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On some Boussinesq systems in two space dimensions: theory and numericalanalysis

Published online by Cambridge University Press:  23 October 2007

Vassilios A. Dougalis ,
Dimitrios E. Mitsotakis  and
Jean-Claude Saut
Vassilios A. Dougalis
Affiliation:
Department of Mathematics, University of Athens, 15784 Zographou, Greece. Institute of Applied and Computational Mathematics, F.O.R.T.H., P.O. Box 1527, 71110 Heraklion, Greece.
Dimitrios E. Mitsotakis
Affiliation:
Department of Mathematics, University of Athens, 15784 Zographou, Greece. Institute of Applied and Computational Mathematics, F.O.R.T.H., P.O. Box 1527, 71110 Heraklion, Greece.
Jean-Claude Saut
Affiliation:
UMR de Mathématiques, Université de Paris-Sud, Bâtiment 425, 91405 Orsay, France. jean-claude.saut@math.u-psud.fr
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Abstract

A three-parameter family of Boussinesq type systems in two space dimensions is considered. These systems approximate the three-dimensional Euler equations, and consist of three nonlinear dispersive wave equations that describe two-way propagation of long surface waves of small amplitude in ideal fluids over a horizontal bottom. For a subset of these systems it is proved that their Cauchy problem is locally well-posed in suitable Sobolev classes. Further, a class of these systems is discretized by Galerkin-finite element methods, and error estimates are proved for the resulting continuous time semidiscretizations. Results of numerical experiments are also presented with the aim of studying properties of line solitary waves and expanding wave solutions of these systems.

Keywords

Boussinesq systems in two space dimensionswater wave theorynonlinear dispersive wave equationsGalerkin-finite element methods for Boussinesq systems.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 5 , September 2007 , pp. 825 - 854
Copyright
© EDP Sciences, SMAI, 2007

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