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

2019/08/27 by Changtao Sheng, Jie Shen, Sheng, Changtao +7 · 1 citation
Mathematics · #41A05 #41A25 #65M70 #65N35 #Differential Equations and Numerical Methods #FOS: Mathematics #Fractional Differential Equations Solutions #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.1908.10029

openalex publication_date 2019/08/27 · openalex created_date 2019/09/05 · openalex updated_date 2026/07/28

Abstract

In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving integral fractional Laplacian in 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((Nlog₂N)^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 Schrödinger equation by using the fourth-order time-splitting method together with the proposed MCF-spectral-Galerkin method.

Citations

• Numerical Methods for the Fractional Laplacian: A Finite Difference-Quadrature Approach
• Reformulation of elasticity theory for discontinuities and long-range forces
• Finite Element Method for the Space and Time Fractional Fokker–Planck Equation
• The random walk's guide to anomalous diffusion: a fractional dynamics approach

Cited by

• Efficient implicit-explicit sparse stochastic method for high dimensional semi-linear nonlocal diffusion equations

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