×

zbMATH — the first resource for mathematics

A primal-dual mimetic finite element scheme for the rotating shallow water equations on polygonal spherical meshes. (English) Zbl 1349.76273
Summary: A new numerical method is presented for solving the shallow water equations on a rotating sphere using quasi-uniform polygonal meshes. The method uses special families of finite element function spaces to mimic key mathematical properties of the continuous equations and thereby capture several desirable physical properties related to balance and conservation. The method relies on two novel features. The first is the use of compound finite elements to provide suitable finite element spaces on general polygonal meshes. The second is the use of dual finite element spaces on the dual of the original mesh, along with suitably defined discrete Hodge star operators to map between the primal and dual meshes, enabling the use of a finite volume scheme on the dual mesh to compute potential vorticity fluxes. The resulting method has the same mimetic properties as a finite volume method presented previously, but is more accurate on a number of standard test cases.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76U05 General theory of rotating fluids
Software:
chammp
PDF BibTeX Cite
Full Text: DOI arXiv
References:
[1] Arnold, D. N.; Boffi, D.; Bonizzoni, F., Finite element differential forms on curvilinear cubic meshes and their approximation properties, Numer. Math., 129, 1-20, (2015) · Zbl 1308.65193
[2] Beirão da Veiga, L.; Brezzi, F.; Cangiani, A.; Manzini, G.; Marini, L. D.; Russo, A., Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23, 199-214, (2013) · Zbl 1416.65433
[3] Beirão da Veiga, L.; Brezzi, F.; Marini, L. D.; Russo, A., Hitchhiker’s guide to the virtual element method, Math. Models Methods Appl. Sci., 24, 1541-1573, (2014) · Zbl 1291.65336
[4] Bochev, P. B.; Ridzal, D., Rehabilitation of the lowest-order Raviart-Thomas element on quadrilateral grids, SIAM J. Numer. Anal., 47, 487-507, (2008) · Zbl 1190.65169
[5] Bossavit, A., Computational electromagnetism: variational formulation, complementarity, edge elements, Academic Press Electromagnetism Series, vol. 2, (1998), Academic Press San Diego · Zbl 0945.78001
[6] Buffa, A.; Christiansen, S. H., A dual finite element complex on the barycentric refinement, Math. Comput., 76, 1743-1769, (2007) · Zbl 1130.65108
[7] Christiansen, S. H., A construction of spaces of compatible differential forms on cellular complexes, Math. Models Methods Appl. Sci., 18, 739-757, (2008) · Zbl 1153.65005
[8] Christiansen, S. H., Minimal mixed finite elements on polyhedra, C. R. Acad. Sci. Paris, 348, 217-221, (2010) · Zbl 1186.65148
[9] Cotter, C. J.; Shipton, J., Mixed finite elements for numerical weather prediction, J. Comput. Phys., 231, 7076-7091, (2012) · Zbl 1284.86005
[10] Cotter, C. J.; Thuburn, J., A finite element exterior calculus framework for the rotating shallow-water equations, J. Comput. Phys., 257, 1506-1526, (2014) · Zbl 1351.76054
[11] Danilov, S., On the utility of triangular C-grid type discretization for numerical modeling of large-scale Ocean flows, Ocean Dyn., 60, 1361-1369, (2010)
[12] Galewsky, J.; Scott, R. K.; Polvani, L. M., An initial value problem for testing numerical models of the global shallow water equations, Tellus A, 56, 429-440, (2004)
[13] Heikes, R.; Randall, D., Numerical integration of the shallow-water equations on a twisted icosahedral grid. part II: a detailed description of the grid and analysis of numerical accuracy, Mon. Weather Rev., 123, 1881-1997, (1995)
[14] Hiptmair, R., Discrete Hodge operators, Numer. Math., 90, 265-289, (2001) · Zbl 0993.65130
[15] Le Roux, D. Y.; 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, 331-360, (2007) · Zbl 1387.76055
[16] Lock, S.-J.; Bitzer, H.-W.; Coals, A.; Gadian, A.; Mobbs, S., Demonstration of a cut-cell representation of 3D orography for studies of atmospheric flows over very steep hills, Mon. Weather Rev., 140, 411-424, (2007)
[17] McRae, A. T.T.; Cotter, C. J., Energy- and enstrophy-conserving schemes for the shallow-water equations, based on mimetic finite elements, Q. J. R. Meteorol. Soc., 140, 2223-2234, (2014)
[18] Melvin, T.; Staniforth, A.; Cotter, C., A two-dimensional mixed finite-element pair on rectangles, Q. J. R. Meteorol. Soc., 140, 930-942, (2014)
[19] Melvin, T.; Staniforth, A.; Thuburn, J., Dispersion analysis of the spectral element method, Q. J. R. Meteorol. Soc., 138, 1934-1947, (2012)
[20] Melvin, T.; Thuburn, J., Wave dispersion properties of compound finite elements, J. Comput. Phys., (2015), submitted for publication
[21] Nédélec, J.-C., Mixed finite elements in \(\mathbb{R}^3\), Numer. Math., 35, 315-341, (1980) · Zbl 0419.65069
[22] Palha, A.; Rebelo, P. P.; Hiemstra, R.; Kreeft, J.; Gerritsma, M., Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms, J. Comput. Phys., 257, 1394-1422, (2014) · Zbl 1352.65629
[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, 3065-3090, (2010) · Zbl 1307.76054
[24] Staniforth, A.; Thuburn, J., Horizontal grids for global weather and climate prediction models: a review, Q. J. R. Meteorol. Soc., 138, 1-26, (2012)
[25] Taylor, M. A.; Fournier, A., A compatible and conservative spectral element method on unstructured grids, J. Comput. Phys., 229, 5879-5895, (2010) · Zbl 1425.76177
[26] Thuburn, J.; Cotter, C. J., A framework for mimetic discretization of the rotating shallow water equations on arbitrary polygonal grids, SIAM J. Sci. Comput., 34, 203-225, (2012) · Zbl 1246.65155
[27] Thuburn, J.; Cotter, C. J.; Dubos, T., A mimetic, semi-implicit, forward-in-time, finite volume shallow water model: comparison of hexagonal-icosahedral and cubed sphere grids, Geosci. Model Dev., 7, 909-929, (2014)
[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] Ullrich, P. A., Understanding the treatment of waves in atmospheric models. part 1: the shortest resolved waves of the 1D linearized shallow-water equations, Q. J. R. Meteorol. Soc., 140, 1426-1440, (2014)
[30] Weller, H., Controlling the computational modes of the arbitrarily structured C-grid, Mon. Weather Rev., 140, 3220-3234, (2012)
[31] Weller, H., Non-orthogonal version of the arbitrary polygonal C-grid and a new diamond grid, Geosci. Model Dev., 7, 779-797, (2014)
[32] Williamson, D. L.; Drake, J. B.; Hack, J. J.; Jakob, R.; Swartztrauber, P. N., A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comput. Phys., 102, 211-224, (1992) · Zbl 0756.76060
[33] Zerroukat, M.; Wood, N.; Staniforth, A.; White, A. A.; Thuburn, J., An inherently mass-conserving semi-implicit semi-Lagrangian discretization of the shallow water equations on the sphere, Q. J. R. Meteorol. Soc., 135, 1104-1116, (2009)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.