  We consider a class of Boussinesq systems of Bona-Smith type in two space
dimensions approximating surface wave flows modelled by the three-dimensional
Euler equations. We show that various initial-boundary-value problems for these
systems, posed on a bounded plane domain are well posed locally in time. In the
case of reflective boundary conditions, the systems are discretized by a
modified Galerkin method which is proved to converge in $L^2$ at an optimal
rate. Numerical experiments are presented with the aim of simulating
two-dimensional surface waves in complex plane domains with a variety of
initial and boundary conditions, and comparing numerical solutions of
Bona-Smith systems with analogous solutions of the BBM-BBM system.
