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. Lagrange– Galerkin methods on spherical geodesic grids: the shallow water equations. Journal of Computational Physics 2000; 160: 336–368). The equations are written in Cartesian co-ordinates that introduce an additional momentum equation, but the pole singularities disappear. This paper represents a departure from previously published work on solving the shallow water equations on 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 three-dimensional equations. A mapping (Song Ch, Wolf JP. The scaled boundary finite element method—alias consistent infinitesimal finite element cell method—for diffusion. International Journal for Numerical Methods in Engineering 1999; 45: 1403–1431) which simplifies these computations is described and is shown to contain the Eulerian version of the method introduced previously by Giraldo (Journal of Computational Physics 2000; 160: 336–368) for the special case of triangular elements. The significance of this mapping is that although the equations are written in Cartesian co-ordinates, the mapping takes into account the curvature of the high-order spectral elements, thereby allowing the elements to lie entirely on the surface of the sphere. In addition, using this mapping simplifies all of the three-dimensional spectral-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 Szabó B, Babuška I. Finite Element Analysis. Wiley, New York, 1991) can be used. Results for six test cases are presented to confirm the accuracy and stability of the new method. Published in 2001 by John Wiley & Sons, Ltd.