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CATEGORIES:Applied and Computational Analysis
SUMMARY:Radial basis functions for solving partial differe
ntial equations - Bengt Fornberg (University of Co
lorado)
DTSTART;TZID=Europe/London:20090611T150000
DTEND;TZID=Europe/London:20090611T160000
UID:TALK18333AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/18333
DESCRIPTION:For the task of solving PDEs\, finite difference (
FD) methods are particularly easy to implement. Fi
nite element (FE) methods are more flexible geomet
rically\, but tend to be difficult to make very ac
curate. Pseudospectral (PS) methods can be seen as
a limit of FD methods if one keeps on increasing
their order of accuracy. They are extremely effect
ive in many situations\, but this strength comes a
t the price of very severe geometric restrictions.
A more standard introduction to PS methods (rathe
r than via FD methods of increasing orders of accu
racy) is in terms of expansions in orthogonal func
tions (such as Fourier\, Chebyshev\, etc.). \n\nRa
dial basis functions (RBFs) were first proposed ar
ound 1970 as a tool for interpolating scattered da
ta. Since then\, both our knowledge about them and
their range of applications have grown tremendous
ly. In the context of solving PDEs\, we can see th
e RBF approach as a major generalization of PS met
hods\, abandoning the orthogonality of the basis f
unctions and in return obtaining much improved sim
plicity and flexibility. Spectral accuracy becomes
now easily available also when using completely u
nstructured meshes\, permitting local node refinem
ents in critical areas. A very counterintuitive pa
rameter range (making all the RBFs very flat) turn
s out to be of special interest.\n \nAs was shown
recently by Dr Natasha Flyer and collaborators\, R
BF discretization competes very favorably against
all previous approaches for solving many convectio
n-dominated PDEs on a sphere or in spherical shell
s - geometries that are ubiquitous in weather\, cl
imate\, and geophysical modeling.
LOCATION:MR14\, CMS
CONTACT:
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