On the standard Galerkin method with explicit RK4 time stepping for the Shallow Water equations

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.