A rational radial basis function method for accurately resolving discontinuities and steep gradients

Radial Basis Function (RBF) methods have become important tools for scattered data interpolation and for solving partial differential equations (PDEs) in complexly shaped domains. When the underlying function is sufficiently smooth, RBF methods can produce exceptional accuracy. However, like other h...

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Bibliographic Details
Published in:Applied numerical mathematics Vol. 130; pp. 131 - 142
Main Authors: Sarra, Scott A., Bai, Yikun
Format: Journal Article
Language:English
Published: Elsevier B.V 01.08.2018
Subjects:
Gibbs phenomenon
Radial Basis Functions
Rational interpolation
Shocks
Time-dependent PDEs
Rational interpolation
Gibbs phenomenon
Time-dependent PDEs
Shocks
Radial Basis Functions
ISSN:0168-9274, 1873-5460
Online Access:Get full text
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Summary:Radial Basis Function (RBF) methods have become important tools for scattered data interpolation and for solving partial differential equations (PDEs) in complexly shaped domains. When the underlying function is sufficiently smooth, RBF methods can produce exceptional accuracy. However, like other high order numerical methods, if the underlying function has steep gradients or discontinuities the RBF method may/will produce solutions with non-physical oscillations. In this work, a rational RBF method is used to approximate derivatives of functions with steep gradients and discontinuities and to solve PDEs with such solutions. The method is non-linear and is more computationally expensive than the standard RBF method. A modified partition of unity method is discussed as an way to implement the rational RBF method in higher dimensions.
ISSN:0168-9274
1873-5460
DOI:10.1016/j.apnum.2018.04.001