  This paper is devoted to the theoretical and numerical investigation of the
initial boundary value problem for a system of equations used for the
description of waves in coastal areas, namely, the Boussinesq-Abbott system in
the presence of topography. We propose a procedure that allows one to handle
very general linear or nonlinear boundary conditions. It consists in reducing
the problem to a system of conservation laws with nonlocal fluxes and coupled
to an ODE. This reformulation is used to propose two hybrid finite
volumes/finite differences schemes of first and second order respectively. The
possibility to use many kinds of boundary conditions is used to investigate
numerically the asymptotic stability of the boundary conditions, which is an
issue of practical relevance in coastal oceanography since asymptotically
stable boundary conditions would allow one to reconstruct a wave field from the
knowledge of the boundary data only, even if the initial data is not known.
