  We consider the Serre system of equations which is a nonlinear dispersive
system that models two-way propagation of long waves of not necessarily small
amplitude on the surface of an ideal fluid in a channel. We discretize in space
the periodic initial-value problem for the system using the standard Galerkin
finite element method with smooth splines on a uniform mesh and prove an
optimal-order $L^{2}$-error estimate for the resulting semidiscrete
approximation. Using the fourth-order accurate, explicit, `classical'
Runge-Kutta scheme for time stepping we construct a highly accurate fully
discrete scheme in order to approximate solutions of the system, in particular
solitary-wave solutions, and study numerically phenomena such as the resolution
of general initial profiles into sequences of solitary waves, and overtaking
collisions of pairs of solitary waves propagating in the same direction with
different speeds.
