On the Use of RBF Interpolation for Flux Reconstruction

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.

Summation-by-parts operators for general function spaces

“…There are also several works on ENO and WENO reconstructions based on non-polynomial function approximations [13,44,39]. Another line of related work deals with global and local radial basis function methods [17,25,24] and flux reconstruction methods based on radial basis functions [77]. Finally, there are some recent efforts on numerical methods for PDEs based on rational function approximations [56,38].…”