On the Use of RBF Interpolation for Flux Reconstruction

TL;DR

This paper investigates the application of radial basis functions in flux reconstruction for solving PDEs, showing that RBFs can outperform polynomials at certain mesh densities and exploring the effects of basis shape and point placement.

Contribution

It introduces RBF interpolation into flux reconstruction, compares its performance with polynomial methods, and analyzes the influence of basis shape and point placement on accuracy.

Findings

RBF flux reconstruction outperforms polynomial methods at specific mesh densities.

Increasing RBF width refines the effective mesh density range.

Gauss--Legendre points are most effective for solution point placement.

Abstract

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.