A Fast Algorithm for Solving the Space-Time Fractional Diffusion Equation

Abstract

In this paper, we propose a fast algorithm for efficient and accurate solution of the space-time fractional diffusion equations defined in a rectangular domain. The spatial discretization is done by using the central finite difference scheme and matrix transfer technique. Due to its nonlocality, numerical discretization of the spectral fractional Laplacian (−Δ)sα/2 results in a large dense matrix. This causes considerable challenges not only for storing the matrix but also for computing matrix-vector products in practice. By utilizing the compact structure of the discrete system and the discrete sine transform, our algorithm avoids to store the large matrix from discretizing the nonlocal operator and also significantly reduces the computational costs. We then use the Laplace transform method for time integration of the semi-discretized system and a weighted trapezoidal method to numerically compute the convolutions needed in the resulting scheme. Various experiments are presented to demonstrate the efficiency and accuracy of our method.

Department(s)

Mathematics and Statistics

Research Center/Lab(s)

Center for High Performance Computing Research

Keywords and Phrases

Caputo fractional derivative; Discrete sine transform; Fractional diffusion equation; Matrix transfer method; Matrix-vector product; Spectral fractional Laplacian

International Standard Serial Number (ISSN)

0898-1221

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 2018 Elsevier Ltd, All rights reserved.

Publication Date

01 Mar 2018

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