A series of papers'-5 dealing with the wave equation formulation of the continuity equation has shown that this procedure is superior to the standard primitive equation method for simulating shallow water flow problems by the finite element method. However, the analysis of the numerical algorithms has concentrated on the wave equation in one dimension in isolation from the momentum equation. Although much information can be ascertained about the behaviour of the numerical scheme, a full analysis requires that the two-dimensional equations be considered, with Coriolis terms included, and attention be paid to the formulation of the momentum equation.
In the current study, we examine the linearized wave continuity and momentum equations in two dimensions. Because the wave equation represents a differentiated primitive equation for conservation of mass, criteria are developed such that the wave equation will not admit solutions which the original equations would not admit. Secondly, the time differencing of the wave and momentum equations is examined to reveal those instances when time-step-to-timestep oscillations will be introduced into a numerical solution. A time-differencing of the momentum equation is proposed which minimizes these effects. Next the discretization in space is considered for uniform rectangular and triangular linear finite element grids. Both lumped and unlumped formulations are considered. Criteria for stability and accuracy of a simulation are obtained from a Fourier analysis and from investigation of the propagation factor. The material derived herein provides effective guidance for successful development and application of a finite element tidal simulation.
EQUATIONS AND PROCEDURE
This study will analyse the linearized, vertically averaged equations for long-wave propagation in two-dimensional flow with Coriolis acceleration. The equations have been linearized in the sense that convective terms are excluded, bottom stress is proportional to the velocity, and the average total depth is much larger than the surface elevation. Various numerical schemes will be analysed to determine how they may best be utilized to simulate long waves. The governing equations for this study are Continuity ay aw av at ax ay L(f; w, V)=-+h-+h-=O