ESAIM: Mathematical Modelling and Numerical Analysis

Research Article

On some Boussinesq systems in two space dimensions: theory and numerical analysis

Dougalis, Vassilios A.a1a2, Mitsotakis, Dimitrios E.a1a2 and Saut, Jean-Claudea3

a1 Department of Mathematics, University of Athens, 15784 Zographou, Greece.

a2 Institute of Applied and Computational Mathematics, F.O.R.T.H., P.O. Box 1527, 71110 Heraklion, Greece.

a3 UMR de Mathématiques, Université de Paris-Sud, Bâtiment 425, 91405 Orsay, France. jean-claude.saut@math.u-psud.fr

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.

(Received June 6 2006)

(Revised November 26 2007)

(Online publication October 23 2007)

Key Words:

  • Boussinesq systems in two space dimensions;
  • water wave theory;
  • nonlinear dispersive wave equations;
  • Galerkin-finite element methods for Boussinesq systems.

Mathematics Subject Classification:

  • 35Q53;
  • 65M60;
  • 76B15
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