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SUMMARY:Radial basis functions for solving partial differential equations 
 - Bengt Fornberg (University of Colorado)
DTSTART:20090611T140000Z
DTEND:20090611T150000Z
UID:TALK18333@talks.cam.ac.uk
CONTACT:6743
DESCRIPTION:For the task of solving PDEs\, finite difference (FD) methods 
 are particularly easy to implement. Finite element (FE) methods are more f
 lexible geometrically\, but tend to be difficult to make very accurate. Ps
 eudospectral (PS) methods can be seen as a limit of FD methods if one keep
 s on increasing their order of accuracy. They are extremely effective in m
 any situations\, but this strength comes at the price of very severe geome
 tric restrictions. A more standard introduction to PS methods (rather than
  via FD methods of increasing orders of accuracy) is in terms of expansion
 s in orthogonal functions (such as Fourier\, Chebyshev\, etc.). \n\nRadial
  basis functions (RBFs) were first proposed around 1970 as a tool for inte
 rpolating scattered data. Since then\, both our knowledge about them and t
 heir range of applications have grown tremendously. In the context of solv
 ing PDEs\, we can see the RBF approach as a major generalization of PS met
 hods\, abandoning the orthogonality of the basis functions and in return o
 btaining much improved simplicity and flexibility. Spectral accuracy becom
 es now easily available also when using completely unstructured meshes\, p
 ermitting local node refinements in critical areas. A very counterintuitiv
 e parameter range (making all the RBFs very flat) turns out to be of speci
 al interest.\n \nAs was shown recently by Dr Natasha Flyer and collaborato
 rs\, RBF discretization competes very favorably against all previous appro
 aches for solving many convection-dominated PDEs on a sphere or in spheric
 al shells - geometries that are ubiquitous in weather\, climate\, and geop
 hysical modeling.
LOCATION:MR14\, CMS
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