  We consider the Shallow Water equations in the supercritical and subcritical
cases in one space variable,posed in a finite spatial interval with
characteristic boundary conditions at the endpoints, which, as is well known,
are transparent,i.e. allow outgoing waves to exit without generating spurious
reflected waves. Assuming that the resulting initial-boundary-value problems
have smooth solutions,we approximate them in space using standard
Galerkin-finite element methods and prove L^2 error estimates for the
semidiscrete problems on quasiuniform meshes.We discretize the problems in the
temporal variable using an explicit,fourth-order accurate Runge-Kutta scheme
and check, by means of numerical experiment, that the resulting fully discrete
schemes have excellent absorption properties.
