  We consider a simple initial-boundary-value problem for the shallow water
equations in one space dimension, and also the analogous problem for a
symmetric variant of the system. Assuming smoothness of solutions, we
discretize these problems in space using standard Galerkin-finite element
methods and prove $L^{2}$-error estimates for the semidiscrete problems for
quasiuniform and uniform meshes. In particular we show that in the case of
spatial discretizations with piecewise linear continuous functions on a uniform
mesh, suitable compatibility conditions at the boundary and superaccuracy
properties of the $L^{2}$ projection on the finite element subspaces lead to an
optimal-order $O(h^{2})$ $L^{2}$-error estimate. We also examine temporal
discretizations of the semidiscrete problems by three explicit Runge-Kutta
methods (the Euler, improved Euler, and the Shu-Osher scheme) and prove
$L^{2}$-error estimates, which are of optimal order in the temporal variable,
under appropriate stability conditions. In a final section of remarks we prove
optimal-order $L^{2}$-error estimates for smooth spline spatial discretizations
of the periodic initial-value problem for the systems. We also prove that
small-amplitude, appropriately transformed solutions of the symmetric system
are close to the corresponding solutions of the usual system while they are
both smooth, thus providing a justification of the symmetric system.
