paris cedex 05
2 Inria Paris, 2 rue Simone Iff, CS 42112, 75589 Paris Cedex 12, France & Sorbonne Université, Université de Paris, CNRS, Laboratoire Jacques-Louis Lions, LJLL, F-75005 Paris
3 Univ. de Paris, Institut de Physique du Globe de Paris, Seismology Group, 1 rue Jussieu, Paris F-75005, France
4 EDANYA, Universidad de Málaga, Campus de Teatinos s/n, 29080 Málaga, Spain
The SMAI Journal of computational mathematics, Volume 6 (2020), pp. 187-226.

We propose a numerical method for a family of two-dimensional dispersive shallow water systems with topography. The considered models consist in shallow water approximations – without the hydrostatic assumption – of the incompressible Euler system with free surface. Hence, the studied models appear as extensions of the classical shallow water system enriched with dispersive terms. The model formulation motivates us to use a prediction-correction scheme for its numerical approximation. The prediction part leads to solving a classical shallow water system with topography while the correction part leads to solving an elliptic-type problem. The numerical approximation of the considered dispersive models in the two-dimensional case over unstructured meshes is described, it requires to combine finite volume and finite element techniques. A special emphasis is given to the formulation and the numerical resolution of the correction step (variational formulation, inf-sup condition, boundary conditions,...). The numerical procedure is confronted with analytical and experimental test cases. Finally, an application to a real tsunami case is given.

Published online:
DOI: 10.5802/smai-jcm.66
Classification: 65M12, 74S10, 76M12, 35L65, 35Q30, 35Q35, 76D05
Keywords: shallow water flows, dispersive effects, prediction-correction scheme, combined finite volume / finite element technique, dispersive wave propagation, tsunami propagation
Nora Aïssiouene 1; Marie-Odile Bristeau 2; Edwige Godlewski 2; Anne Mangeney 3; Carlos Parés Madroñal 4; Jacques Sainte-Marie 2

1 Sorbonne Université, Institut Carnot Smiles, 4 Place Jussieu, F-75252 Paris cedex 05
2 Inria Paris, 2 rue Simone Iff, CS 42112, 75589 Paris Cedex 12, France & Sorbonne Université, Université de Paris, CNRS, Laboratoire Jacques-Louis Lions, LJLL, F-75005 Paris
3 Univ. de Paris, Institut de Physique du Globe de Paris, Seismology Group, 1 rue Jussieu, Paris F-75005, France
4 EDANYA, Universidad de Málaga, Campus de Teatinos s/n, 29080 Málaga, Spain
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     title = {A two-dimensional method for a family of dispersive shallow water models},
     journal = {The SMAI Journal of computational mathematics},
     pages = {187--226},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
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Nora Aïssiouene; Marie-Odile Bristeau; Edwige Godlewski; Anne Mangeney; Carlos Parés Madroñal; Jacques Sainte-Marie. A two-dimensional method for a family of dispersive shallow water models. The SMAI Journal of computational mathematics, Volume 6 (2020), pp. 187-226. doi : 10.5802/smai-jcm.66. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.66/

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