The Taylor–Galerkin Method for the Shallow-Water Equations on the Sphere

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Abstract
In this paper numerical solutions to the shallow-water equations on the sphere are obtained using a method that has already proved its worth in other areas of computational fluid dynamics (CFD), but has yet to make an impact in environmental- or meteorological-type flows. This is the Taylor–Galerkin finite-element method. This method offers the flexibility in mesh refinement associated with the finite-element method in general, together with the accuracy of the Lax-Wendroff method (although with fewer of the well-known problems of that method). Here the method is formulated in a form suitable for solving advection problems on the sphere, and its potential is explored on a well-known test problem. The problems are solved in Cartesian geometry, avoiding the singularities associated with the poles in the usual spherical polar transformation.

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