Solving PDEs with radial basis functions

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 spect...

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Bibliographic Details
Published in:	Acta numerica Vol. 24; pp. 215 - 258
Main Authors:	Fornberg, Bengt, Flyer, Natasha
Format:	Journal Article
Language:	English
Published:	Cambridge, UK Cambridge University Press 01.05.2015
Subjects:	Mathematical functions Numerical analysis
ISSN:	0962-4929, 1474-0508
Online Access:	Get full text
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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.
AbstractList	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.
Author	Fornberg, Bengt Flyer, Natasha
Author_xml	– sequence: 1 givenname: Bengt surname: Fornberg fullname: Fornberg, Bengt email: fornberg@colorado.edu organization: Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA E-mail: fornberg@colorado.edu – sequence: 2 givenname: Natasha surname: Flyer fullname: Flyer, Natasha email: flyer@ucar.edu organization: Institute for Mathematics Applied to Geosciences, National Center for Atmospheric Research, Boulder, CO 80305, USA E-mail: flyer@ucar.edu
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Snippet	Finite differences provided the first numerical approach that permitted large-scale simulations in many applications areas, such as geophysical fluid dynamics....
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SubjectTerms	Mathematical functions Numerical analysis
Title	Solving PDEs with radial basis functions
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