In the second part we tackle the numerical solution of time dependent partial
differential equations in spherical geometry. First we describe in detail the
discretization and parametrization of differential operators using the splines
on the reduced grid for the scalar linear advection equation. This model is
considered rather as a test equation, but essentially all difficulties we are
faced with in more realistic models also arise in this simplified context. We
can prove stability and first order convergence of a spline collocation scheme.
Basing on these results, the method is extended to the shallow water equations.
They are known to describe large scale phenomena of horizontal dynamics of global atmospheric motion to a good approximation, and since they are the core of most global models, they serve as a widespread test for numerical methods for numerical weather forecasts. Our scheme is applied to a set of problems that now has been established as a de facto benchmark test. This set contains seven particular tests of increasing order of complexity from simple code checks up to realistic simulations with observed and analyzed initial height and wind fields. We observe convergence of the approximations, when the resolution is refined, however, at the present a strict mathematical proof seems out of reach.
A comparison of our method to a preliminary version of the new scheme that in future will be in operational use by Deutscher Wetterdienst for the integration of its global model shows that our approach is competitive. Finally we develop a multiscale algorithm for spherical PDEs. The simultaneous localization properties of wavelets in space and in frequency are exploited to control the spatial resolution adaptively. We describe it in detail for the advection equation, present its extension to the shallow water equations, and show that its accuracy and efficiency can be estimated a priori in terms of the control parameters of the update algorithm. In the adaptive approximations, the lacunary grids reflect the structure of the fields, but no influence of the irregular grid structure of the full approximations nor of the parametrization is observed. Compared to the spline collocation scheme with uniform spatial resolution, for fields with sufficiently localized features there is a gain in computational complexity by performing the time marching procedure in the lacunary multiscale bases.