A Novel and Simple Spectral Method for Nonlocal Pdes with the Fractional Laplacian

Abstract

We propose a novel and simple spectral method based on the semi-discrete Fourier transforms to discretize the fractional Laplacian [Formula presented]. Numerical analysis and experiments are provided to study its performance. Our method has the same symbol |ξ|α as the fractional Laplacian [Formula presented] at the discrete level, and thus it can be viewed as the exact discrete analogue of the fractional Laplacian. This unique feature distinguishes our method from other existing methods for the fractional Laplacian. Note that our method is different from the Fourier pseudospectral methods in the literature which are usually limited to periodic boundary conditions (see Remark 1.1). Numerical analysis shows that our method can achieve a spectral accuracy. The stability and convergence of our method in solving the fractional Poisson equations were analyzed. Our scheme yields a multilevel Toeplitz stiffness matrix, and thus fast algorithms can be developed for efficient matrix-vector multiplications. The computational complexity is O(2Nlog⁡(2N)), and the memory storage is O(N) with N the total number of points. Extensive numerical experiments verify our analytical results and demonstrate the effectiveness of our method in solving various problems.

Department(s)

Mathematics and Statistics

Comments

National Science Foundation, Grant DMS–1913293

Keywords and Phrases

Anomalous diffusion; Fractional Laplacian; Fractional Poisson equations; Hypergeometric functions; Semi-discrete Fourier transform; Spectral methods

International Standard Serial Number (ISSN)

0898-1221

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 2024 Elsevier, All rights reserved.

Publication Date

15 Aug 2024

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