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

2021/08/09 by Pui Ho Lam, Lam, Pui Ho, Hing Cheung So +3
Computer Science · Mathematics · #26A33 #65B99 #65M06 #65R10 #FOS: Mathematics #Numerical Analysis (math.NA) #cs.NA #math.NA #msc:26A33 #msc:65B99 #msc:65M06 #msc:65R10

paper · pdf · doi:10.48550/arxiv.2108.03772

14 pages, 2 figures

arxiv created 2021/08/09 · arxiv updated 2021/08/10

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 Grünwald-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(N²) operations to evaluate for O(h^N) 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 Grünwald-Letnikov type difference scheme is also kept because of the application of filtering. The higher convergence rate is verified through numerical experiments.

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