二维分数阶发展型方程的正式的二阶BDF交替方向隐式紧致差分格式

数学物理学报 ›› 2017, Vol. 37 ›› Issue (5): 976-992.

• 论文 • 上一篇

二维分数阶发展型方程的正式的二阶BDF交替方向隐式紧致差分格式

陈红斌^1,2, 甘四清¹, 徐大³, 彭玉龙²

1. 中南大学数学与统计学院长沙 410083;
2. 中南林业科技大学理学院长沙 410004;
3. 湖南师范大学数学与计算机科学学院, 高性能计算与随机信息处理省部共建教育部重点实验室长沙 410081

收稿日期:2016-08-03 修回日期:2017-02-03 出版日期:2017-10-26 发布日期:2017-10-26
作者简介:陈红斌,E-mail:chb8080@eyou.com;甘四清,E-mail:sqgan@csu.edu.cn;徐大,E-mail:daxu@hunnu.edu.cn
基金资助:
国家自然科学基金（11571373，11671131）和湖南省重点学科建设项目

A Formally Second-Order BDF Compact ADI Difference Scheme for the Two-Dimensional Fractional Evolution Equation

Chen Hongbin^1,2, Gan Siqing¹, Xu Da³, Peng Yulong²

1. School of Mathematics and Statistics, Central South University, Changsha 410083;
2. College of Science, Central South University of Forestry and Technology, Changsha 410004;
3. HPCSIP Key Laboratory, Ministry of Education, College of Mathematics and Computer Science, Hunan Normal University, Changsha 410081

Received:2016-08-03 Revised:2017-02-03 Online:2017-10-26 Published:2017-10-26
Supported by:
Supported by the NSFC (11571373, 11671131) and the Construct Program of the Key Discipline in Hunan Province

摘要/Abstract

摘要： 该文将研究二维分数阶发展型方程的正式的二阶向后微分公式（BDF）的交替方向隐式（ADI）紧致差分格式.在时间方向上用二阶向后微分公式离散一阶时间导数，积分项用二阶卷积求积公式近似，在空间方向上用四阶精度的紧致差分离散二阶空间导数得到全离散紧致差分格式.基于与卷积求积相对应的实二次型的非负性，利用能量方法研究了差分格式的稳定性和收敛性，理论结果表明紧致差分格式的收敛阶为O（k^α+1+h₁⁴+h₂⁴），其中k为时间步长，h₁和h₂分别是空间x和y方向的步长.最后，数值算例验证了理论分析的正确性.

关键词: 二维分数阶发展型方程, 二阶BDF ADI, 紧致差分格式, 稳定性, 收敛性

Abstract: In this paper, we will consider a formally second-order backward differentiation formula (BDF) compact alternating direction implicit (ADI) difference scheme for the twodimensional fractional evolution equation. To obtain a fully discrete implicit scheme, the integral term is treated by means of the second order convolution quadrature suggested by Lubich and the second order space derivatives are approximated by the fourth-order accuracy compact finite difference. The stability and convergence of the compact difference scheme in a new norm are proved by the energy method. The verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature. A numerical experiment in total agreement with our analysis is reported.

Key words: Two-dimensional fractional evolution equation, Second-order BDF ADI, Compact difference scheme, Stability, Convergence

中图分类号:

O241.82

陈红斌, 甘四清, 徐大, 彭玉龙. 二维分数阶发展型方程的正式的二阶BDF交替方向隐式紧致差分格式[J]. 数学物理学报, 2017, 37(5): 976-992.

Chen Hongbin, Gan Siqing, Xu Da, Peng Yulong. A Formally Second-Order BDF Compact ADI Difference Scheme for the Two-Dimensional Fractional Evolution Equation[J]. Acta mathematica scientia,Series A, 2017, 37(5): 976-992.

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二维分数阶发展型方程的正式的二阶BDF交替方向隐式紧致差分格式

A Formally Second-Order BDF Compact ADI Difference Scheme for the Two-Dimensional Fractional Evolution Equation

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