On the role of polynomials in RBF-FD approximations: III. Behavior near domain boundaries

•Large PHS+poly based RBF-FD stencils can lead to high orders of accuracy without numerical ill-conditioning.•It can also combine high orders of accuracy near boundaries with an absence of Runge-phenomenon-type boundary errors.•Numerical explanations to this behavior are provided based on a closed-f...

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Published in:Journal of computational physics Vol. 380; pp. 378 - 399
Main Authors: Bayona, Víctor, Flyer, Natasha, Fornberg, Bengt
Format: Journal Article
Language:English
Published: Cambridge Elsevier Inc 01.03.2019
Elsevier Science Ltd
Subjects:
Basis functions
Boundaries
Computational physics
Cubic polyharmonic splines
Finite difference method
Ill-conditioned problems (mathematics)
PHS
Polynomials
Radial basis function
Radial basis functions
RBF
RBF-FD
Runge's phenomenon
Splines
Radial basis functions
RBF-FD
PHS
Runge's phenomenon
RBF
Cubic polyharmonic splines
ISSN:0021-9991, 1090-2716
Online Access:Get full text
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Summary:•Large PHS+poly based RBF-FD stencils can lead to high orders of accuracy without numerical ill-conditioning.•It can also combine high orders of accuracy near boundaries with an absence of Runge-phenomenon-type boundary errors.•Numerical explanations to this behavior are provided based on a closed-form expression for the RBF+poly cardinal functions.•It explains the role of polynomials and RBFs in RBF+poly approximations. Radial basis function generated finite difference (RBF-FD) approximations generalize grid-based regular finite differences to scattered node sets. These become particularly effective when they are based on polyharmonic splines (PHS) augmented with multi-variate polynomials (PHS+poly). One key feature is that high orders of accuracy can be achieved without having to choose an optimal shape parameter and without having to deal with issues related to numerical ill-conditioning. The strengths of this approach were previously shown to be especially striking for approximations near domain boundaries, where the stencils become highly one-sided. Due to the polynomial Runge phenomenon, regular FD approximations of high accuracy will in such cases have very large weights well into the domain. The inclusion of PHS-type RBFs in the process of generating weights makes it possible to avoid this adverse effect. With that as motivation, this study aims at gaining a better understanding of the behavior of PHS+poly generated RBF-FD approximations near boundaries, illustrating it in 1-D, 2-D and 3-D.
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ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2018.12.013