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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 spacedimensions is considered. These systems approximate thethree-dimensional Euler equations, and consist of three nonlineardispersive wave equations that describe two-way propagation oflong surface waves of small amplitude in ideal fluids over ahorizontal bottom. For a subset of these systems it is proved thattheir Cauchy problem is locally well-posed in suitable Sobolevclasses. Further, a class of these systems is discretized byGalerkin-finite element methods, and error estimates are provedfor the resulting continuous time semidiscretizations. Results ofnumerical experiments are also presented with the aim of studyingproperties of line solitary waves and expanding wave solutions of these systems.

Keywords

Boussinesq systems in two space dimensionswater wavetheorynonlinear dispersive wave equationsGalerkin-finiteelement 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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