High-order fractional central difference method for multi-dimensional integral fractional Laplacian and its applications

In order to change the current situation where the numerical accuracy of existing fractional central difference (FCD) methods for integral fractional Laplacian (IFL) does not exceed second- order no matter how smooth the solution is. A simple and easy-to-implement high-order FCD scheme on uniform meshes is proposed for multi-dimensional IFL. The new generating functions are constructed to accommodate the discretization of the classical and integral fractional Laplacian in a unified framework. Compared to other finite difference methods, the weights or coefficients of high-order FCD can be easily calculated using fast Fourier transform (FFT). And our scheme inherits the merits of the existing FCD method, such as the FFT efficiency and low storage costs. Furthermore, it can be extended to arbitrary bounded domains via the fictitious domain method, which allow the FFT algorithm. The stability and convergence analysis of our method are given in solving the fractional Poisson equations. Extensive numerical experiments are provided to verify our theoretical results. The new method can even achieve eighth order accuracy when the solution is sufficiently smooth. Utilizing its efficiency, our method is applied to solve the time-dependent problems with IFL that included of fractional Schr & ouml;dinger equation, fractional Allen-Cahn equation and anomalous diffusion problems, some new observations are discovered from our numerical simulations.