Editorial Type:
Article
Article Type:
Research Article

Finite Elements for Shallow-Water Equation Ocean Models

Daniel Y. Le Roux Department of Atmospheric and Oceanic Sciences, Center for Climate and Global Change Research, McGill University, and Centre de Recherche en Calcul Appliqué, Montreal, Quebec, Canada

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Andrew Staniforth Meteorological Research Branch, Environment Canada, Dorval, Quebec, Canada

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Charles A. Lin Department of Atmospheric and Oceanic Sciences, Center for Climate and Global Change Research, McGill University, and Centre de Recherche en Calcul Appliqué, Montreal, Quebec, Canada

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Print Publication:
01 Jul 1998
DOI:
https://doi.org/10.1175/1520-0493(1998)126<1931:FEFSWE>2.0.CO;2
Page(s):
1931–1951
Restricted access

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.

Corresponding author address: Dr. Daniel Y. Le Roux, Dept. of Atmospheric and Oceanic Sciences, Center for Climate and Global Change Research, McGill University, Montreal, PQ H3A 2K6, Canada.

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.

Corresponding author address: Dr. Daniel Y. Le Roux, Dept. of Atmospheric and Oceanic Sciences, Center for Climate and Global Change Research, McGill University, Montreal, PQ H3A 2K6, Canada.

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