Fast Implicit Difference Schemes for Time-Space Fractional Diffusion Equations with the Integral Fractional Laplacian


In this paper, we develop two fast implicit difference schemes for solving a class of variable-coefficient time-space fractional diffusion equations with integral fractional Laplacian (IFL). The proposed schemes utilize the graded L1 formula for the Caputo fractional derivative and a special finite difference discretization for IFL, where the graded mesh can capture the model problem with a weak singularity at initial time. The stability and convergence are rigorously proved via the M-matrix analysis, which is from the spatial discretized matrix of IFL. Moreover, the proposed schemes use the fast sum-of-exponential approximation and Toeplitz matrix algorithms to reduce the computational cost for the nonlocal property of time and space fractional derivatives, respectively. The fast schemes greatly reduce the computational work of solving the discretized linear systems from π’ͺ(MN3 + M2N) by a direct solver to π’ͺ(MN(log N+ Nexp)) per preconditioned Krylov subspace iteration and a memory requirement from π’ͺ(MN2) to π’ͺ(NNexp), where N and (Nexp β‰ͺ) M are the number of spatial and temporal grid nodes, respectively. The spectrum of preconditioned matrix is also given for ensuring the acceleration benefit of circulant preconditioners. Finally, numerical results are presented to show the utility of the proposed methods.


Mathematics and Statistics

Research Center/Lab(s)

Center for High Performance Computing Research

Keywords and Phrases

Caputo derivative; circulant preconditioner; fractional diffusion equations; integral fractional Laplacian; Krylov subspace solvers

International Standard Serial Number (ISSN)

0170-4214; 1099-1476

Document Type

Article - Journal

Document Version


File Type





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Publication Date

15 Jan 2021