A spectral element shallow water model on spherical geodesic grids

The spectral element method for the two-dimensional shallow water equations on the sphere is presented. The equations are written in conservation form and the domains are discretized using quadrilateral elements obtained from the generalized icosahedral grid introduced previously (Giraldo FX. Lagran ge-Galerkin methods on spherical geodesic grids: the shallow water equation s. Journal of Computational Physics 2000; 160: 336-368). The equations are written in Cartesian co-ordinates that introduce an additional momentum equ ation, but the pole singularities disappear. This paper represents a depart ure from previously published work on solving the shallow water equations o n the sphere in that the equations are all written, discretized, and solved in three-dimensional Cartesian space. Because the equations are written in a three-dimensional Cartesian co-ordinate system, the algorithm simplifies into the integration of surface elements on the sphere from the fully thre e-dimensional equations. A mapping (Song Ch, Wolf JP. The scaled boundary f inite element method-alias consistent infinitesimal finite element cell met hod-for diffusion. International Journal for Numerical Methods in Engineeri ng 1999; 45: 1403-1431) which simplifies these computations is described an d is shown to contain the Eulerian version of the method introduced previou sly by Giraldo (Journal of Computational Physics 2000; 160: 336-368) for th e special case of triangular elements. The significance of this mapping is that although the equations are written in Cartesian co-ordinates, the mapp ing takes into account the curvature of the high-order spectral elements, t hereby allowing the elements to lie entirely on the surface of the sphere. In addition, using this mapping simplifies all of the three-dimensional spe ctral-type finite element surface integrals because any of the typical two- dimensional planar finite element or spectral element basis functions found in any textbook (for example, Huebner et al. The Finite Element Method for Engineers. Wiley, New York, 1995; Karniadakis GE, Sherwin SJ. Spectral/hp Element Methods for CFD. Oxford University Press, New York, 1999; and Szabo B, Babuska I. Finite Element Analysis. Wiley, New York, 1991) can be used. Results for six test cases are presented to confirm the accuracy and stabi lity of the new method. Published in 2001 by John Wiley & Sons, Ltd.