A study on fractional centered difference scheme for high-dimensional integral fractional Laplacian operator with {ω}-circulant preconditioner

Lot-Kei Chou, Wei Qu, Yuan-Yuan Huang and Siu-Long Lei

Mathematics and Computers in Simulation (MATCOM), 2025, vol. 231, issue C, 128-143

Abstract: Fundamental properties for the coefficients of a second-order finite difference approximation of the fractional Laplacian in d≥2 dimensions are derived in this paper. The obtained decay rate of the coefficients implies that the coefficients (with no closed-form expression) can be approximated via d-dimensional inverse fast Fourier transform of size K per dimension with accuracy O(K−d−α), where α∈(0,2) is the order of the fractional Laplacian. For solving fractional partial differential equations on regular grids, the coefficient matrix is a d-level Toeplitz matrix that can be preconditioned by the d-level {ω}-circulant matrix. Here, a spectral analysis of the difference matrix is derived. The purpose of this work is also to justify some observations presented by Hao et al. (2021). Numerical experiments in two-dimension and three-dimension illustrate that {ω}-circulant preconditioner has better performance over T. Chan’s circulant preconditioner.

Keywords: High-dimensional integral fractional Laplacian operator; Fractional centered difference; Krylov subspace methods; Generating function; d-level {ω}-circulant preconditioner (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:231:y:2025:i:c:p:128-143

DOI: 10.1016/j.matcom.2024.12.002

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