时间分数阶慢扩散方程的一类有效差分方法

数学物理学报 ›› 2018, Vol. 38 ›› Issue (6): 1122-1134.

时间分数阶慢扩散方程的一类有效差分方法

赵雅迪,吴立飞,杨晓忠*(),孙淑珍

华北电力大学数理学院信息与计算研究所北京 102206

收稿日期:2017-04-11 出版日期:2018-12-26 发布日期:2018-12-27
通讯作者: 杨晓忠 E-mail:yxiaozh@ncepu.edu.cn
基金资助:
国家自然科学基金(11371135);中央高校基本科研业务费专项资金(2018MS168)

A Kind of Efficient Difference Method for the Time Fractional Sub-Diffusion Equation

Yadi Zhao,Lifei Wu,Xiaozhong Yang*(),Shuzhen Sun

Institute of Information and Computation, Mathematics and Physics Department, North China Electric Power University, Beijing 102206

Received:2017-04-11 Online:2018-12-26 Published:2018-12-27
Contact: Xiaozhong Yang E-mail:yxiaozh@ncepu.edu.cn
Supported by:
the NSFC(11371135);the Fundamental Research Funds for the Central Universities(2018MS168)

摘要/Abstract

摘要：

对时间分数阶慢扩散方程提出一类数值差分方法：显-隐（Explicit-Implicit，E-I）和隐-显（Implicit-Explicit，I-E）差分方法.它是将古典显式格式与古典隐式格式相结合构造出的一类有效差分格式.理论证明了格式解的存在唯一性，用傅里叶方法证明了格式的稳定性和收敛性.数值试验验证了理论分析，表明E-I格式和I-E格式在具有良好的精度且无条件稳定的情况下，计算速度比隐式格式提高了75%.从而用此格式解决分数阶慢扩散方程是可行的.

关键词: 时间分数阶慢扩散方程, 显-隐(隐-显)差分格式, 稳定性, 收敛性, 数值试验

Abstract:

In this paper, a kind of numerical difference method of explicit-implicit(E-I) scheme and implicit-explicit(I-E) scheme was constructed for solving the time fractional sub-diffusion equation. It is based on the combination of the explicit scheme and implicit scheme. Theoretical analyses have shown that the solution of E-I(I-E) scheme is uniquely solvable. At the same time the stability and convergence of the scheme were proved by the Fourier method. Numerical experiments verified the theoretical analyses and showed the computational efficiency of E-I scheme is 75% higher than the implicit scheme under the premise of unconditional stability and having better accuracy. Therefore it is feasible to use the scheme for solving the time fractional diffusion equation.

Key words: Time fractional sub-fractional equation, Explicit-implicit (implicit-explicit) difference scheme, Stability, Convergence, Numerical experiments

中图分类号:

O241.8

赵雅迪, 吴立飞, 杨晓忠, 孙淑珍. 时间分数阶慢扩散方程的一类有效差分方法[J]. 数学物理学报, 2018, 38(6): 1122-1134.

Yadi Zhao, Lifei Wu, Xiaozhong Yang, Shuzhen Sun. A Kind of Efficient Difference Method for the Time Fractional Sub-Diffusion Equation[J]. Acta mathematica scientia,Series A, 2018, 38(6): 1122-1134.

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