Error estimates for Galerkin approximations of the “classical” Boussinesq system

Antonopoulos, D.; Dougalis, V.

doi:10.1090/S0025-5718-2012-02663-9

Error estimates for Galerkin approximations of the “classical” Boussinesq system
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by D. C. Antonopoulos and V. A. Dougalis;

Math. Comp. 82 (2013), 689-717

DOI: https://doi.org/10.1090/S0025-5718-2012-02663-9

Published electronically: December 12, 2012

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Abstract:

We consider the “classical” Boussinesq system in one space dimension and its symmetric analog. These systems model two-way propagation of nonlinear, dispersive long waves of small amplitude on the surface of an ideal fluid in a uniform horizontal channel. We discretize an initial-boundary-value problem for these systems in space using Galerkin-finite element methods and prove error estimates for the resulting semidiscrete problems and also for their fully discrete analogs effected by explicit Runge-Kutta time-stepping procedures. The theoretical orders of convergence obtained are consistent with the results of numerical experiments that are also presented.

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