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Mixed finite elements for numerical weather prediction. (English) Zbl 1284.86005
Summary: We show how mixed finite element methods that satisfy the conditions of finite element exterior calculus can be used for the horizontal discretisation of dynamical cores for numerical weather prediction on pseudo-uniform grids. This family of mixed finite element methods can be thought of in the numerical weather prediction context as a generalisation of the popular polygonal C-grid finite difference methods. There are a few major advantages: the mixed finite element methods do not require an orthogonal grid, and they allow a degree of flexibility that can be exploited to ensure an appropriate ratio between the velocity and pressure degrees of freedom so as to avoid spurious mode branches in the numerical dispersion relation. These methods preserve several properties of the C-grid method when applied to linear barotropic wave propagation, namely: (a) energy conservation, (b) mass conservation, (c) no spurious pressure modes, and (d) steady geostrophic modes on the \(f\)-plane. We explain how these properties are preserved, and describe two examples that can be used on pseudo-uniform grids: the recently-developed modified RTk-Q(k-1) element pairs on quadrilaterals and the BDFM1-P1\(_{DG}\) element pair on triangles. All of these mixed finite element methods have an exact 2:1 ratio of velocity degrees of freedom to pressure degrees of freedom. Finally we illustrate the properties with some numerical examples.

86A10 Meteorology and atmospheric physics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI arXiv
[1] Arakawa, A.; Lamb, V., Computational design of the basic dynamical processes of the UCLA general circulation model, (), 173-265
[2] D. Arnold, Differential complexes and numerical stability, in: L. Tatsien (Ed.), Proceedings of the ICM, Beijing 2002, vol. 1, 2002, pp. 137-157. · Zbl 1023.65113
[3] Arnold, D.; Brezzi, F.; Fortin, M., A stable finite element for the Stokes equations, Calcolo, 21, 4, 337-344, (1984) · Zbl 0593.76039
[4] Arnold, D.; Falk, R.; Winther, R., Finite element exterior calculus homological techniques and applications, Acta numer., 15, 1-155, (2006) · Zbl 1185.65204
[5] Boffi, D.; Brezzi, F.; Gastaldi, L., On the convergence of eigenvalues for mixed formulations, Ann. scuola norm. sup. Pisa cl. sci., XXV, 131-154, (1997) · Zbl 1003.65052
[6] Boffi, D.; Gastaldi, L., Some remarks on quadrilateral mixed finite elements, Comput. struct., 87, 11-12, 751-757, (2009)
[7] Brezzi, F.; Douglas, J.; Fortin, M.; Marini, L., Efficient rectangular mixed finite elements in two and three space variables, Modélisation mathématique et analyse numérique, 21, 4, 581-604, (1987) · Zbl 0689.65065
[8] Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods, (1991), Springer-Verlag · Zbl 0788.73002
[9] Brezzi, F.; Jr, J.D.; Marini, L., Two families of mixed finite elements for second order elliptic problems, Numer. math., 47, 2, 217-235, (1985) · Zbl 0599.65072
[10] Cockburn, B.; Shu, C.-W., Runge – kutta discontinuous Galerkin methods for convection-dominated problems, J. sci. comput., 16, 3, 173-261, (2001) · Zbl 1065.76135
[11] 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. methods fluids, 63, 6, 701-724, (2010) · Zbl 1423.76220
[12] Cotter, C.; Ham, D., Numerical wave propagation for the triangular p1dg-p2 finite element pair, J. comput. phys., 230, 8, 2806-2820, (2011) · Zbl 1316.76019
[13] 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)
[14] Danilov, S., On utility of triangular C-grid type discretization for numerical modeling of large-scale Ocean flows, Ocean dyn., 60, 6, 1361-1369, (2010)
[15] Davies, T.; Cullen, M.J.P.; Malcolm, A.J.; Mawson, M.H.; Staniforth, A.; White, A.A.; Wood, N., A new dynamical core for the met office’s global and regional modelling of the atmosphere, Q. J. roy. metero. soc., 131, 608, 1759-1782, (2005)
[16] Ham, D.A.; Kramer, S.C.; Stelling, G.S.; Pietrzak, J., The symmetry and stability of unstructured mesh c-grid shallow water models under the influence of Coriolis, Ocean model., 16, 1-2, 47-60, (2007)
[17] Le Roux, D.; 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
[18] Le Roux, D.; Séne, A.; Rostand, V.; Hanert, E., On some spurious mode issues in shallow-water models using a linear algebra approach, Ocean model., 83-94, (2005)
[19] Le Roux, D.; Staniforth, A.; Lin, C.A., Finite elements for shallow-water equation Ocean models, Mon. weather rev., 126, 7, 1931-1951, (1998)
[20] 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)
[21] Putman, W.; Lin, S.-J., Finite-volume transport on various cubed sphere grids, J. comput. phys., 227, 55-78, (2007) · Zbl 1126.76038
[22] 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)
[23] Ringler, T.D.; Thuburn, J.; Klemp, J.B.; Skamarock, W.C., A unified approach to energy conservation and potential vorticity dynamics for arbitrarily-structured C-grids, J. comput. phys., 229, 9, 3065-3090, (2010) · Zbl 1307.76054
[24] Rostand, V.; Le Roux, D., Raviart – thomas and brezzi – douglas – marini finite-element approximations of the shallow-water equations, Int. J. numer. methods fluids, 57, 8, 951-976, (2008) · Zbl 1140.76022
[25] 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
[26] A. Staniforth, Personal communication.
[27] Thuburn, J., Numerical wave propagation on the hexagonal C-grid, J. comput. phys., 227, 11, 5836-5858, (2008) · Zbl 1220.76018
[28] 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
[29] 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
[30] White, L.; Legat, V.; Deleersnijder, E., Tracer conservation for three-dimensional finite element free-surface Ocean modeling on moving prismatic meshes, Mon. weather rev., 136, 420-442, (2008)
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