Fractional centered difference scheme for high-dimensional integral fractional Laplacian

Abstract

In this work we study the finite difference method for the fractional diffusion equation with high-dimensional hyper-singular integral fractional Laplacian. We first propose a simple and easy-to-implement discrete approximation, i.e., fractional centered difference scheme with γth-order (γ≤2) convergence for the fractional operator. Based on the established approximation, we then construct a finite difference scheme to solve fractional diffusion equations and analyze the stability and convergence in discrete energy norm (0<α≤2) and in discrete maximum norm (1<α≤2). We further present a fast solver for the linear system which is obtained by discretization on rectangular domain and use the fictitious domain method to extend the fast solver to the non-rectangular one. Several numerical results are provided to support our theoretical results.