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A conservative discretization of the shallow-water equations on triangular grids. (English) Zbl 1416.86003
Summary: A structure-preserving discretization of the shallow-water equations on unstructured spherical grids is introduced. The unstructured grids that we consider have triangular cells with a C-type staggering of variables, where scalar variables are located at centres of grid cells and normal components of velocity are placed at cell boundaries. The staggering necessitates reconstructions and these reconstructions are build into the algorithm such that the resulting discrete equations obey a weighted weak form. This approach, combined with a mimetic discretization of the differential operators of the shallow-water equations, provides a conservative discretization that preserves important aspects of the mathematical structure of the continuous equations, most notably the simultaneous conservation of quadratic invariants such as energy and enstrophy. The structure-preserving nature of our discretization is confirmed through theoretical analysis and through numerical experiments on two different triangular grids, a symmetrized icosahedral grid of nearly uniform resolution and a non-uniform triangular grid whose resolution increases towards the poles.

MSC:
86A05 Hydrology, hydrography, oceanography
86A10 Meteorology and atmospheric physics
86-08 Computational methods for problems pertaining to geophysics
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
Software:
chammp; ICON; MPAS-Ocean
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References:
[1] Arakawa, A.; Lamb, V. R., A energy and potential-enstrophy conserving scheme for the shallow water equations, Mon. Weather Rev., 109, 18-36, (1981)
[2] Arakawa, A.; Hsu, Y.-J., Energy conserving and potential-enstrophy dissipating schemes for the shallow water equations, Mon. Weather Rev., 118, 1960-1969, (1990)
[3] Bonaventura, L.; Ringler, T., Analysis of discrete shallow-water models on geodesic Delaunay grids with c-type staggering, Mon. Weather Rev., 133, 2351-2373, (2005)
[4] Bonelle, J.; Di Petro, D. A.; Ern, A., Low-order reconstruction operators on polyhedral meshes: application to compatible discrete operator schemes, Comput. Aided Geom. Des., 35/36, 27-41, (2015)
[5] Brezzi, F.; Lipnikov, K.; Shashkov, M., Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes, SIAM J. Numer. Anal., 43, 1872-1896, (2005) · Zbl 1108.65102
[6] Brezzi, F.; Buffa, A.; Manzini, G., Mimetic scalar products of discrete differential forms, J. Comput. Phys., 257, 1228-1259, (2014) · Zbl 1352.65417
[7] Campin, J.-M.; Adcroft, A.; Hill, C.; Marshall, J., Conservation of properties in a free-surface model, Ocean Model., 6, 221-224, (2004)
[8] Chen, Q.; Gunzburger, M.; Ringler, T., A scale-invariant formulation of the anticipated potential vorticity method, Mon. Weather Rev., 139, 2614-2629, (2011)
[9] Cotter, C.; Shipton, J., Mixed finite elements for numerical weather prediction, J. Comput. Phys., 231, 7076-7091, (2012) · Zbl 1284.86005
[10] Danilov, S., On utility of triangular C-grid type discretization for numerical modeling of large-scale ocean flows, Ocean Dyn., 60, 1361-1369, (2010)
[11] Dubinkina, S.; Frank, J., Statistical mechanics of Arakawa’s discretization, J. Comput. Phys., 227, 1286-1305, (2007) · Zbl 1130.82005
[12] Düben, P.; Korn, P.; Aizinger, V., A discontinuous/continuous low order finite element shallow water model on the sphere, J. Comput. Phys., 231, 2396-2413, (2012) · Zbl 1426.76252
[13] Fringer, O. B.; Gerritsen, M.; Street, R. L., An unstructured-grid, finite-volume, nonhydrostatic, parallel coastal ocean simulator, Ocean Model., 14, 139-173, (2006)
[14] A. Gassmann, Inspection of hexagonal and triangular C-grid discretizations of the shallow water equations, J. Comput. Phys. 230, 2706-2721. · Zbl 1316.76069
[15] J. Galewsky, R.K. Scott, L.M. Polvani, An initial-value problem for testing numerical models of the global shallow-water equations, Tellus A 56, 429-440.
[16] Gresho, P.; Chan, S., On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix, part 2: implementation, Int. J. Numer. Methods Fluids, 11, 621-659, (1990) · Zbl 0712.76036
[17] 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, 47-60, (2007)
[18] Harlow, F. H.; Welch, J. E., Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids, 8, 2182-2189, (1965) · Zbl 1180.76043
[19] Heikes, R.; Randall, D., Numerical integration of the shallow-water equations on a twisted icosahedral grid. Part I: basic design and results of tests, Mon. Weather Rev., 123, 1862-1880, (1995)
[20] Heikes, R. P.; Randall, D. A.; Konor, C. S., Optimized icosahedral grids: performance of finite-difference operators and multigrid solver, Mon. Weather Rev., 141, 4450-4469, (2013)
[21] Hughes, T. J.R., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, (1997), Prentice-Hall
[22] Hyman, J. M.; Shashkov, M., Adjoint operators for the natural discretizations of the divergence, gradient and curl on logically rectangular grids, Appl. Numer. Math., 25, 413-442, (1997) · Zbl 1005.65024
[23] Ketefian, G. S.; Jacobson, M. Z., A mass, energy, vorticity, and potential enstrophy conserving lateral fluid-land boundary scheme for the shallow water equations, J. Comput. Phys., 228, 1-32, (2009) · Zbl 1194.76208
[24] Koren, B.; Abgrall, R.; Bochev, P.; Frank, J.; Perot, B., Physics compatible numerical methods, J. Comput. Phys., 257, 1039-1526, (2014)
[25] Korn, P., Formulation of an unstructured grid model for global ocean dynamics, J. Comput. Phys., 339, 525-552, (2017) · Zbl 1380.65275
[26] Korn, P.; Danilov, S., Elementary dispersion analysis of some mimetic discretizations on triangular C-grids, J. Comput. Phys., 330, 156-172, (2017) · Zbl 1380.65220
[27] Läuter, M.; Giraldo, F. X.; Handorf, D.; Dethloff, K., A discontinuous Galerkin method for the shallow water equations in spherical triangular coordinates, J. Comput. Phys., 227, 10226-10242, (2008) · Zbl 1218.76028
[28] Le Roux, D. Y.; 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. Methods Fluids, 59, 767-790, (2009) · Zbl 1156.76035
[29] Lipnikov, K.; Manzini, G.; Shashkov, M., Mimetic finite difference method, J. Comput. Phys., 257, 1163-1227, (2014) · Zbl 1352.65420
[30] Marshall, J.; Adcroft, A.; Hill, C.; Perelmann, L.; Heisey, C., A finite-volume, incompressible Navier-Stokes model for studies of the ocean on parallel computers, J. Geophys. Res., 102, C3, 5753-5766, (1997) · Zbl 0907.58089
[31] 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)
[32] Miura, H.; Kimoto, M., A comparison of grid quality of optimized spherical hexagonal-pentagonal geodesic grids, Mon. Weather Rev., 133, 2817-2833, (2005)
[33] Pedlosky, J., Geophysical Fluid Dynamics, (1987), Springer · Zbl 0713.76005
[34] Perot, B., Conservation properties of unstructured staggered mesh schemes, J. Comput. Phys., 159, 58-89, (2000) · Zbl 0972.76068
[35] Peixoto, P. S., Accuracy analysis of mimetic finite volume operators on geodesic grids and a consistent alternative, J. Comput. Phys., 310, 127-160, (2016) · Zbl 1349.76376
[36] Peixoto, P. S.; Barros, S. R.M., Analysis of grid imprinting on geodesic spherical icosahedral grids, J. Comput. Phys., 237, 61-78, (2013) · Zbl 1286.65118
[37] Peixoto, P. S.; Barros, S. R.M., On vector field reconstructions for semi-Lagrangian transport methods on geodesic staggered grids, J. Comput. Phys., 273, 185-211, (2014) · Zbl 1351.86005
[38] Peixoto, P. S.; Thuburn, J.; Bell, M. J., Numerical instabilities of spherical shallow-water models considering small equivalent depths, Q. J. R. Meteorol. Soc., 144, 156-171, (2017)
[39] Raviart, P. A.; Thomas, J. M., A mixed finite-element method for 2nd order elliptic problems, (Galligani, I.; Magenes, I., Mathematical Aspects of the Finite-Element Methods, Lect. Notes Math., (1977), Springer: Springer Berlin), 292-315 · Zbl 0362.65089
[40] Ringler, T.; Petersen, M.; Higdon, R. L.; Jacobsen, D.; Jones, P. W.; Maltrud, M., A multi-resolution approach to global ocean modeling, Ocean Model., 69, 211-232, (2013)
[41] Ringler, T. D.; Thuburn, J.; Klemp, J. B.; Skamarock, W. C., A unified approach to energy and potential vorticity dynamics for arbitrarily structured C-grids, J. Comput. Phys., 229, 3065-3090, (2010) · Zbl 1307.76054
[42] Ripodas, P.; Gassmann, A.; Förstner, J.; Majewski, D.; Giorgetta, M.; Korn, P.; Kornblueh, L.; Wan, H.; Zängl, G.; Bonaventura, L.; Heinze, T., Icosahedral shallow water model (ICOSWM): results of shallow water test cases and sensitivity to model parameters, Geosci. Model Dev., 2, 231-251, (2009)
[43] Skamarock, W. C.; Klemp, J. B.; Duda, M. G.; Fowler, L. D.; Park, S.; Ringler, T. D., A multiscale nonhydrostatic atmosphere model using centroidal Voronoi tesselations and C-grid staggering, Mon. Weather Rev., 140, 3090-3105, (2012)
[44] Staniforth, A.; Thuburn, J., Horizontal grids for global weather prediction and climate prediction models: a review, Q. J. R. Meteorol. Soc., 138, 1-26, (2012)
[45] Sadourny, R., Conservative finite-differencing approximations of the primitive equations on quasi-uniform spherical grids, Mon. Weather Rev., 100, 136-144, (1972)
[46] Sadourny, R.; Arakawa, A.; Mintz, Y., Integration of the nondivergent barotropic vorticity equation with an icosahedral-hexagonal grid for the sphere, Mon. Weather Rev., 96, 351-356, (1968)
[47] Sadourny, R.; Basdevant, C., Parameterization of subgrid-scale barotropic and baroclinic eddies in quasi-geostrophic models: anticipated potential vorticity method, J. Atmos. Sci., 42, 1353-1363, (1985)
[48] Salmon, R., A general method for conserving energy and potential enstrophy in shallow-water model, J. Atmos. Sci., 64, 515-531, (2007)
[49] Stuhne, G. R.; Peltier, W. B., New icosahedral grid-point discretizations of the shallow water equations on the sphere, J. Comput. Phys., 148, 23-58, (1999) · Zbl 0930.76067
[50] Stuhne, G. R.; Peltier, W. B., A robust unstructured grid discretization for 3-dimensional hydrostatic flows in spherical geometry: a new numerical structure for ocean general circulation modeling, J. Comput. Phys., 213, 704-729, (2006) · Zbl 1136.86303
[51] Stuhne, G. R.; Peltier, W. B., An unstructured C-grid based method for 3-D global ocean dynamics: free-surface formulations and tidal test cases, Ocean Model., 28, 97-105, (2009)
[52] Taylor, M. A.; Fournier, A., A compatible and conservative spectral finite element method on unstructured grids, J. Comput. Phys., 229, 704-729, (2010), 5879-5895
[53] Thuburn, J.; Cotter, C., 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
[54] 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)
[55] Thuburn, J.; Cotter, C., A primal-dual mimetic finite element scheme for the rotating shallow water equations on polygonal spherical meshes, J. Comput. Phys., 290, 274-297, (2015) · Zbl 1349.76273
[56] 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
[57] Thuburn, J., Some conservation issues for the dynamical cores of NWP and climate models, J. Comput. Phys., 227, 3715-3730, (2008) · Zbl 1132.86314
[58] Tomita, H.; Tsugawa, M.; Satoh, M.; Goto, K., Shallow water model on a modified icosahedral geodesic grid by using spring dynamics, J. Comput. Phys., 174, 579-613, (2001) · Zbl 1056.76058
[59] Vallis, G., Atmospheric and Oceanic Fluid Dynamics, (2006), Cambridge University Press
[60] da Veiga, L. B.; Lipnikov, K.; Manzini, G., The Mimetic Finite Difference Method for Elliptic Problems, (2014), Springer · Zbl 1286.65141
[61] Weller, H.; Thuburn, J.; Cotter, C. J., Computational modes and grid imprinting on five quasi-uniform spherical C-grids, Mon. Weather Rev., 140, 2734-2755, (2012)
[62] Weller, H.; Weller, H. G.; Fournier, A., Voronoi, delaunay, and block-structured mesh refinement for solution of the shallow-water equations on the sphere, Mon. Weather Rev., 137, 4208-4224, (2009)
[63] Williamson, D. L.; Drake, J. B.; Hack, J. J.; Jacob, R.; Swarztrauber, 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
[64] Wolfram, P. J.; Fringer, O. B., Mitigating horizontal divergence “checker-board” oscillations on unstructured triangular C-grids for nonlinear hydrostatic and nonhydrostatic flows, Ocean Model., 69, 64-78, (2013)
[65] Zängl, G.; Reinert, D.; Ripodas, P.; Baldauf, M., The ICON (ICOsahedral non-hydrostatic) modelling framework of DWD and MPI-M: description of the non-hydrostatic dynamical core, Q. J. R. Meteorol. Soc., 141, 563-579, (2015)
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