  We consider a simple initial-boundary-value problem for the shallow water
equations in one space dimension. We discretize the problem in space by the
standard Galerkin finite element method on a quasiuniform mesh and in time by
the classical 4-stage, 4th order, explicit Runge-Kutta scheme. Assuming
smoothness of solutions, a Courant number restriction, and certain hypotheses
on the finite element spaces, we prove L2 error estimates that are of
fourth-order accuracy in the temporal variable and of the usual, due to the
nonuniform mesh, suboptimal order in space. We also make a computational study
of the numerical spatial and temporal orders of convergence, and of the
validity of a hypothesis made on the finite element spaces.
