  We consider the `classical' Boussinesq system in one space dimension and its
symmetric analog. These systems model two-way propagation of nonlinear,
dispersive long waves of small amplitude on the surface of an ideal fluid in a
uniform horizontal channel. We discretize an initial-boundary-value problem for
these systems in space using Galerkin-finite element methods and prove error
estimates for the resulting semidiscrete problems and also for their fully
discrete analogs effected by explicit Runge-Kutta time-stepping procedures. The
theoretical orders of convergence obtained are consistent with the results of
numerical experiments that are also presented.

Numerical Analysis