A Simple Staggered Finite Element Scheme for Simulation of Shallow Water Free Surface Flows

This chapter describes a simple staggered finite element scheme for simulation of shallow water free surface flows. Numerical schemes for simulation of free surface flows that make use of the idea of replacing the first order continuity equation with the second order wave equation have received considerable attention in recent years. The computational attraction of the staggered finite element scheme is that in the absence of convective terms and for a given approximation to the elevation values the momentum equations can be integrated independently within the elements. By treating the discontinuities in the approximate velocity values across element boundaries appropriately, it is none the less found that the scheme is sufficiently accurate for many simulation purposes. With the simplifications referred to above the matrix to be inverted when advancing one time step with the global system will be positive definite and independent of time so that one only has to carry out a Cholesky factorization at the beginning, and when the time step is changed. The adequacy of this approach has been demonstrated in a number of test runs.