Fast Fourier-like Mapped Chebyshev Spectral-Galerkin Methods for PDEs with Integral Fractional Laplacian in Unbounded Domains

Abstract

In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving integral fractional Laplacian in ${\mathbb{R}}^d$, which is built upon two essential components: (i) the Dunford-Taylor formulation of the fractional Laplacian; and (ii) Fourier-like bi-orthogonal mapped Chebyshev functions (MCFs) as basis functions. As a result, the fractional Laplacian can be fully diagonalised, and the complexity of solving an elliptic fractional PDE is quasi-optimal, i.e., $O((N{\log }_{2}N{)}^d)$ with $N$ being the number of modes in each spatial direction. Ample numerical tests for various decaying exact solutions show that the convergence of the fast solver perfectly matches the order of theoretical error estimates. With a suitable time-discretization, the fast solver can be directly applied to a large class of nonlinear fractional PDEs. As an example, we solve the fractional nonlinear Schr{ö}dinger equation by using the fourth-order time-splitting method together with the proposed MCF-spectral-Galerkin method.

arXiv commentThis article has a total of 24 pages and including 22 figures

References in corpus (7)

• The scaling laws of human travel
• A short FE implementation for a 2d homogeneous Dirichlet problem of a Fractional Laplacian
• Numerical study of fractional Nonlinear Schrödinger equations
• Aspects of an adaptive finite element method for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver
• What Is the Fractional Laplacian?
• Computing the ground and first excited states of the fractional Schrodinger equation in an infinite potential well
• Jacobi-Galerkin spectral method for eigenvalue problems of Riesz fractional differential equations

Cited by in corpus (1)

• Numerical methods for nonlocal and fractional models