TY - JOUR
T1 - A Fast Temporal Second Order Difference Scheme for the Fractional Sub-Diffusion Equations on One Dimensional Space Unbounded Domain
AU - Qi , Ren-Jun
AU - Sun , Zhi-Zhong
JO - Journal of Mathematical Study
VL - 2
SP - 173
EP - 205
PY - 2023
DA - 2023/06
SN - 56
DO - http://doi.org/10.4208/jms.v56n2.23.05
UR - https://global-sci.org/intro/article_detail/jms/21837.html
KW - Fractional sub-diffusion equations, space unbounded domain, high-order local artificial boundary conditions, difference scheme, fast algorithm, energy method.
AB - <p style="text-align: justify;">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.</p>