Mathematics and Statistics Faculty Research & Creative Works

Accurate Numerical Methods for Two and Three Dimensional Integral Fractional Laplacian with Applications

Abstract

In this paper, we propose accurate and efficient finite difference methods to discretize the two- and three-dimensional fractional Laplacian (-Δ)α/2 (0 < α < 2) in hypersingular integral form. The proposed methods provide a fractional analogue of the central difference schemes to the fractional Laplacian. As α → 2-, they collapse to the central difference schemes of the classical Laplace operator −Δ. We prove that our methods are consistent if 𝑢 ∈ C[α], α –[α]+ ε (ℝd), and the local truncation error is 𝓞 (hε), with ε > 0 a small constant and [· } denoting the floor function. If 𝑢 ∈ C2+[α], α -[ α]+ ε (ℝd), they can achieve the second order of accuracy for any α ∈ (0,2). These results hold for any dimension d ≥ 1 and thus improve the existing error estimates of the one-dimensional cases in the literature. Extensive numerical experiments are provided and confirm our analytical results. We then apply our method to solve the fractional Poisson problems and the fractional Allen-Cahn equations. Numerical simulations suggest that to achieve the second order of accuracy, the solution of the fractional Poisson problem should at most satisfy 𝑢 ∈ C1,1 (ℝd). One merit of our methods is that they yield a multilevel Toeplitz stiffness matrix, an appealing property for the development of fast algorithms via the fast Fourier transform (FFT). Our studies of the two- and three-dimensional fractional Allen-Cahn equations demonstrate the efficiency of our methods in solving the high-dimensional fractional problems.

Department(s)

Mathematics and Statistics

Research Center/Lab(s)

Center for High Performance Computing Research

This work was supported by the US National Science Foundation under Grant Number DMS-1620465.

Keywords and Phrases

Consistency; Finite difference methods; Fractional Allan-Cahn equation; Fractional Poisson equation; Integral fractional Laplacian; Montgomery identity

0045-7825

Document Type

Article - Journal

Citation

text

English