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Solving PDEs with radial basis functions*

Published online by Cambridge University Press:  27 April 2015

Bengt Fornberg  and
Natasha Flyer
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
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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
Information
Acta Numerica , Volume 24 , 01 May 2015 , pp. 215 - 258
Copyright
© Cambridge University Press, 2015 

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