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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Veröffentlicht in:	Journal of computational physics Jg. 380; S. 378 - 399
Hauptverfasser:	Bayona, Víctor, Flyer, Natasha, Fornberg, Bengt
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Cambridge Elsevier Inc 01.03.2019 Elsevier Science Ltd
Schlagworte:	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
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Abstract	•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.
AbstractList	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. •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.
Author	Bayona, Víctor Fornberg, Bengt Flyer, Natasha
Author_xml	– sequence: 1 givenname: Víctor surname: Bayona fullname: Bayona, Víctor email: victor.bayona.revilla@gmail.com organization: Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés, Madrid, Spain – sequence: 2 givenname: Natasha surname: Flyer fullname: Flyer, Natasha email: flyer@ucar.edu organization: Analytics and Integrated Machine Learning, National Center for Atmospheric Research, Boulder, CO 80305, USA – sequence: 3 givenname: Bengt surname: Fornberg fullname: Fornberg, Bengt email: fornberg@colorado.edu organization: Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA
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Snippet	•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... Radial basis function generated finite difference (RBF-FD) approximations generalize grid-based regular finite differences to scattered node sets. These become...
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SubjectTerms	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
Title	On the role of polynomials in RBF-FD approximations: III. Behavior near domain boundaries
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