  We consider the one-dimensional shallow water equations (SW) in a finite
channel with variable bottom topography. We pose several initial-boundary-value
problems for the SW system, including problems with transparent
(characteristic) boundary conditions in the supercritical and the subcritical
case. We discretize these problems in the spatial variable by standard
Galerkin-finite element methods and prove L^2-error estimates for the resulting
semidiscrete approximations. We couple the schemes with the 4th order-accurate,
explicit, classical Runge-Kutta time stepping procedure and use the resulting
fully discrete methods in numerical experiments of shallow water wave
propagation over variable bottom topographies with several kinds of boundary
conditions. We discuss issues related to the attainment of a steady state of
the simulated flows, including the good balance of the schemes.
