Quadratic Finite Elements in Shallow Water Problems

Two quadratic finite element models of the shallow water equations are presented, one using an explicit fourth-order Runge Kutta time integration scheme, the other an implicit trapezoidal rule scheme. The models are applied to rectangular channel problems which are similar to the tidal behavior of an estuary. Stability, accuracy, and the influence on the results of the friction and advective terms are covered. Bottom topography is shown to influence the results by changing the nature of the wave in the channel. Advective terms should be included in the model if there is significant bottom slope. In real situations the model is started with the water still and the surface flat, i.e., the cold start. In the frictionless situation, this produces spurious wave forms related to the natural frequencies of the channel. These may be damped out by the specification of a high level of friction.