Notes on error estimates for the standard Galerkin-finite element method for the Shallow Water equations

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.