Radial Basis Function Methods for Solving Partial Differential Equations
Bengt Fornberg
Date and time:Â
Friday, January 15, 2010 - 4:30pm
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For the task of solving PDEs, finite difference (FD) methods are particularly easy to implement. Finite element methods are more flexible geometrically, but tend to be difficult to make very accurate. Pseudospectral (PS) methods can be seen as a limit of FD methods of increasing orders of accuracy. They can be extremely effective in many situations, but this strength comes at the price of very severe geometric restrictions. A more standard introduction to PS methods (rather than via FD methods of increasing orders of accuracy) is in terms of expansions in orthogonal functions (such as Fourier modes, Chebyshev polynomials, etc.)
Radial basis functions (RBFs) were first proposed around 1970 as a tool for interpolating scattered data, and in 1990 as a new possibility for solving PDEs. Since then, both our knowledge about RBFs and their range of applications have grown tremendously. In the context of solving PDEs, we can see the RBF approach as a major generalization of PS methods, abandoning the orthogonality of the basis functions and in return obtaining much improved simplicity and geometric flexibility. Spectral accuracy becomes now easily available also when using completely unstructured node layouts, permitting local node refinements in critical areas. A very counterintuitive parameter range (making all the RBFs very flat) turns out to be of special interest. Computational cost and numerical stability were initially seen as potential difficulties, but major progress have recently been made also in these areas. The first major PDE applications for which RBFs have been shown to compete very successfully against the best previously available numerical approaches can be found in the geosciences. Several such cases, due to the work of Dr. Natasha Flyer, NCAR, will be described. These include vortex roll-ups, idealized cyclogenesis, unsteady nonlinear flows described by the shallow water equations, and 3-D convection in the earth's mantle.