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We consider two `Classical' Boussinesq type systems modelling two-waypropagation of long surface waves in a finite channel with variable bottomtopography. Both systems are derived from the 1-d Serre-Green-Naghdi (SGN)system; one of them is valid for stronger bottom variations, and coincides withPeregrine's system, and the other is valid for smaller bottom variations. Wediscretize in the spatial variable simple initial-boundary-value problems(ibvp's) for both systems using standard Galerkin-finite element methods andprove $L^2$ error estimates for the ensuing semidiscrete approximations. Wecouple the schemes with the 4th order-accurate, explicit, classical Runge-Kuttatime-stepping procedure and use the resulting fully discrete methods innumerical simulations of dispersive wave propagation over variable bottoms withseveral kinds of boundary conditions, including absorbing ones. We describe indetail the changes that solitary waves undergo when evolving under each systemover a variety of variable-bottom environments. We assess the efficacy of bothsystems in approximating these flows by comparing the results of theirsimulations with each other, with simulations of the SGN-system, and withavailable experimental data from the literature.

Galerkin finite element methods for the numerical solution of two classical-Boussinesq type systems over variable bottom topography