×

Raviart-Thomas and Brezzi-Douglas-Marini finite element approximations of the shallow-water equations. (English) Zbl 1140.76022

Summary: We present an analysis of discrete shallow-water equations using Raviart-Thomas and Brezzi-Douglas-Marini finite elements. For inertia-gravity waves, the discrete formulations are obtained and the dispersion relations are computed in order to quantify the dispersive nature of the schemes on two meshes made up of equilateral and biased triangles. A linear algebra approach is also used to ascertain the possible presence of spurious modes arising from the discretization. The geostrophic balance is examined, and the smallest representable vortices are characterized on both structured and unstructured meshes. Numerical solutions of two test problems to simulate gravity and Rossby modes are in good agreement with analytical results.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B65 Rossby waves (MSC2010)

Software:

Gmsh; SPARSKIT
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] CareyGF (ed.). Finite Element Modeling of Environmental Problems. Wiley: New York, 1995.
[2] Danilov, A finite-element ocean model: principles and evaluation, Ocean Modelling 6 pp 125– (2004)
[3] Hua, A noise-free finite-element scheme for the two-layer shallow water equations, Tellus 36 pp 157– (1984)
[4] Le Provost, Tidal Hydrodynamics pp 41– (1991)
[5] Le Roux, Finite element for shallow-water equation ocean models, Monthly Weather Review 126 pp 1931– (1998)
[6] Hanert, A comparison of three finite elements to solve the linear shallow water equations, Ocean Modelling 5 pp 17– (2002)
[7] Lynch, A wave-equation model for finite-element tidal computations, Computers and Fluids 7 pp 207– (1979) · Zbl 0421.76013
[8] Côté, An accurate and efficient finite-element global model of the shallow-water equations, Monthly Weather Review 118 pp 2707– (1990)
[9] Le Roux, A semi-implicit semi-Lagrangian finite-element shallow-water ocean model, Monthly Weather Review 128 pp 1384– (2000)
[10] Walters, Accuracy of an estuarine hydrodynamic model using smooth elements, Water Resources Research 16 pp 187– (1980)
[11] Agoshkov, Finite Elements in Fluids pp 1001– (1993)
[12] Iskandarani, A staggered spectral finite-element model for the shallow-water equations, International Journal for Numerical Methods in Fluids 20 pp 393– (1995) · Zbl 0870.76057
[13] Williams, Improved finite-element forms for the shallow-water wave equations, International Journal for Numerical Methods in Fluids 1 pp 81– (1981) · Zbl 0459.76010
[14] Kinnmark, A two-dimensional analysis of the wave equation model for finite element tidal computations, International Journal for Numerical Methods in Engineering 20 pp 369– (1984) · Zbl 0562.76020
[15] Hughes, A new finite element formulation for computational fluid dynamics: V. Circumventing the Babu\~ska–Brezzi condition: a stable Petrov–Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Computer Methods in Applied Mechanics and Engineering 59 pp 85– (1986) · Zbl 0622.76077
[16] Le Roux, Stability/dispersion analysis of the discontinuous Galerkin linearized shallow-water system, International Journal for Numerical Methods in Fluids 48 pp 325– (2005) · Zbl 1065.76142
[17] Le Roux, Dispersion relation analysis of the P1NC-P1 finite element pair in shallow-water ocean models, SIAM Journal on Scientific Computing 27 pp 394– (2005)
[18] Le Roux, Analysis of numerically-induced oscillations in 2D finite-element shallow-water models, part I: inertia–gravity waves, SIAM Journal on Scientific Computing 29 pp 331– (2007) · Zbl 1387.76055
[19] Le Roux, Analysis of numerically-induced oscillations in 2D finite-element shallow-water models, part II: free planetary waves, SIAM Journal on Scientific Computing · Zbl 1191.35024
[20] Walters, Analysis of spurious oscillation modes for the shallow water and Navier–Stokes equations, Computers and Fluids 11 pp 51– (1983) · Zbl 0521.76017
[21] Le Roux, On some spurious mode issues in shallow-water models using a linear algebra approach, Ocean Modelling 10 pp 83– (2005)
[22] Rostand, Kernel analysis of the discretized finite difference and finite element shallow-water models, SIAM Journal on Scientific Computing · Zbl 1191.35025
[23] Brezzi, Vistas in Applied Mathematics, Numerical Analysis, Atmospheric Sciences, Immunology (1986)
[24] LeBlond, Waves in the Ocean (1978)
[25] Foreman, A two-dimensional dispersion analysis of selected methods for solving the linearized shallow-water equations, Journal of Computational Physics 56 pp 287– (1984) · Zbl 0557.76004
[26] Adcroft, A new treatment of the coriolis term in C-grid models at both high and low resolutions, Monthly Weather Review 127 pp 1928– (1999)
[27] Mesinger, Numerical Methods Used in Atmospheric Models (1976)
[28] Saad Y. SPARSKIT: a basic tool kit for sparse matrix computations, 2005. http://www.cs.umn.edu/saad/software/SPARSKIT/sparskit.html.
[29] Geuzaine C, Remacle JF. GMSH: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities, 2007. http://www.geuz.org/gmsh/. · Zbl 1176.74181
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.