A Compact Scheme Combining the Fast Time Stepping Method for Solving 2D Fractional Subdiffusion Equations

Fractal and Fractional

article

Yibin Xu¹  Yanqin Liu²  Xiuling Yin²  Libo Feng³  Zihua Wang² 

[1] School of Mathematics and Statistics, Qilu University of Technology ,(Shandong Academy of Sciences);School of Mathematics and Big Data, Dezhou University;School of Mathematical Sciences, Queensland University of Technology

关键词: time stepping method;    compact difference scheme;    fractional nonlinear subdiffusion equation;    fast algorithm;    correction terms;   

DOI  :  10.3390/fractalfract7020186

学科分类：社会科学、人文和艺术（综合）

来源: mdpi

【 摘 要 】

In this paper, in order to improve the calculation accuracy and efficiency of α-order Caputo fractional derivative (0 < α ≤ 1), we developed a compact scheme combining the fast time stepping method for solving 2D fractional nonlinear subdiffusion equations. In the temporal direction, a time stepping method was applied. It can reach second-order accuracy. In the spatial direction, we utilized the compact difference scheme, which can reach fourth-order accuracy. Some properties of coefficients are given, which are essential for the theoretical analysis. Meanwhile, we rigorously proved the unconditional stability of the proposed scheme and gave the sharp error estimate. To overcome the intensive computation caused by the fractional operators, we combined a fast algorithm, which can reduce the computational complexity from O(N2) to O(Nlog(N)), where N represents the number of time steps. Considering that the solution of the subdiffusion equation is weakly regular in most cases, we added correction terms to ensure that the solution can achieve the optimal convergence accuracy.

【 授权许可】

CC BY