A simplified nonlinear dispersive system of BBM-type, initially derived by D. Mitsotakis, is employed here in order to model the generation and propagation of surface water waves over variable bottom. The simplification consists in applying the so-called Boussinesq approximation. Using the finite element method and the FreeFem++ software, we solve numerically this system for three different complexities for the bathymetry function: a flat bottom case, a variable bottom in space, and a variable bottom both in space and in time. The last case is illustrated with the Java 2006 tsunami event. This article is designed rather as a tutorial paper even if it contains the description of completely new adaptation techniques.