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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. [email protected]
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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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References

Alazman, A.A., Albert, J.P., Bona, J.L., Chen, M. and Comparisons, J. Wu between the BBM equation and a Boussinesq system. Adv. Differential Equations 11 (2006) 121–166.
D.C. Antonopoulos, The Boussinesq system of equations: Theory and numerical analysis. Ph.D. Thesis, University of Athens, 2000 (in Greek).
D.C. Antonopoulos, V.A. Dougalis and D.E. Mitsotakis, Theory and numerical analysis of the Bona-Smith type systems of Boussinesq equations. (to appear).
Bona, J.L. and Chen, M., Boussinesq, A system for two-way propagation of nonlinear dispersive waves. Physica D 116 (1998) 191–224. CrossRef
Bona, J.L. and Smith, R., A model for the two-way propagation of water waves in a channel. Math. Proc. Camb. Phil. Soc. 79 (1976) 167–182. CrossRef
Bona, J.L., Chen, M. and Saut, J.-C., Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: I. Derivation and Linear Theory. J. Nonlinear Sci. 12 (2002) 283–318. CrossRef
Bona, J.L., Chen, M. and Saut, J.-C., Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: II. The nonlinear theory. Nonlinearity 17 (2004) 925–952. CrossRef
Bona, J.L., Colin, T. and Lannes, D., Long wave approximations for water waves. Arch. Rational Mech. Anal. 178 (2005) 373–410. CrossRef
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (1994).
Chen, M., Exact traveling-wave solutions to bi-directional wave equations. Int. J. Theor. Phys. 37 (1998) 1547–1567. CrossRef
Chen, M., Solitary-wave and multi pulsed traveling-wave solutions of Boussinesq systems. Applic. Analysis 75 (2000) 213–240. CrossRef
V.A. Dougalis and D.E. Mitsotakis, Solitary waves of the Bona-Smith system, in Advances in scattering theory and biomedical engineering, D. Fotiadis and C. Massalas Eds., World Scientific, New Jersey (2004) 286–294.
V.A. Dougalis, D.E. Mitsotakis and J.-C. Saut, On initial-boundary value problems for some Boussinesq systems in two space dimensions. (to appear).
P. Grisvard, Quelques proprietés des espaces de Sobolev, utiles dans l'étude des équations de Navier-Stokes (I). Problèmes d'évolution, non linéaires, Séminaire de Nice (1974–1976).
Kincaid, D.R., Respess, J.R., Young, D.M. and Grimes, R.G., ITPACK 2C: A Fortran package for solving large sparse linear systems by adaptive accelerated iterative methods. ACM Trans. Math. Software 8 (1982) 302–322. CrossRef
Rannacher, R. and Scott, R., Some optimal error estimates for piecewise linear finite element approximations. Math. Comp. 38 (1982) 437–445. CrossRef
Schatz, A.H. and Wahlbin, L.B., On the quasi-optimality in L∞ of the $H^{\circ}_1$ -projection into finite elements spaces. Math. Comp. 38 (1982) 1–22.
Schultz, M.H., L ∞ Multivariate approximation theory. SIAM J. Numer. Anal. 6 (1969) 161–183. CrossRef
Schultz, M.H., Approximation theory of multivatiate spline functions in Sobolev spaces. SIAM J. Numer. Anal. 6 (1969) 570–582. CrossRef
Toland, J.F., Existence of symmetric homoclinic orbits for systems of Euler-Lagrange equations. A.M.S. Proc. Symposia in Pure Mathematics 45 (1986) 447–459. CrossRef
G.B. Whitham, Linear and Non-linear Waves. Wiley, New York (1974).