A fast convolution method for the fractional Laplacian in $\mathbb{R}$

TL;DR

This paper introduces a fast, second-order accurate numerical method for approximating the fractional Laplacian on the real line, utilizing domain mapping, a modified midpoint rule, and FFT-based convolution for efficiency.

Contribution

The authors develop a novel convolution-based approach that efficiently computes the fractional Laplacian without domain truncation, with proven second-order accuracy and demonstrated application to fractional Schrödinger equations.

Findings

Method achieves second-order accuracy.

Efficient computation via FFT-based convolution.

Successfully applied to fractional Schrödinger equation.

Abstract

In this article, we develop a new method to approximate numerically the fractional Laplacian of functions defined on $\mathbb R$, as well as some more general singular integrals. After mapping $\mathbb R$ into a finite interval, we discretize the integral operator using a modified midpoint rule. The result of this procedure can be cast as a discrete convolution, which can be evaluated efficiently using the Fast-Fourier Transform (FFT). The method provides an efficient, second order accurate, approximation to the fractional Laplacian, without the need to truncate the domain. We first prove that the method gives a second-order approximation for the fractional Laplacian and other related singular integrals; then, we detail the implementation of the method using the fast convolution, and give numerical examples that support its efficacy and efficiency; finally, as an example of its applicability to an evolution problem, we employ the method for the discretization of the nonlocal part of the one-dimensional cubic fractional Schr\"odinger equation in the focusing case.