  This paper is concerned with the numerical approximation of
initial-boundary-value problems of a three-parameter family of Bona-Smith
systems, derived as a model for the propagation of surface waves under a
physical Boussinesq regime. The work proposed here is focused on the
corresponding problem with Dirichlet boundary conditions and its approximation
in space with spectral methods based on Jacobi polynomials, which are defined
from the orthogonality with respect to some weighted $L^{2}$ inner product.
Well-posedness of the problem on the corresponding weighted Sobolev spaces is
first analyzed and existence and uniqueness of solution, locally in time, are
proved. Then the spectral Galerkin semidiscrete scheme and some detailed
comments on its implementation are introduced. The existence of numerical
solution and error estimates on those weighted Sobolev spaces are established.
Finally, the choice of the time integrator to complete the full discretization
takes care of different stability issues that may be relevant when
approximating the semidiscrete system. Some numerical experiments illustrate
the results.
