Numerical methods: Application of radial basis functions to modeling




		RBFs were used to produce this 23-day simulation of a steadily translating low pressure system, mode led by the forced nonlinear shallow water equations. This simulation used simple algoritms on 3,136 nodes with a time step of 10 minutes. This result demonstrates that RBFs can be applied to geophysical models, and that this approach shows promise for efficiently using thousands of processors to produce highly accurate Earth system simulations. (See 8 MB 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).

Building on the accomplishments of FY2006, the IMAGe Computational Mathematics Group together with the University of Utah, University of Colorado-Boulder, University Of Michigan-Ann Arbor, Arizona State University, and Uppsala University, Sweden continue research in the developing area of RBFs for climate and solar modeling.

Results from the deformational flow test case reported in FY2006 are compared to a discontinuous Galerkin spectral element scheme (DGSE) in the table that follows. Because this test case has just been developed, comparison is limited to this method since it is the only one for which results are available in the literature. The table shows that to achieve a given accuracy for this test case, the RBF method needs a much lower resolution while being able to take much larger time steps than the DGSE method.

Method	N	L2 error	Time step
RBF	4096	1e-5	600 minutes
DGSE	55296	1e-5	6 minutes

In 2007, our efforts furthered the topic by concentrating on solving the shallow water wave equations on the sphere using RBFs. This research concentrates on the performance of RBFs for three test cases of the shallow water equations with known solutions. The first test case is a steady-state solution for global nonlinear zonal geostrophic flow. The second is similar to the first except that the wind field is compactly supported (i.e. nonzero in a limited band region), emitting a more complex solution. The last is a forced nonlinear system with a translating low pressure center in the Northern Hemisphere that is superimposed on a westerly jet stream.

RBFs numerically solved the first test case trivially. The solution is second-order spherical harmonics that RBFs can reproduce exactly with just nine nodes on the sphere. We could stably integrate at machine precision for decades. For the second test case, the solution is not an exact spherical harmonic expansion, and the errors compare well against other spectral methods as shown in the table. For the third test case, the comparision analysis to other spectral methods is currently being performed. However, the RBF solution can be seen in the animation above.

The following table summarizes the results for the second test case to compare the performance of RBFs with spherical harmonics (SH), double Fourier series (DF), and spectral elements on a cube (SE). It shows that the RBF method achieves superior accuracy with significantly fewer processors.

Method	N	L2 error	Time step
RBF	1849	1e-8	10 minutes
SH	2048	2e-6	6 minutes
DF	2048	2e-6	6 minutes
SE	6144	8e-7	1.5 minutes

In the coming year, we will focus on applying RBFs to unsteady fluid flows. This will require mathematical breakthroughs in the areas of local node refinement and filtering for RBFs.

This work advances NCAR's strategic priority of "Conducting computer science, computational science, applied mathematics, statistics, and numerical methods R&D" and is supported by NSF grant ATM-0620100.