Testing of Finite Element Schemes for Linear Shallow Water Equations

Abstract

This chapter discusses the problem of testing finite element schemes for shallow water free surface flow, including the effects of friction, wind stress, and Coriolis force. By neglecting non- linear terms in the shallow water equations, they can be reduced to a Helmholtz equation by considering only periodic or time in dependent solutions. In the case when only effects of friction and wind stress are included there exist analytical solutions for simple geometries that lend themselves very well for comparisons with solutions obtained by finite element schemes. When the effect of Coriolis force is also present, analytical solutions, however, become much more complicated even in simple geometries due to awkward boundary conditions. In this case, therefore, it is proposed the use of a simple finite difference scheme for comparisons with the finite element solutions. As all the difference approximations can be made to be of second order for simple geometries, a posteriori error estimates for the finite difference solutions are available using extrapolation.