Finite Elements for Shallow-Water Equation Ocean Models

The finite-element spatial discretization of the linear shallow-water equations on unstructured triangular meshes is examined in the context of a semi-implicit temporal discretization. Triangular finite elements are attractive for ocean modeling because of their flexibility for representing irregular boundaries and for local mesh refinement. The semi-implicit scheme is beneficial because it slows the propagation of the high-frequency small-amplitude surface gravity waves, thereby circumventing a severe time step restriction. High-order computationally expensive finite elements are, however, of little benefit for the discretization of the terms responsible for rapidly propagating gravity waves in a semi-implicit formulation. Low-order velocity/surface-elevation finite-element combinations are therefore examined here. Ideally, the finite-element basis-function pair should adequately represent approximate geostrophic balance, avoid generating spurious computational modes, and give a consistent discretization of the governing equations. Existing finite-element combinations fail to simultaneously satisfy all of these requirements and consequently suffer to a greater or lesser extent from noise problems. An unconventional and largely unknown finite-element pair, based on a modified combination of linear and constant basis functions, is shown to be a good compromise and to give good results for gravity-wave propagation.