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Lagrange-Galerkin methods on spherical geodesic grids: The shallow water equations. (English) Zbl 0977.76045
Summary: We present a weak Lagrange-Galerkin finite element method for two-dimensional shallow water equations on the sphere. This method offers stable and accurate solutions because the equations are integrated along the characteristics. The equations are written in three-dimensional Cartesian conservation form, and the domains are discretized using linear triangular elements. The use of linear triangular elements permits the construction of accurate (by virtue of second-order spatial and temporal accuracies of the scheme) and efficient (by virtue of less stringent CFL condition of Lagrangian methods) schemes on unstructured domains. Using linear triangles in three-dimenisonal Cartesian space allows for the explicit construction of area coordinate basis functions, thereby simplifying the calculation of the finite element integrals. The triangular grids are constructed by a generalization of icosahedral grids that have been typically used in recent papers. An efficient searching strategy for the departure points is also presented for these generalized icosahedral grids which involves very few floating point operations. In addition, a high-order scheme for computing the characteristic curves in three-dimensional Cartesian space is presented: a general family of Runge-Kutta schemes. Results for six test cases are reported in order to confirm the accuracy of the scheme.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
chammp
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##### References:
 [1] Chukapalli, G., Weather and climate numerical algorithms: A unified approach to an efficient, parallel implementation, (1997) [2] Côté, J., A Lagrange multiplier approach for the metric terms of semi-Lagrangian models on the sphere, Quart. J. roy. meteorological soc., 114, 1347, (1988) [3] Giraldo, F.X., Lagrange – galerkin methods on spherical geodesic grids, J. comput. phys., 136, 197, (1997) · Zbl 0909.65066 [4] Giraldo, F.X., Trajectory calculations for spherical geodesic grids in Cartesian space, Monthly weather rev., 127, 1651, (1999) [5] F. X. Giraldo, The Lagrange-Galerkin method for the 2D shallow water equations on adaptive grids, Internat. J. Numer. Methods Fluids, in press. · Zbl 0989.76047 [6] 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 rev., 123, 1862, (1995) [7] McDonald, A.; Bates, J.R., Semi-Lagrangian integration of a gridpoint shallow water model on the sphere, Monthly weather rev., 117, 130, (1989) [8] Neta, B.; Giraldo, F.X.; Navon, I.M., Analysis of the turkel – zwas scheme for the 2D shallow water equations in spherical coordinates, J. comput. phys., 133, 102, (1997) · Zbl 0883.76060 [9] Priestley, A., The taylor – galerkin method for the shallow-water equations on the sphere, Monthly weather rev., 120, 3003, (1992) [10] R. J. Purser, Non-standard grids, in, Proceedings of the European Center for Medium-Range Weather Forecasting Annual Seminar, September 7-11, 1998. [11] Taylor, M.; Tribbia, J.; Iskandarani, M., The spectral element method for the shallow water equations on the sphere, J. comput. phys., 130, 92, (1997) · Zbl 0868.76072 [12] 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, (1992) · Zbl 0756.76060
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