• Arakawa, A., 1966: Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow. Part I. J. Comput. Phys.,1, 119–143.

  • ——, and V. R. Lamb, 1977: Computational design of the basic dynamical processes of the UCLA General Circulation Model. Methods Comput. Phys.,17, 173–265.

  • Babuska, I., 1971: Error bounds for finite-element method. Numer. Math.,16, 322–333.

  • Batteen, M. L., and Y. J. Han, 1981: On the computational noise of finite-difference schemes used in ocean models. Tellus,33, 387–396.

  • Bercovier, M., and O. Pironneau, 1979: Error estimates for the finite element method solution of the Stokes problem in the primitive variables. Numer. Math.,33, 211–224.

  • Brebbia, C. A., and P. W. Partridge, 1976a: Finite element models for circulation studies. Mathematical Models for Environmental Problems, C. A. Brebbia, Ed., Wiley, 141.

  • ——, and ——, 1976b: Finite element simulation of water circulation in the North Sea. Appl. Math. Model.,1, 101–107.

  • Brezzi, F., 1974: On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers. R.A.I.R.O.,8, 129–151.

  • ——, and M. Fortin, 1991: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, Vol. 15, Springer-Verlag, 350 pp.

  • Crouzeix, M., and P. A. Raviart, 1973: Conforming and non-conforming finite-element methods for solving the stationary Stokes equations. R.A.I.R.O. Anal. Numer.,7, 33–76.

  • Cullen, M. J. P., 1976: On the use of artificial smoothing in Galerkin and finite-difference solutions of the primitive equations. Quart. J. Roy. Meteor. Soc.,102, 77–93.

  • ——, and C. D. Hall, 1979: Forecasting and general circulation results from finite-element models. Quart. J. Roy. Meteor. Soc.,105, 571–592.

  • Dutto, L. C., 1993: On the iterative methods for solving linear systems of equations. Eur. J. Finite Elem.,2, 423–448.

  • Engelman, M. S., R. L. Sani, and P. M. Gresho, 1982: The implementation of normal and/or tangential boundary conditions in finite-element codes for incompressible fluid flow. Int. J. Numer. Methods Fluids,2, 225–238.

  • Foreman, M. G. G., 1984: A two-dimensional dispersion analysis of selected methods for solving the linearized shallow-water equations. J. Comput. Phys.,56, 287–323.

  • Fortin, M., and R. Pierre, 1992: Stability analysis of discrete generalized Stokes problems. Numer. Methods Partial Differential Equations,8, 303–323.

  • ——, H. Manouzi, and A. Soulaimani, 1993: On finite-element approximation and stabilization methods for compressible viscous flows. Int. J. Numer. Methods Fluids,17, 477–499.

  • Gates, W. L., 1968: A numerical study of transient Rossby waves in a wind-driven homogeneous ocean. J. Atmos. Sci.,25, 3–22.

  • Gill, A. E., 1982: Atmosphere-Ocean Dynamics. International Geophysical Series, Vol. 31, Academic Press, 662 pp.

  • Girault, V., and P. A. Raviart, 1986: Finite Element Methods for Navier–Stokes Equations. Springer Series in Computational Mathematics, Vol. 5, Springer-Verlag, 374 pp.

  • Gray, W. G., 1980: Do finite-element models simulate surface flow? Finite Elements in Water Resources: Proceedings of the Third Int. Conf. on Finite Elements in Water Resources. University, MS, University of Mississippi, 1.122–1.136.

  • ——, 1984: On normal flow boundary conditions in finite element codes for two-dimensional shallow-water flow. Int. J. Numer. Methods Fluids,4, 99–104.

  • ——, and D. R. Lynch, 1979: On the control of noise in finite-element tidal computations: A semi-implicit approach. Comput. Fluids,7, 47–67.

  • Gunzburger, M. D., 1989: Finite Element Methods for Viscous Incompressible Flows: A Guide to the Theory, Practice and Algorithms. Academic Press, 269 pp.

  • Hood, P., and C. Taylor, 1974: Navier–Stokes equations using mixed interpolation. Finite Elements in Flow Problems, J. T. Oden et al., Eds., University of Alabama in Huntsville (UAH) Press, 121–132.

  • Hua, B. L., and F. Thomasset, 1984: A noise-free finite-element scheme for the two-layer shallow-water equations. Tellus,36A, 157–165.

  • Kelley, R. G., and R. T. Williams, 1976: A finite-element prediction model with variable element sizes. Naval Postgraduate School Rep. NPS-63Wu76101, 109 pp. [Available from Naval Postgraduate School, Monterey, CA 93943.].

  • Kellogg, R. B., and B. Liu, 1996: A finite-element method for the compressible Stokes equations. SIAM J. Numer. Anal.,33, 780–788.

  • Ladyzhenskaya, O. A., 1969: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, 224 pp.

  • LeBlond, P. H., and L. A. Mysak, 1978: Waves in the Ocean. Elsevier, 602 pp.

  • Le Roux, D. Y., C. A. Lin, and A. Staniforth, 1997: An accurate interpolating scheme for semi-Lagrangian advection on an unstructured mesh for ocean modeling. Tellus,49A, 119–138.

  • Lynch, D. R., and W. G. Gray, 1979: A wave-equation model for finite-element tidal computations. Comput. Fluids,7, 207–228.

  • Ozell, B., R. Camarero, A. Garon, and F. Guibault, 1995: Analysis and visualization tools in CFD, part I: A configurable data extraction environment. Finite Elem. in Anal. Des.,19, 295–307.

  • Pierre, R., 1988: Simple C0-approximations for the computation of incompressible flows. Comput. Methods Appl. Mech. Eng.,68, 205–227.

  • Robert, A., 1981: A stable numerical integration scheme for the primitive meteorological equations. Atmos.Ocean,19, 35–46.

  • ——, 1982: A semi-Lagrangian and semi-implicit numerical integration scheme for the primitive meteorological equations. J. Meteor. Soc. Japan,60, 319–325.

  • ——, J. Henderson, and C. Turnbull, 1972: An implicit time integration scheme for baroclinic models of the atmosphere. Mon. Wea. Rev.,100, 329–335.

  • Sani, R. L., P. M. Gresho, R. L. Lee, and D. Griffiths, 1981a: The cause and cure (!) of the spurious pressures generated by certain FEM solutions of the incompressible Navier–Stokes equations. Part 1. Int. J. Numer. Methods Fluids,1, 17–43.

  • ——, ——, ——, and ——, 1981b: The cause and cure (!) of the spurious pressures generated by certain FEM solutions of the incompressible Navier–Stokes equations. Part 2. Int. J. Numer. Methods Fluids,1, 171–204.

  • Schoenstadt, A., 1980: A transfer function analysis of numerical schemes used to simulate geostrophic adjustment. Mon. Wea. Rev.,108, 1248–1259.

  • Staniforth, A., and H. L. Mitchell, 1977: A semi-implicit finite-element barotropic model. Mon. Wea. Rev.,105, 154–169.

  • ——, and J. Côté, 1991: Semi-Lagrangian integration schemes for atmospheric models: A review. Mon. Wea. Rev.,119, 2206–2223.

  • Temam, R., 1977: Navier-Stokes Equations: Theory and Numerical Analysis. North Holland Publishing Co., 500 pp.

  • Temperton, C., and A. Staniforth, 1987: An efficient two-time level semi-Lagrangian semi-implicit integration scheme. Quart. J. Roy. Meteor. Soc.,113, 1025–1039.

  • Thomasset, F., 1981: Implementation of Finite Element Methods for NavierStokes Equations. Springer-Verlag, 159 pp.

  • Walters, R. A., 1983: Numerically induced oscillations in finite-element approximations to the shallow-water equations. Int. J. Numer. Methods Fluids,3, 591–604.

  • ——, and R. T. Cheng, 1980: Accuracy of an estuarine hydrodynamic model using smooth elements. Water Resour. Res.,16, 187–195.

  • ——, and G. F. Carey, 1983: Analysis of spurious oscillation modes for the shallow-water and Navier–Stokes equations. Comput. Fluids,11, 51–68.

  • Williams, R. T., 1981: On the formulation of the finite-element prediction models. Mon. Wea. Rev.,109, 463–466.

  • ——, and O. C. Zienkiewicz, 1981: Improved finite-element forms for the shallow-water wave equations. Int. J. Numer. Methods Fluids,1, 81–97.

  • Winninghoff, F., 1968: On the adjustment toward a geostrophic balance in a simple primitive equation model with application to the problem on initialization and objective analysis. Ph.D. thesis, University of California, Los Angeles, 161 pp. [Available from Dept. of Meteorology, UCLA, Los Angeles, CA 90095-1361.].

  • Zeng, Q. C., 1986: Some numerical ocean-atmospheric coupling models. Integrated Global Ocean Monitoring: Proceedings of the First International Symposium, Tallin, U.S.S.R.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 303 103 5
PDF Downloads 262 71 4

Finite Elements for Shallow-Water Equation Ocean Models

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

Search for other papers by Daniel Y. Le Roux in
Current site
Google Scholar
PubMed
Close
,
Andrew StaniforthMeteorological Research Branch, Environment Canada, Dorval, Quebec, Canada

Search for other papers by Andrew Staniforth in
Current site
Google Scholar
PubMed
Close
, and
Charles A. LinDepartment of Atmospheric and Oceanic Sciences, Center for Climate and Global Change Research, McGill University, and Centre de Recherche en Calcul Appliqué, Montreal, Quebec, Canada

Search for other papers by Charles A. Lin in
Current site
Google Scholar
PubMed
Close
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.

Email: dleroux@cerca.umontreal.ca

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.

Email: dleroux@cerca.umontreal.ca

Save