Arbitrary order of convergence for Riesz fractional derivative via central difference method

TL;DR

This paper introduces a new finite difference method for Riesz derivatives that achieves arbitrary convergence rates by applying a pre-filter to the central difference stencil, improving efficiency and adaptability.

Contribution

A novel filtering approach for Gr"unwald-Letnikov type schemes enabling customizable convergence rates and efficient computation for Riesz derivatives.

Findings

Higher convergence verified through numerical experiments.

Filtering reduces the number of nodal points needed for a given error.

Method supports adaptive grid sizes for dynamic problems.

Abstract

We propose a novel method to compute a finite difference stencil for Riesz derivative for artibitrary speed of convergence. This method is based on applying a pre-filter to the Gr\"unwald-Letnikov type central difference stencil. The filter is obtained by solving for the inverse of a symmetric Vandemonde matrix and exploiting the relationship between the Taylor's series coefficients and fast Fourier transform. The filter costs O\left(N^{2}\right) operations to evaluate for O\left(h^{N}\right) of convergence, where h is the sampling distance. The higher convergence speed should more than offset the overhead with the requirement of the number of nodal points for a desired error tolerance significantly reduced. The benefit of progressive generation of the stencil coefficients for adaptive grid size for dynamic problems with the Gr\"unwald-Letnikov type difference scheme is also kept because of the application of filtering. The higher convergence rate is verified through numerical experiments.