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

Dougalis, Vassilios A.; Mitsotakis, Dimitrios E.; Saut, Jean-Claude

ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 5, pp. 825-854.

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.

Classification : 35Q53, 65M60, 76B15
Mots clés : Boussinesq systems in two space dimensions, water wave theory, nonlinear dispersive wave equations, Galerkin-finite element methods for Boussinesq systems

[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

[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

[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

[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

[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

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

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

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

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

[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

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

[17] A.H. Schatz and L.B. Wahlbin, On the quasi-optimality in $L_{\infty }$ of the $H_1^{\circ }$-projection into finite elements spaces. Math. Comp. 38 (1982) 1-22. | Zbl

[18] M.H. Schultz, $L^{\infty }$ Multivariate approximation theory. SIAM J. Numer. Anal. 6 (1969) 161-183. | Zbl

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

[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

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