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A spectral element semi-Lagrangian (SESL) method for the spherical shallow water equations. (English) Zbl 1076.76058
Summary: A spectral element semi-Lagrangian (SESL) method for the shallow water equations on the sphere is presented. The sphere is discretized using a hexahedral grid although any grid imaginable can be used as long as it is comprised of quadrilaterals. The equations are written in Cartesian coordinates to eliminate the pole singularity which plagues the equations in spherical coordinates. In a previous paper of the first author [Int. J. Numer. Methods Fluids 35, 869–901 (2001; Zbl 1030.76045)] we showed how to construct an explicit Eulerian spectral element (SE) model on the sphere; we now extend this work to a semi-Lagrangian formulation. The novelty of the Lagrangian formulation presented is that the high order SE basis functions are used as the interpolation functions for evaluating the values at the Lagrangian departure points. This makes the method not only high order accurate but quite general and thus applicable to unstructured grids and portable to distributed memory computers. The equations are discretized fully implicitly in time in order to avoid having to interpolate derivatives at departure points. By incorporating the Coriolis terms into the Lagrangian derivative, the block LU decomposition of the equations results in a symmetric positive-definite pseudo-Helmholtz operator which we solve using the generalized minimum residual method (GMRES) with a fast projection method [P. F. Fischer, Comput. Methods Appl. Mech. Eng. 163, 193–204 (1998; Zbl 0960.76063)]. Results for eight test cases are presented to confirm the accuracy and stability of the method. These results show that SESL yields the same accuracy as an Eulerian spectral element semi-implicit (SESI) while allowing for time-steps 10 times as large and being up to 70% more efficient.

76M22 Spectral methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
86-08 Computational methods for problems pertaining to geophysics
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[1] Bermejo, R.; Staniforth, A., The conversion of semi-Lagrangian advection schemes to quasi-monotone schemes, Monthly weather review, 120, 2622-2632, (1992)
[2] Boyd, J., The erfc-log filter and the asymptotics of the Euler and vandeven sequence accelerations, Houston journal of mathematics, (1996)
[3] G. Chukapalli, Weather and climate numerical algorithms: a unified approach to an efficient, parallel implementation, Ph.D. Dissertation, University of Toronto, 1997
[4] Côté, J., A Lagrange multiplier approach for the metric terms of semi-Lagrangian models on the sphere, Quarterly journal of the royal meteorological society, 114, 1347-1352, (1988)
[5] Côté, J.; Roch, M.; Staniforth, A.; Fillion, L., A variable-resolution semi-Lagrangian finite-element global model of the shallow-water equations, Monthly weather review, 121, 231-243, (1993)
[6] Fischer, P.F., Projection techniques for iterative solution of A{\bfx}={\bfb} with successive right-hand sides, Computer methods in applied mechanics and engineering, 163, 193-204, (1998) · Zbl 0960.76063
[7] Falcone, M.; Ferretti, R., Convergence analysis for a class of high-order semi-Lagrangian advection schemes, SIAM journal of numerical analysis, 35, 909-940, (1998) · Zbl 0914.65097
[8] Giraldo, F.X., Lagrange – galerkin methods on spherical geodesic grids, Journal of computational physics, 136, 197-213, (1997) · Zbl 0909.65066
[9] Giraldo, F.X., The lagrange – galerkin spectral element method on unstructured quadrilateral grids, Journal of computational physics, 147, 114-146, (1998) · Zbl 0920.65070
[10] Giraldo, F.X., Lagrange – galerkin methods on spherical geodesic grids: the shallow water equations, Journal of computational physics, 160, 336-368, (2000) · Zbl 0977.76045
[11] Giraldo, F.X., A spectral element shallow water on spherical geodesic grids, International journal for numerical methods in fluids, 35, 869-901, (2001) · Zbl 1030.76045
[12] Giraldo, F.X.; Hesthaven, J.S.; Warburton, T., Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations, Journal of computational physics, 181, 499-525, (2002) · Zbl 1178.76268
[13] Giraldo, F.X.; Rosmond, T.E., A scalable spectral element Eulerian atmospheric model (SEE-AM) for fnwp: dynamical core test, Monthly weather review, (2003), in press
[14] J.J. Hack, B.A. Boville, B.P. Briegleb, J.T. Kiehl, P.J. Rasch, D.L. Williamson, Description of the NCAR community climate model (CCM2), NCAR Technical Note NCAR/TN-382+STR, National Center for Atmospheric Research, Climate Modeling Section, P.O. Box 3000, Boulder, CO 80307, 1992
[15] Heikes, R.; Randall, D.A., Numerical integration of the shallow water equations on a twisted icosahedral grid. part I: basic design and results of tests, Monthly weather review, 123, 1862-1880, (1995)
[16] Heinze, T.; Hense, A., The shallow water equations on the sphere and their lagrange – galerkin solution, Meteorology and atmospheric physics, 81, 129-137, (2002)
[17] Hogan, T.F.; Rosmond, T.E., The description of the navy global operational prediction system’s spectral forecast model, Monthly weather review, 119, 1786-1815, (1991)
[18] Iskandarani, M.; Haidvogel, D.B.; Boyd, J.P., Staggered spectral element model with application to the oceanic shallow water equations, International journal for numerical methods in fluids, 20, 393-414, (1995) · Zbl 0870.76057
[19] Kwizak, M.; Robert, A.J., A semi-implicit scheme for grid point atmospheric models of the primitive equations, Monthly weather review, 99, 32-36, (1971)
[20] Lin, S.J.; Rood, R.B., An explicit flux-form semi-Lagrangian shallow-water model on the sphere, Quarterly journal of the royal meteorological society, 123, 2477-2498, (1997)
[21] Malesky, A.V.; Thomas, S.J., Parallel algorithms for semi-Lagrangian advection, International journal for numerical methods in fluids, 25, 455-473, (1997) · Zbl 0910.76063
[22] Maday, Y.; Patera, A.T.; Ronquist, E.M., An operator-integration-factor splitting method for time-dependent problems: application to incompressible fluid flow, Journal of scientific computing, 5, 263-292, (1990) · Zbl 0724.76070
[23] McDonald, A.; Bates, J.R., Semi-Lagrangian integration of a gridpoint shallow water model on the sphere, Monthly weather review, 117, 130-137, (1989)
[24] Neta, B.; Giraldo, F.X.; Navon, I.M., Analysis of the turkel – zwas scheme for the 2D shallow water equations in spherical coordinates, Journal of computational physics, 133, 102-122, (1997) · Zbl 0883.76060
[25] Ritchie, H., Semi-Lagrangian advection on a Gaussian grid, Monthly weather review, 115, 608-619, (1987)
[26] Robert, A., A semi-Lagrangian and semi-implicit numerical integration scheme for the primitive meteorological equations, Journal of the meteorological society of Japan, 60, 319-325, (1982)
[27] M. Rochas, ARPEGE Documentation, Part 2, Chapter 6 (available from Meteo-France), 1990
[28] Ronchi, C.; Iacono, R.; Paolucci, P.S., The “cubed sphere”: a new method for the solution of partial differential equations in spherical geometry, Journal of computational physics, 124, 93-114, (1996) · Zbl 0849.76049
[29] Sela, J.G., Spectral modeling at the national meteorological center, Monthly weather review, 108, 1279-1292, (1980)
[30] Simmons, A.J.; Burridge, D.M.; Jarraud, M.; Girard, C.; Wergen, W., The ECMWF medium-range prediction models development of the numerical formulations and the impact of increased resolution, Meteorology and atmospheric physics, 40, 28-60, (1989)
[31] Swarztrauber, P.N.; Williamson, D.L.; Drake, J.B., The Cartesian method for solving partial differential equations in spherical geometry, Dynamics of atmospheres and oceans, 27, 679-706, (1997)
[32] Taylor, M.; Tribbia, J.; Iskandarani, M., The spectral element method for the shallow water equations on the sphere, Journal of computational physics, 130, 92-108, (1997) · Zbl 0868.76072
[33] Temperton, C.; Staniforth, A., An efficient two time-level semi-Lagrangian semi-implicit integration scheme, Quarterly journal of the royal meteorological society, 113, 1025-1039, (1987)
[34] Thomas, S.J.; Loft, R.D.; Dennis, J.M., Parallel implementation issues: global versus local methods, Computing in science and engineering, 4, 26-31, (2002)
[35] E. Turkel, G. Zwas, Explicit large time-step schemes for the shallow water equations, in: R. Vichnevetsky, R.S. Stepleman (Eds.), Advances in Computer Methods for Partial Differential Equations, IMACS, Lehigh University, 1979, p. 65
[36] 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, Journal of computational physics, 102, 211-224, (1992) · Zbl 0756.76060
[37] Xiu, D.; Karniadakis, G.E., A semi-Lagrangian high-order method for the navier – stokes equations, Journal of computational physics, 172, 658-684, (2001) · Zbl 1028.76026
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