Finite element solution of the shallow water wave equations has found increasing use by researchers and practitioners in the modelling of oceans and coastal areas. Wave equation models, most of which use equal-order C? interpolants for both the velocity and the surface elevation, do not introduce spurious oscillation modes, hence avoiding the need for artificial or numerical damping. An important question for both primitive equation and wave equation models is the interpretation of boundary conditions. Analysis of the characteristics of the governing equations shows that for most geophysical flows a single condition at each boundary is sufficient, yet there is not a consensus in the literature as to what that boundary condition must be or how it should be implemented in a finite element code. Traditionally (partly because of limited data), surface elevation is specified at open ocean boundaries while the normal flux is specified as zero at land boundaries. In most finite element wave equation models both of these boundary conditions are implemented as essential conditions. Our recent work focuses on alternative ways to numerically implement normal flow boundary conditions with an eye towards improving the mass-conserving properties of wave equation models. A unique finite element formulation using generalized functions demonstrates that boundary conditions should be implemented by treating normal fluxes as natural conditions with the flux interpreted as external to the computational domain. Results from extensive numerical experiments show that the scheme does conserve mass for all parameter values. Furthermore, convergence studies demonstrate that the algorithm is consistent, as residual errors at the boundary diminish as the grid is refined.
KEY WORDS: shallow water equations; wave continuity equation; boundary conditions; finite elements; generalized functions BACKGROUND Shallow water equations are obtained by vertically averaging the time-averaged microscopic mass and momentum balances over the depth of the water column. Early finite element solutions of the shallow water equations were often plagued by spurious oscillations.' Various methods were introduced to eliminate the oscillations, but all included some type of artificial damping. Lynch and Gray2 and Gray3 present the wave continuity equation as a means to avoid spurious oscillations without resorting to numerical or artificial damping of the solution. Since the inception of the wave continuity formulation in 1979, the original algorithm has been modified in a number of substantial ways: a numerical