Numerical Methods: Application of Radial Basis Functions to Modeling
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:
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:
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.