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