A fully discrete GL-ADI scheme for 2D time-fractional reaction-subdiffusion equation

IF 4.3 3区材料科学 Q1 ENGINEERING, ELECTRICAL & ELECTRONIC ACS Applied Electronic Materials Pub Date : 2024-10-24 DOI:10.1016/j.amc.2024.129147

Yubing Jiang , Hu Chen , Chaobao Huang , Jian Wang

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Abstract

Alternating direction implicit (ADI) difference method for solving a 2D reaction-subdiffusion equation whose solution behaves a weak singularity at

t = 0

is studied in this paper. A Grünwald-Letnikov (GL) approximation is used for the discretization of Caputo fractional derivative (of order α, with

0 < α < 1

) on a uniform mesh. Stability and convergence of the fully discrete ADI scheme are rigorously established. With the help of a discrete fractional Gronwall inequality, we get the sharp error estimate. The stability in

L^{2}

norm and the convergence of the GL-ADI scheme are strictly proved, where the convergent order is

O (τ t_{s}^{α - 1} + τ^{2 α} + h_{1}^{2} + h_{2}^{2})

. Numerical experiments are given to verify the theoretical analysis.

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二维时间分数反应-次扩散方程的完全离散 GL-ADI 方案

本文研究了交替方向隐式（ADI）差分法求解二维反应-次扩散方程，该方程的解在 t=0 处表现为弱奇点。在均匀网格上对 Caputo 分数导数（阶数 α，0<α<1）的离散化采用了 Grünwald-Letnikov (GL) 近似方法。严格确定了完全离散 ADI 方案的稳定性和收敛性。在离散分式 Gronwall 不等式的帮助下，我们得到了尖锐的误差估计。严格证明了 GL-ADI 方案在 L2 规范下的稳定性和收敛性，其中收敛阶数为 O(τtsα-1+τ2α+h12+h22)。数值实验验证了理论分析。

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