Numerical Methods: Application of Radial Basis Functions to Modeling

Twelve-day simulation of deformational flow or idealized cyclogenesis on the sphere using radial basis functions (RBFs) to approximate the true solution. RBFs were used for the spatial discretization with 3,136 node points (shown as solid circles) and a time step of 10 hours. The animation demonstrates the very high accuracy of RBFs and indicates their value for improving simulations of global processes. (Press Shift and click the Reload button on your browser to restart the animation.)

While computer technology has advanced dramatically in recent years, numerical schemes currently used for climate and solar modeling fall drastically short of scientists' expectations. Spherical harmonics require large grids to resolve small features, and this is computationally impractical. Spectral element methods can resolve small features, but they require higher resolution near artificial boundaries to achieve high accuracy. Both methods involve high algorithmic complexity and are impossible or awkward to apply to irregular geometries. As a result, geoscientists and computational mathematicians are searching for new options.

Radial basis functions (RBFs) offer the geosciences community a new and efficient numerical approach for solving time-dependent partial differential equations (PDEs). Their attributes are very attractive and include:

1. Spectrally accurate for arbitrary node locations
2. Naturally permit local mesh refinement
3. Can be applied to irregular geometries, as RBFs do not depend on any grid
4. Extreme algorithmic simplicity
5. Generally much higher accuracy than spectral methods for a given number of nodes

However, RBFs are still in a developmental stage, and much research is needed before they can be applied to large-scale production models. But the outlook is exceptionally promising.

Building on the accomplishments of 2005, CSS, together with the University of Utah, continues research in the developing area of radial basis functions. In FY 2006, our efforts were concentrated on:

1. Furthering the mathematical understanding of why RBFs perform so well compared to other spectral methods for test cases such as pure advection of a cosine bell directly over the poles. (See table.)

2. Determining how RBFs perform on a new numerical test case for spectral methods, deformational flow, or idealized cyclogenesis, where accuracy is temporally limited by the formation of fine structures in the solution (see animation above).

Results from this second test case were very rewarding in that 10-hour time steps could be taken rather than the 6-minute time steps needed for a discontinuous Galerkin method. A comparison of the results from both efforts was submitted to Journal of Computational Physics. In the coming year, a shallow water model will be built using RBFs, and its performance will be gauged on a benchmark (the classic Williamson's test suite).

This work is supported in FY 2006 by an NSF Collaboration in Mathematical Geosciences grant that involves NCAR, the University of Utah, the University of Colorado at Boulder, the University of Michgan at Ann Arbor, and Arizona State University.