Numerical solution of Boussinesq systems of the Bona-Smith family

• Autores: D.C. Antonopoulos, Vassilios A. Dougalis, D. E. Mitsotakis
• Localización: Applied numerical mathematics, ISSN-e 0168-9274, Vol. 60, Nº. 4, 2010, pág. 314336
• Idioma: inglés
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• Resumen

  • In this paper we consider the one-parameter family of Bona�Smith systems, which belongs to the class of Boussinesq systems modelling two-way propagation of long waves of small amplitude on the surface of water in a channel. We study numerically three initial-boundary-value problems for these systems, corresponding, respectively, to homogeneous Dirichlet, reflection, and periodic boundary conditions posed at the endpoints of a finite spatial interval. We approximate these problems using the standard Galerkin-finite element method for the spatial discretization and a fourth-order, explicit Runge�Kutta scheme for the time stepping, and analyze the convergence of the fully discrete schemes. We use these numerical methods as exploratory tools in a series of numerical experiments aimed at illuminating interactions of solitary-wave solutions of the Bona�Smith systems, such as head-on and overtaking collisions, and interactions of solitary waves with the boundaries.