Construction of Splines and Wavelets on the Sphere and Numerical Solutions to the Shallow Water Equations of Global Atmospheric Dynamics

Jochen Göttelmann

ISBN 978-3-89722-040-9
180 pages, year of publication: 1998
price: 40.00 €

The present work is divided into two main parts. The first part is devoted to the construction of wavelets on a manifold with special emphasis on the two-dimensional unit sphere . We present a general principle for the construction of locally supported wavelets on manifolds, once a multiresolution analysis of is given. Since we wish to cover as wide as possible a class of manifolds, the terminology is used in a broad sense, e.g. we do not require vanishing moments or orthogonality of the wavelets. Essentially we use Sweldens' lifting scheme, but we can prove stability of the wavelets in and in a scale of Sobolev spaces , by abstract techniques. Next we construct a multiresolution analysis of . We introduce a quasi-uniform longitude-latitude based reduced grid for the definition of continuous and piecewise bilinear splines that fulfill the stability conditions for singlescale bases. The corresponding wavelets constructed by the general principle are proved to be stable bases of the Sobolev spaces for . Numerical examples confirm that the poles are not exceptional points. We conclude the first part with a brief description how spherical wavelets can be used to compress atmospheric data. We show theoretically and by some numerical examples that the efficiency of the wavelet compression strongly depends on the smoothness of the data.

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.