On Generalisation of Isotropic Central Difference for Higher Order Approximation of Fractional Laplacian

TL;DR

This paper investigates the generalisation of central difference methods for fractional Laplacian approximation, proposing new quadrature and lattice Boltzmann techniques to improve convergence and error isotropy.

Contribution

It introduces a composite quadrature rule and a higher order lattice Boltzmann method for more accurate and isotropic fractional Laplacian approximation.

Findings

Fast Fourier transform-based stencils hinder convergence.

Proposed methods achieve higher order convergence and error isotropy.

Numerical experiments confirm the effectiveness of the new approaches.

Abstract

The study of generalising the central difference for integer order Laplacian to fractional order is discussed in this paper. Analysis shows that, in contrary to the conclusion of a previous study, difference stencils evaluated through fast Fourier transform prevents the convergence of the solution of fractional Laplacian. We propose a composite quadrature rule in order to efficiently evaluate the stencil coefficients with the required convergence rate in order to guarantee convergence of the solution. Furthermore, we propose the use of generalised higher order lattice Boltzmann method to generate stencils which can approximate fractional Laplacian with higher order convergence speed and error isotropy. We also review the formulation of the lattice Boltzmann method and discuss the explicit sparse solution formulated using Smolyak's algorithm, as well as the method for the evaluation of the Hermite polynomials for efficient generation of the higher order stencils. Numerical experiments are carried out to verify the error analysis and formulations.