Error estimates for finite element methods for the shallow water equations

We consider a simple initial–boundary–value problem for the system of the shallow water equations and its symmetric variant in one space dimension. We discretize the problem in space by the standard Galerkin–finite element method and prove error estimates for the resulting semidiscretizations for quasiuniform and uniform meshes. We also discretize the problem in time using the third–order Shu–Osher Runge–Kutta scheme and prove error estimates of optimal temporal order under a Courant stability condition.