A compatible and conservative spectral element method on unstructured grids. (English) Zbl 1425.76177

Summary: We derive a formulation of the spectral element method which is compatible on very general unstructured three-dimensional grids. Here compatible means that the method retains discrete analogs of several key properties of the divergence, gradient and curl operators: the divergence and gradient are anti-adjoints (the negative transpose) of each other, the curl is self-adjoint and annihilates the gradient operator, and the divergence annihilates the curl. The adjoint relations hold globally, and at the element level with the inclusion of a natural discrete element boundary flux term.
We then discretize the shallow-water equations on the sphere using the cubed-sphere grid and show that compatibility allows us to locally conserve mass, energy and potential vorticity. Conservation is obtained without requiring the equations to be in conservation form. The conservation is exact assuming exact time integration.


76M22 Spectral methods applied to problems in fluid mechanics
76M10 Finite element methods applied to problems in fluid mechanics
86A10 Meteorology and atmospheric physics
35Q35 PDEs in connection with fluid mechanics


chammp; GASpAR
Full Text: DOI


[1] Maday, Y.; Patera, A.T., Spectral element methods for the incompressible navier – stokes equations, (), 71-143
[2] Patera, A., A spectral element method for fluid dynamics: laminar flow in a channel expansion, J. comput. phys., 54, 468-488, (1984) · Zbl 0535.76035
[3] Taylor, M.; Tribbia, J.; Iskandarani, M., The spectral element method for the shallow water equations on the sphere, J. comput. phys., 130, 92-108, (1997) · Zbl 0868.76072
[4] Giraldo, F.X., A spectral element shallow water model on spherical geodesic grids, Int. J. numer. methods fluids, 35, 869-901, (2001) · Zbl 1030.76045
[5] Giraldo, F.X.; Rosmond, T.E., A scalable spectral element Eulerian atmospheric model (SEE-AM) for NWP: dynamical core tests, Mon. wea. rev., 132, 133-153, (2004)
[6] Fournier, A.; Taylor, M.; Tribbia, J., The spectral element atmosphere model (SEAM): high-resolution parallel computation and localized resolution of regional dynamics, Mon. wea. rev., 132, 726-748, (2004)
[7] Thomas, S.; Loft, R., The NCAR spectral element climate dynamical core: semi-implicit Eulerian formulation, J. sci. comput., 25, 307-322, (2005) · Zbl 1203.86013
[8] Wang, H.; Tribbia, J.J.; Baer, F.; Fournier, A.; Taylor, M.A., A spectral element version of CAM, Mon. wea. rev., 135, 3825G3840, (2007)
[9] St-Cyr, A.; Jablonowski, C.; Dennis, J.M.; Tufo, H.M.; Thomas, S.J., A comparison of two shallow water models with non-conforming adaptive grids, Mon. wea. rev., 136, 1898-1922, (2008)
[10] Haidvogel, D.; Curchitser, E.N.; Iskandarani, M.; Hughes, R.; Taylor, M.A., Global modeling of the Ocean and atmosphere using the spectral element method, Atmos. Ocean spec., 35, 505-531, (1997)
[11] Iskandarani, M.; Haidvogel, D.; Levin, J.; Curchitser, E.N.; Edwards, C.A., Multiscale geophysical modeling using the spectral element method, Comput. sci. eng., 4, 42-48, (2002)
[12] Molcard, A.; Pinardi, N.; Iskandarani, M.; Haidvogel, D., Wind driven circulation of the mediterranean sea simulated with a spectral element Ocean model, Dyn. atmos. oceans, 35, 97-130, (2002)
[13] Komatitsch, D.; Tromp, J., Spectral-element simulations of global seismic wave propagation - I. validation, Geophys. J. int., 149, 390-412, (2002)
[14] Tromp, J.; Komatitsch, D.; Liu, Q., Spectral-element and adjoint methods in seismology, Commun. comput. phys., 3, 1, 1-32, (2008) · Zbl 1183.74320
[15] Hughes, T.J.R.; Engel, G.; Mazzei, L.; Larson, M.G., The continuous Galerkin method is locally conservative, J. comput. phys., 163, 467-488, (2000) · Zbl 0969.65104
[16] Samarskiı˘, A.A.; Tishkin, V.F.; Favorskiı˘, A.P.; Shashkov, M.Y., Operator-difference schemes, Differentsial’nye uravneniya, 17, 7, 1317-1327, (1981), p. 1344
[17] Margolin, L.G.; Tarwater, A.E., A diffusion operator for Lagrangian codes, (), 1252-1260
[18] Nicolaides, R., Direct discretization of planar div-curl problems, SIAM J. numer. anal., 32-56, (1992) · Zbl 0745.65063
[19] Shashkov, M.; Steinberg, S., Support-operator finite-difference algorithms for general elliptic problems, J. comput. phys., 118, 131-151, (1995) · Zbl 0824.65101
[20] Shashkov, M., Conservative finite difference methods on general grids, (1996), CRC-Press Boca Raton, FL, p. 384 · Zbl 0844.65067
[21] 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
[22] Hyman, J.M.; Shashkov, M., Natural discretizations for the divergence, gradient and curl on logically rectangular grids, Int. J. appl. numer. math., 33, 81-104, (1997) · Zbl 0868.65006
[23] Bochev, P.; Hyman, M., Principles of compatible discretizations, (), 89-120 · Zbl 1110.65103
[24] Gassmann, A.; Herzog, H.J., Towards a consistent numerical compressible non-hydrostatic model using generalized Hamiltonian tools, Q.J.R. meteorol. soc., 134, 635, 1597-1613, (2008)
[25] Salmon, R., Poisson-bracket approach to the construction of energy- and potential-enstrophy-conserving algorithms for the shallow-water equations, J. atmo. sci., 61, 2016-2036, (2004)
[26] Salmon, R., A general method for conserving energy and potential enstrophy in shallow-water models, J. atmo. sci., 64, 515-531, (2007)
[27] Boyd, J.P., Chebyshev and Fourier spectral methods: second revised edition, (2001), Dover Publications
[28] Margolin, L.G.; Shashkov, M., Finite volume methods and the equations of finite scale: a mimetic approach, Int. J. numer. meth. fluids, 56, 8, 991-1002, (2008) · Zbl 1155.76041
[29] Simmons, A.J.; Burridge, D.M., An energy and angular momentum conserving vertical finite-difference scheme and hybrid vertical coordinates, Mon. wea. rev., 109, 758-766, (1981)
[30] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T., Spectral methods: evolution to complex geometries and applications to fluid dynamics, (2007), Springer · Zbl 1121.76001
[31] Fournier, A.; Rosenberg, D.; Pouquet, A., Dynamically adaptive spectral-element simulations of 2D incompressible navier – stokes vortex decays, Geophys. astrophys. fluid dyn., 103, 2, 245-268, (2009)
[32] Karniadakis, G.E.; Sherwin, S.J., Spectral/hp element methods for computational fluid dynamics (numerical mathematics and scientific computation), (2005), Oxford University Press USA
[33] Deville, M.O.; Fischer, P.F.; Mund, E.H., High order methods for incompressible fluid flow, (2002), Cambridge University Press · Zbl 1007.76001
[34] Solin, P.; Segeth, K.; Dolezel, I., Higher-order finite element methods, (2004), Chapman & Hall/CRC Press
[35] Rosenberg, D.; Fournier, A.; Fischer, P.; Pouquet, A., Geophysical-astrophysical spectral-element adaptive refinement (gaspar): object-oriented h-adaptive fluid dynamics simulation, J. comp. phys., 215, 59-80, (2006) · Zbl 1140.86300
[36] Heinbockel, J.H., Introduction to tensor calculus and continuum mechanics, (2001), Trafford Publishing Victoria, B.C.
[37] Thomas, S.; Loft, R., Parallel semi-implicit spectral element methods for atmospheric general circulation models, J. sci. comput., 15, 499-518, (2000) · Zbl 1048.76043
[38] Sadourny, R., Conservative finite-difference approximations of the primitive equations on quasi-uniform spherical grids, Mon. wea. rev., 100, 2, 136-144, (1972)
[39] Rančić, M.; Purser, R.; Mesinger, F., A global shallow-water model using an expanded spherical cube: gnomonic versus conformal coordinates, Q.J.R. meteorol. soc., 122, 959-982, (1996)
[40] Vos, P.E.J.; Sherwin, S.J.; Kirby, R.M., From h to p efficiently: implementing finite and spectral/hp element methods to achieve optimal performance for low and high order discretisations, J. comput. phys., 229, 5161-5181, (2010) · Zbl 1194.65138
[41] Dennis, J.; Fournier, A.; Spotz, W.F.; -Cyr, A. St.; Taylor, M.A.; Thomas, S.J.; Tufo, H., High resolution mesh convergence properties and parallel efficiency of a spectral element atmospheric dynamical core, Int. J. high perf. comput. appl., 19, 225-235, (2005)
[42] 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, 211-224, (1992) · Zbl 0756.76060
[43] Jakob-Chien, R.; Hack, J.J.; Williamson, D.L., Spectral transform solutions to the shallow water test set, J. comput. phys., 119, 164-187, (1995) · Zbl 0878.76059
[44] Durran, D.R., The third-order Adams-bashforth method: an attractive alternative to leapfrog time differencing, Mon. wea. rev., 119, 702-720, (1991)
[45] R. Jakob, J.J. Hack, D.L. Williamson, Solutions to the shallow water test set using the spectral transform method, Technical Report, NCAR/TN-388+STR, National Center for Atmospheric Research, 1993. · Zbl 0878.76059
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.