  A highly accurate numerical scheme is presented for the Serre system of
partial differential equations, which models the propagation of dispersive
shallow water waves in the fully-nonlinear regime. The fully-discrete scheme
utilizes the Galerkin / finite-element method based on smooth periodic splines
in space, and an explicit fourth-order Runge-Kutta method in time. Computations
compared with exact solitary and cnoidal wave solutions show that the scheme
achieves the optimal orders of accuracy in space and time. These computations
also show that the stability of this scheme does not impose restrictive
conditions on the temporal step size. In addition, solitary, cnoidal, and
dispersive shock waves are studied in detail using this numerical scheme for
the Serre system and compared with the 'classical' Boussinesq system for
small-amplitude shallow water waves. The results show that the interaction of
solitary waves in the Serre system is more inelastic. The efficacy of the
numerical scheme for modeling dispersive shocks is shown by comparison with
asymptotic results. These results have application to the modeling of shallow
water waves of intermediate or large amplitude, such as occurs in the nearshore
zone.
