A Fast Temporal Second Order Difference Scheme for the Fractional Sub-Diffusion Equations on One Dimensional Space Unbounded Domain
Year: 2023
Author: Ren-Jun Qi, Zhi-Zhong Sun
Journal of Mathematical Study, Vol. 56 (2023), Iss. 2 : pp. 173–205
Abstract
The numerical solution of the fractional sub-diffusion equations on one dimensional space unbounded domain is considered. Based on the high-order local artificial boundary conditions proposed in [Zhang W., et al., J. Math. Study., 2017, 50(1): 28-53], the original space unbounded problem can be reformulated to an initial-boundary value problem on a bounded computational domain. By Alikhanov’s $L2-1_σ$ formula and sum-of-exponentials approximation, a fast temporal second order difference scheme for the reduced problem is presented. The unique solvability, stability and convergence order $O(τ^2+h^2 )$ of the proposed method are proved by means of energy method, where $τ$ and $h$ denote the time and space step sizes, respectively. Some numerical examples are included to validate the theoretical results. To the best of our knowledge, this is the first work that combines the high order numerical method with the artificial boundary method for the time fractional diffusion problems on spatial unbounded domains.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jms.v56n2.23.05
Journal of Mathematical Study, Vol. 56 (2023), Iss. 2 : pp. 173–205
Published online: 2023-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 33
Keywords: Fractional sub-diffusion equations space unbounded domain high-order local artificial boundary conditions difference scheme fast algorithm energy method.