×

Numerical wave propagation for the triangular \(P1_{DG}-P2\) finite element pair. (English) Zbl 1316.76019

Summary: The \(f\)-plane and \(\beta \)-plane wave propagation properties are examined for discretisations of the linearised rotating shallow-water equations using the \(P1_{DG}-P2\) finite element pair on arbitrary triangulations in planar geometry. A discrete Helmholtz decomposition of the functions in the velocity space based on potentials taken from the pressure space is used to provide a complete description of the numerical wave propagation for the discretised equations. In the \(f\)-plane (planar geometry, Coriolis force independent of space) case, this decomposition is used to obtain decoupled equations for the geostrophic modes, the inertia-gravity modes, and the inertial oscillations. As has been noticed previously, the geostrophic modes are steady. The Helmholtz decomposition is used to show that the resulting inertia-gravity wave equation is third-order accurate in space. In general the \(P1_{DG}-P2\) finite element pair is second-order accurate, so this leads to very accurate wave propagation. It is further shown that the only spurious modes supported by this discretisation are spurious inertial oscillations which have frequency \(f\), and which do not propagate. A restriction of the \(P1_{DG}\) velocity space is proposed in which these modes are not present, leading to a finite element discretisation which is completely free of spurious modes. The Helmholtz decomposition also allows a simple derivation of the quasi-geostrophic limit of the discretised \(P1_{DG}-P2\) equations in the \(\beta \)-plane (planar geometry, Coriolis force linear in space) case resulting in a Rossby wave equation which is also third-order accurate. This means that the dispersion relation for the wave propagation is very accurate; an illustration of this is provided by a numerical dispersion analysis in the case of a triangulation consisting of equilateral triangles.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76M10 Finite element methods applied to problems in fluid mechanics
86A10 Meteorology and atmospheric physics

Software:

SymPy; NICAM
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Arakawa, A.; Lamb, V., Computational design of the basic dynamical processes of the UCLA general circulation model, (Chang, J., Methods in Computational Physics, vol. 17 (1977), Academic Press), 173-265
[2] Brenner, S.; Scott, R., The Mathematical Theory of Finite Element Methods (1994), Springer-Verlag
[3] Comblen, R.; Lambrechts, J.; Remacle, J.-F.; Legat, V., Practical evaluation of five partly discontinuous finite element pairs for the non-conservative shallow water equations, Int. J. Numer. Meth. Fluid., 63, 6, 701-724 (2010) · Zbl 1423.76220
[4] Cotter, C. J.; Ham, D. A.; Pain, C. C., A mixed discontinuous/continuous finite element pair for shallow-water ocean modelling, Ocean Model., 26, 86-90 (2009)
[5] Cotter, C. J.; Ham, D. A.; Pain, C. C.; Reich, S., LBB stability of a mixed finite element pair for fluid flow simulations, J. Comput. Phys., 228, 3, 336-348 (2009) · Zbl 1409.76058
[6] Fox-Rabinovitz, M., Computational dispersion properties of 3D staggered grids for a nonhydrostatic anelastic system, Mon. Weather Rev., 124, 498-510 (1996)
[7] Gresho, P. M.; Sani, R. L., Incompressible Flow and the Finite Element Method, Isothermal Laminar Flow, vol. 2 (2000), Wiley · Zbl 0988.76005
[8] Kossevich, A. M., Geometry of Crystal Lattices (2005), Wiley
[9] Le Roux, D.; Hanert, E.; Rostand, V.; Pouliot, B., Impact of mass lumping on gravity and Rossby waves in 2D finite-element shallow-water models, Int. J. Numer. Meth. Fluid, 59, 7, 767-790 (2008) · Zbl 1156.76035
[10] Le Roux, D.; Staniforth, A.; Lin, C. A., Finite elements for shallow-water equation ocean models, Mon. Weather Rev., 126, 7, 1931-1951 (1998)
[11] Majewski, D.; Liermann, D.; Prohl, P.; Ritter, B.; Buchhold, M.; Hanisch, T.; Paul, G.; Wergen, W.; Baumgardner, J., The operational global icosahedral-hexagonal gridpoint model GME: description and high-resolution tests, Mon. Weather Rev., 130, 319-338 (2002)
[12] Randall, D., Geostrophic adjustment and the finite-difference shallow-water equations, Mon. Weather Rev., 122, 1371-1377 (1994)
[13] Raviart, Thomas, A mixed finite element method for 2nd order elliptic problems, (Mathematical Aspects of the Finite Element Method. Lecture Notes in Mathematics (1977), Springer: Springer Berlin), 292-315
[14] Ringler, T. D.; Heikes, R.; Randall, D., Modeling the atmospheric general circulation using a spherical geodesic grid: a new class of dynamical cores, Mon. Weather Rev., 128, 2471-2490 (2000)
[15] Roux, D. Y.L.; Pouliot, B., Analysis of numerically induced oscillations in two-dimensional finite-element shallow-water models part II: free planetary waves, SIAM J. Sci. Comput., 30, 4, 1971-1991 (2008) · Zbl 1191.35024
[16] Roux, D. Y.L.; Rostand, V.; Pouliot, B., Analysis of numerically induced oscillations in 2d finite-element shallow-water models part I: inertia-gravity waves, SIAM J. Sci. Comput., 29, 1, 331-360 (2007) · Zbl 1387.76055
[17] Satoh, M.; Matsuno, T.; Tomita, H.; Miura, H.; Nasuno, T.; Iga, S., Nonhydrostatic icosahedral atmospheric model (NICAM) for global cloud resolving simulations, J. Comput. Phys., 227, 7, 3486-3514 (2008) · Zbl 1132.86311
[18] SymPy Development Team, SymPy: Python library for symbolic mathematics, 2009. <http://www.sympy.org>; SymPy Development Team, SymPy: Python library for symbolic mathematics, 2009. <http://www.sympy.org>
[19] Thuburn, J., Numerical wave propagation on the hexagonal C-grid, J. Comput. Phys., 227, 11, 5836-5858 (2008) · Zbl 1220.76018
[20] Thuburn, J.; Ringler, T. D.; Skamarock, W. C.; Klemp, J. B., Numerical representation of geostrophic modes on arbitrarily structured C-grids, J. Comput. Phys., 228, 8321-8335 (2009) · Zbl 1173.86304
[21] Umgiesser, G.; Canu, D. M.; Cucco, A.; Solidoro, C., A finite element model for the Venice Lagoon. Development, set up, calibration and validation, J. Marine Syst., 51, 1-4, 123-145 (2004)
[22] Walters, R.; Casulli, V., A robust, finite element model for hydrostatic surface water flows, Commun. Numer. Methods Eng., 14, 931-940 (1998) · Zbl 0915.76056
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.