Finite Elements for Shallow-Water Equation Ocean Models

Abstract

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.

A finite-element ocean model: principles and evaluation

“…Not every choice of velocity-pressure pairs of FE functional spaces is suitable for reproducing this balance. The problem was thoroughly examined by Le Roux et al (1998) who concluded that the space of piecewise linear basis functions X for velocity and pressure is one of the most convenient from this point of view. Second, the piecewise linear basis functions require only nodal values of respective fields and thus provide compact storage for the fields and, more importantly, the matrices associated with equations.…”

“…Namely, using the former one redefines the differential model operator while the latter allows to preserve the operator by changing the space where the solution is searched for. With special care in choosing the functional spaces for unknown model variables it is possible to resolve the first problem (Le Roux et al, 1998). The second problem can be mainly addressed to OGCMs employing curvilinear coordinates (POM, HOPE, etc.)…”

A discontinuous finite element baroclinic marine model on unstructured prismatic meshes

“…However, the search for an equivalent optimal finite element pair for the shallow water equations is still an open issue. Le Roux et al (1998) gave the first review of available choices. More recent mathematical and numerical analysis of finite element pairs for gravity and Rossby waves are provided in Le Roux et al (2007, , and .…”

A diagnostic stabilized finite-element ocean circulation model

“…This choice was made because the use of higher-order FEs may generate spurious numerical modes which affect accuracy and stability of the numerical scheme. At the same time P 1 -P 1 approximation is computationally efficient and was recently shown to be among the most accurate pairs in representing the geostrophic balance (Le Roux et al, 1998). To satisfy the LBB condition we employ the RFBF stabilization technique (Brezzi et al, 1996a,b) which may be considered as a universal recipe for avoiding spurious pressure modes and numerical instabilities associated with inadequate spatial resolution.…”

“…Le Roux et al (1998) found a low-order element pair satisfying the Babus ska-Brezzi (LBB) compatibility condition (Babus ska, 1971), and having perfect properties in representing the geostrophic balance. Later this FE formulation was combined with a semi-Lagrangian scheme to produce an efficient shallow water ocean model (Le Roux et al, 2000).…”