The shallow water equations describe large scale horizontal phenomena of the global atmospheric motion to good approximation. Thus they provide a widely accepted primary test for numerical methods for global atmospheric modelling before proceeding to complete 3D baroclinic models [7]. They are prognostic equations for the (horizontal) velocity field and the depth of a shallow homogeneous, incompressible, hydrostatic, and inviscid fluid layer on a rotating sphere.
Currently there is a controversy on the question which of the different approaches to the integration of global models is preferable. For example, the European Centre for Medium Range Weather Forecasts (ECMWF) and the National Center for Atmospheric Research (NCAR) run spectral transform methods, which use spherical harmonics for the spatial discretization, whereas currently at German Weather Service (DWD) a finite difference scheme on a uniform triangular grid is developed. Although spectral methods are superior at today's resolutions, it seems that grid point schemes will be competitive in the future, since they allow for adaptivity and they appear more appropriate for massively parallel computer systems. In this research note we present a spline collocation scheme for the shallow water equations, a preliminary version of which already has been studied in detail for the scalar advection equation in [1,2].
Let := {(x, y, z) ∈ R 3 , x 2 + y 2 + z 2 = 1} for the two-dimensional unit sphere. From a theoretical point of view it is convenient to formulate spherical differential equations coordinate-free. But for the parametrization of partial differential operators and vector fields on the sphere we need local coordinates. We will use standard polar coordinates (λ, ϑ) ∈ [0, 2π] × [-π/2, π/2] on the strip {(x, y, z) ∈ , |z| ≤ 1/ √ 2} around the equator and two stereographic coordinate systems (ξ, η) ∈ R 3 on the polar caps {(x, y, z) ∈ , z ≥ 1/ √ 2} and {(x, y, z) ∈ , z ≤ -1/ √ 2}. Of course, this particular partition is somewhat 291