On some Boussinesq systems in two space dimensions : theory and numerical analysis
Dougalis, Vassilios A. ; Mitsotakis, Dimitrios E. ; Saut, Jean-Claude
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007), p. 825-854 / Harvested from Numdam
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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.

Publié le : 2007-01-01
DOI : https://doi.org/10.1051/m2an:2007043
Classification:  35Q53,  65M60,  76B15 
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     author = {Dougalis, Vassilios A. and Mitsotakis, Dimitrios E. and Saut, Jean-Claude},
     title = {On some Boussinesq systems in two space dimensions : theory and numerical analysis},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {41},
     year = {2007},
     pages = {825-854},
     doi = {10.1051/m2an:2007043},
     mrnumber = {2363885},
     zbl = {1140.76314},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2007__41_5_825_0}
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Dougalis, Vassilios A.; Mitsotakis, Dimitrios E.; Saut, Jean-Claude. On some Boussinesq systems in two space dimensions : theory and numerical analysis. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007) pp. 825-854. doi : 10.1051/m2an:2007043. http://gdmltest.u-ga.fr/item/M2AN_2007__41_5_825_0/

[1] A.A. Alazman, J.P. Albert, J.L. Bona, M. Chen and J. Wu, Comparisons between the BBM equation and a Boussinesq system. Adv. Differential Equations 11 (2006) 121-166. | Zbl 1104.35039

[2] D.C. Antonopoulos, The Boussinesq system of equations: Theory and numerical analysis. Ph.D. Thesis, University of Athens, 2000 (in Greek).

[3] 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).

[4] J.L. Bona and M. Chen, A Boussinesq system for two-way propagation of nonlinear dispersive waves. Physica D 116 (1998) 191-224. | Zbl 0962.76515

[5] J.L. Bona and R. Smith, A model for the two-way propagation of water waves in a channel. Math. Proc. Camb. Phil. Soc. 79 (1976) 167-182. | Zbl 0332.76007

[6] J.L. Bona, M. Chen and J.-C. Saut, 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. | Zbl 1022.35044

[7] J.L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: II. The nonlinear theory. Nonlinearity 17 (2004) 925-952. | Zbl 1059.35103

[8] J.L. Bona, T. Colin and D. Lannes, Long wave approximations for water waves. Arch. Rational Mech. Anal. 178 (2005) 373-410. | Zbl 1108.76012

[9] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (1994). | MR 1278258 | Zbl 0804.65101

[10] M. Chen, Exact traveling-wave solutions to bi-directional wave equations. Int. J. Theor. Phys. 37 (1998) 1547-1567. | Zbl 1097.35115

[11] M. Chen, Solitary-wave and multi pulsed traveling-wave solutions of Boussinesq systems. Applic. Analysis 75 (2000) 213-240. | Zbl 1034.35108

[12] 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.

[13] 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).

[14] 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).

[15] D.R. Kincaid, J.R. Respess, D.M. Young and R.G. Grimes, ITPACK 2C: A Fortran package for solving large sparse linear systems by adaptive accelerated iterative methods. ACM Trans. Math. Software 8 (1982) 302-322. | Zbl 0485.65025

[16] R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximations. Math. Comp. 38 (1982) 437-445. | Zbl 0483.65007

[17] A.H. Schatz and L.B. Wahlbin, On the quasi-optimality in L ∞ of the H 1 ∘ -projection into finite elements spaces. Math. Comp. 38 (1982) 1-22. | Zbl 0483.65006

[18] M.H. Schultz, L ∞ Multivariate approximation theory. SIAM J. Numer. Anal. 6 (1969) 161-183. | Zbl 0202.15901

[19] M.H. Schultz, Approximation theory of multivatiate spline functions in Sobolev spaces. SIAM J. Numer. Anal. 6 (1969) 570-582. | Zbl 0211.18803

[20] J.F. Toland, Existence of symmetric homoclinic orbits for systems of Euler-Lagrange equations. A.M.S. Proc. Symposia in Pure Mathematics 45 (1986) 447-459. | Zbl 0589.34029

[21] G.B. Whitham, Linear and Non-linear Waves. Wiley, New York (1974). | Zbl 0373.76001