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Energy-enstrophy conserving compatible finite element schemes for the rotating shallow water equations with slip boundary conditions. (English) Zbl 1416.65335

Summary: We describe an energy-enstrophy conserving discretisation for the rotating shallow water equations with slip boundary conditions. This relaxes the assumption of boundary-free domains (periodic solutions or the surface of a sphere, for example) in the energy-enstrophy conserving formulation of A. T. T. McRae and the second author [“Energy- and enstrophy-conserving schemes for the shallow-water equations, based on mimetic finite elements”, Q. J. R. Meteorol. Soc. 140, No. 684, 2223–2234 (2014; doi:10.1002/qj.2291)]. This discretisation requires extra prognostic vorticity variables on the boundary in addition to the prognostic velocity and layer depth variables. The energy-enstrophy conservation properties hold for any appropriate set of compatible finite element spaces defined on arbitrary meshes with arbitrary boundaries. We demonstrate the conservation properties of the scheme with numerical solutions on a rotating hemisphere.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
86A05 Hydrology, hydrography, oceanography
76U05 General theory of rotating fluids

Software:

Firedrake; chammp
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References:

[1] Arakawa, A.; Hsu, Y.-J. G., Energy conserving and potential-enstrophy dissipating schemes for the shallow water equations, Mon. Weather Rev., 118, 10, 1960-1969 (1990)
[2] Arakawa, A.; Lamb, V. R., A potential enstrophy and energy conserving scheme for the shallow water equations, Mon. Weather Rev., 109, 1, 18-36 (1981)
[3] Cohen, D.; Hairer, E., Linear energy-preserving integrators for Poisson systems, BIT Numer. Math., 51, 1, 91-101 (2011) · Zbl 1216.65175
[4] 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
[5] Eldred, C.; Dubos, T.; Kritsikis, E., High-order mimetic finite elements for the hydrostatic primitive equations on a cubed-sphere grid using Hamiltonian methods, (EGU General Assembly Conference Abstracts, vol. 18 (2016)), 17338
[6] Eldred, C.; Randall, D., Total energy and potential enstrophy conserving schemes for the shallow water equations using Hamiltonian methods - part 1: derivation and properties, Geosci. Model Dev., 10, 2, 791 (2017)
[7] Gibson, T.; Mitchell, L., A domain-specific language for the hybridization and static condensation of finite element methods (2018), submitted for publication
[8] Hallberg, R.; Rhines, P., Buoyancy-driven circulation in an ocean basin with isopycnals intersecting the sloping boundary, J. Phys. Oceanogr., 26, 6, 913-940 (1996)
[9] Hirani, A. N., Discrete Exterior Calculus (2003), California Institute of Technology, Ph.D. thesis
[10] Holm, D. D.; Marsden, J. E.; Ratiu, T. S., The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137, 1, 1-81 (1998) · Zbl 0951.37020
[11] Hughes, T. J.R., Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Eng., 127, 1, 387-401 (1995) · Zbl 0866.76044
[12] Ketefian, G.; Jacobson, M., A mass, energy, vorticity, and potential enstrophy conserving lateral fluid-land boundary scheme for the shallow water equations, J. Comput. Phys., 228, 1, 1-32 (2009) · Zbl 1194.76208
[13] Lee, D.; Palha, A.; Gerritsma, M., Discrete conservation properties for shallow water flows using mixed mimetic spectral elements (2017), arXiv preprint · Zbl 1382.65318
[14] McRae, A. T.T., Compatible Finite Element Methods for Atmospheric Dynamical Cores (2015)
[15] 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, 684, 2223-2234 (2014)
[16] Natale, A.; Cotter, C. J., Scale-selective dissipation in energy-conserving finite element schemes for two-dimensional turbulence, Q. J. R. Meteorol. Soc., 143, 705, 1734-1745 (2017)
[17] Palha, A.; Gerritsma, M., A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier-Stokes equations, J. Comput. Phys., 328, 200-220 (2017) · Zbl 1406.76064
[18] Rathgeber, F.; Ham, D. A.; Mitchell, L.; Lange, M.; Luporini, F.; McRae, A. T.; Bercea, G.-T.; Markall, G. R.; Kelly, P. H., Firedrake: automating the finite element method by composing abstractions, ACM Trans. Math. Softw. (TOMS), 43, 3, 24 (2016) · Zbl 1396.65144
[19] 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
[20] Sadourny, R., The dynamics of finite-difference models of the shallow-water equations, J. Atmos. Sci., 32, 4, 680-689 (1975)
[21] Salmon, R., Poisson-bracket approach to the construction of energy- and potential-enstrophy-conserving algorithms for the shallow-water equations, J. Atmos. Sci., 61, 16, 2016-2036 (2004)
[22] Salmon, R., A general method for conserving quantities related to potential vorticity in numerical models, Nonlinearity, 18, 5, R1 (2005) · Zbl 1213.76143
[23] Salmon, R., A general method for conserving energy and potential enstrophy in shallow-water models, J. Atmos. Sci., 64, 2, 515-531 (2007)
[24] Salmon, R., A shallow water model conserving energy and potential enstrophy in the presence of boundaries, J. Mar. Res., 67, 6, 779-814 (2009)
[25] Stewart, A. L.; Dellar, P. J., An energy and potential enstrophy conserving numerical scheme for the multi-layer shallow water equations with complete Coriolis force, J. Comput. Phys., 313, 99-120 (2016) · Zbl 1349.65335
[26] Tezduyar, T., Finite element formulation for the vorticity-stream function form of the incompressible Euler equations on multiply-connected domains, Comput. Methods Appl. Mech. Eng., 73, 331-339 (1989) · Zbl 0687.76021
[27] Tezduyar, T.; Glowinski, R.; Liou, J., Petrov-Galerkin methods on multiply connected domains for the vorticity-stream function formulation of the incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids, 8, 10, 1269-1290 (1988) · Zbl 0667.76046
[28] 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, 3, B203-B225 (2012) · Zbl 1246.65155
[29] Williamson, D. L.; Drake, J. B.; Hack, J. J.; Jakob, R.; Swarztrauber, P. N., A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comput. Phys., 102, 1, 211-224 (1992) · Zbl 0756.76060
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