Finite-element simulation of the shallow-water equations model on a limited-area domain

A Galerkin finite-element model of the shallow-water equations on a limited domain is presented. The evolutionary equations of continuity and momentum are coupled at each time step using an extrapolated Crank-Nicolson method to quasilinearize the nonlinear advective terms. The coupling allows time steps to be used larger than those possible with an uncoupled model. A linear one-dimensional stability analysis of the finite-element model is presented. Three mass matrix schemes, the consistent mass (CM), the lumped mass (LM) and a generalized mixed mass (GMM) scheme were used for numerical tests and for comparing the accuracy of the finiteelement model both against a refined mesh solution and a highly accurate nonlinear ADI finite difference model when the same test problem was solved. The accuracy of the GMM mass matrix scheme was found to be greater than that of both the LM and CM schemes. Integral invariants of the shallow-water equations conserved almost perfectly for long-term runs. Extensive comparisons with results of other investigators using two different initial conditions for the shallow-water equations showed the results with the GMM mass scheme to have fourthorder accuracy in both amplitude and phase. A compact storage scheme is provided in which advantage has been taken of the sparsity of the global matrices.