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In this paper we consider the numerical solution of Boussinesq-Peregrine typesystems by the application of the Galerkin finite element method. The structureof the Boussinesq systems is explained and certain alternative nonlinear anddispersive terms are compared. A detailed study of the convergence propertiesof the standard Galerkin method, using various finite element spaces onunstructured triangular grids, is presented.  Along with the study of the Peregrine system, a new Boussinesq system ofBBM-BBM type is derived. The new system has the same structure in its momentumequation but differs slightly in the mass conservation equation compared to thePeregrine system. Further, the finite element method applied to the new systemhas better convergence properties, when used for its numerical approximation.  Due to the lack of analytical formulas for solitary wave solutions for thesystems under consideration, a Galerkin finite element method combined with thePetviashvili iteration is proposed for the numerical generation of accurateapproximations of line solitary waves. Various numerical experiments related tothe propagation of solitary and periodic waves over variable bottom topographyand their interaction with the boundaries of the domains are presented. Weconclude that both systems have similar accuracy when approximate long waves ofsmall amplitude while the Galerkin finite element method is more effective whenapplied to BBM-BBM type systems.

Boussinesq-Peregrine water wave models and their numerical approximation