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Solving PDEs with radial basis functions*

Published online by Cambridge University Press:  27 April 2015

Bengt Fornberg
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA E-mail: fornberg@colorado.edu
Natasha Flyer
Affiliation:
Institute for Mathematics Applied to Geosciences, National Center for Atmospheric Research, Boulder, CO 80305, USA E-mail: flyer@ucar.edu

Abstract

Finite differences provided the first numerical approach that permitted large-scale simulations in many applications areas, such as geophysical fluid dynamics. As accuracy and integration time requirements gradually increased, the focus shifted from finite differences to a variety of different spectral methods. During the last few years, radial basis functions, in particular in their ‘local’ RBF-FD form, have taken the major step from being mostly a curiosity approach for small-scale PDE ‘toy problems’ to becoming a major contender also for very large simulations on advanced distributed memory computer systems. Being entirely mesh-free, RBF-FD discretizations are also particularly easy to implement, even when local refinements are needed. This article gives some background to this development, and highlights some recent results.

Type
Research Article
Copyright
© Cambridge University Press, 2015 

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