Abstract
We consider a class of Boussinesq systems of Bona-Smith type in two space dimensions approximating surface wave flows modelled by the three-dimensional Euler equations. We show that various initial-boundary-value problems for these systems, posed on a bounded plane domain are well posed locally in time. In the case of reflective boundary conditions, the systems are discretized by a modified Galerkin method which is proved to converge in L 2 at an optimal rate. Numerical experiments are presented with the aim of simulating two-dimensional surface waves in realistic (plane) domains with a variety of initial and boundary conditions, and comparing numerical solutions of Bona-Smith systems with analogous solutions of the BBM-BBM system.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Antonopoulos, D.C., Dougalis, V.A., Mitsotakis, D.E.: Initial-boundary-value problems for the Bona-Smith family of Boussinesq systems. Adv. Differ. Equ. 14, 27–53 (2009)
Antonopoulos, D.C., Dougalis, V.A., Mitsotakis, D.E.: Numerical solution of Boussinesq systems of the Bona-Smith type. Appl. Numer. Math. To appear
Bona, J.L., Chen, M.: A Boussinesq system for two-way propagation of nonlinear dispersive waves. Physica D 116, 191–224 (1998)
Bona, J.L., Smith, R.: A model for the two-way propagation of water waves in a channel. Math. Proc. Camb. Philos. Soc. 79, 167–182 (1976)
Bona, J.L., Chen, M., 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, 283–318 (2002)
Bona, J.L., Chen, M., Saut, J.-C.: Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: II. The nonlinear theory. Nonlinearity 17, 925–952 (2004)
Bona, J.L., Colin, T., Lannes, D.: Long wave approximations for water waves. Arch. Ration. Mech. Anal. 178, 373–410 (2005)
Chen, M.: Numerical investigation of a two-dimensional Boussinesq system. Discrete Continuous Dyn. Syst. 23, 1169–1190 (2009)
Chen, M., Iooss, G.: Periodic wave patterns of two-dimensional Boussinesq systems. Eur. J. Mech. B, Fluids 25, 393–405 (2006)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holand, Amsterdam/New York/Oxford (1978)
Crouzeix, M., Thomée, V.: The stability in L p and \(W_{p}^{1}\) of the L 2-projection onto finite element function spaces. Math. Comput. 48, 521–532 (1987)
Dougalis, V.A., Mitsotakis, D.E., Saut, J.-C.: On some Boussinesq systems in two space dimensions: theory and numerical analysis. M2AN Math. Model. Numer. Anal. 41, 825–854 (2007)
Dougalis, V.A., Mitsotakis, D.E., Saut, J.-C.: On initial-boundary value problems for a Boussinesq system of BBM-BBM type in a plane domain. Discrete Continuous. Dyn. Syst. 23, 1191–1204 (2009)
Scott, R.: Optimal L ∞ estimates for the finite element method on irregular meshes. Math. Comput. 30, 681–697 (1976)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported in part by a French-Greek scientific cooperation grant for the period 2006–2008, funded jointly by EGIDE, France, and the General Secretariat of Research and Technology, Greece. D. Mitsotakis was also supported by Marie Curie Fellowship No. PIEF-GA-2008-219399 of the European Commission.
Rights and permissions
About this article
Cite this article
Dougalis, V.A., Mitsotakis, D.E. & Saut, JC. Boussinesq Systems of Bona-Smith Type on Plane Domains: Theory and Numerical Analysis. J Sci Comput 44, 109–135 (2010). https://doi.org/10.1007/s10915-010-9368-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-010-9368-z