On the Use of RBF Interpolation for Flux Reconstruction

Flux reconstruction provides a framework for solving partial differential equations in which functions are discontinuously approximated within elements. Typically, this is done by using polynomials. Here, the use of radial basis functions as a methods for underlying functional approximation is explo...

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Bibliographic Details
Published inarXiv.org
Main Authors Watson, Rob, Trojak, Will
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 05.01.2022
Subjects
Approximation
Burgers equation
Functionals
Interpolation
Mathematical analysis
Numerical methods
Partial differential equations
Polynomials
Radial basis function
Reconstruction
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Summary:Flux reconstruction provides a framework for solving partial differential equations in which functions are discontinuously approximated within elements. Typically, this is done by using polynomials. Here, the use of radial basis functions as a methods for underlying functional approximation is explored in one dimension, using both analytical and numerical methods. At some mesh densities, RBF flux reconstruction is found to outperform polynomial flux reconstruction, and this range of mesh densities becomes finer as the width of the RBF interpolator is increased. A method which avoids the poor conditioning of flat RBFs is used to test a wide range of basis shapes, and at very small values, the polynomial behaviour is recovered. Changing the location of the solution points is found to have an effect similar to that in polynomial FR, with the Gauss--Legendre points being the most effective. Altering the location of the functional centres is found to have only a very small effect on performance. Similar behaviours are determined for the non-linear Burgers' equation.
ISSN:2331-8422