计算高维带弱奇异核发展型方程的交替方向隐式欧拉方法

doi:10.12286/jssx.j2021-0818

计算数学 ›› 2023, Vol. 45 ›› Issue (1): 39-56.DOI: 10.12286/jssx.j2021-0818

计算高维带弱奇异核发展型方程的交替方向隐式欧拉方法

杨雪花, 刘艳玲, 张海湘

湖南工业大学理学院, 株洲 412007

收稿日期:2021-05-22 出版日期:2023-02-14 发布日期:2023-02-13
通讯作者: 张海湘, Email: hassenzhang@163.com
基金资助:
国家自然科学天元基金(12226337,12226340,12126307,12126321),湖南省教育厅科研项目(21B0550),湖南工业大学研究生创新项目(CX2114)资助.

PDF ( 31 ) 引用

杨雪花, 刘艳玲, 张海湘. 计算高维带弱奇异核发展型方程的交替方向隐式欧拉方法[J]. 计算数学, 2023, 45(1): 39-56.

Yang Xuehua, Liu Yanling, Zhang Haixiang. ALTERNATING DIRECTION IMPLICIT EULER METHOD FOR THE HIGH-DIMENSIONAL EVOLUTION EQUATIONS WITH WEAKLY SINGULAR KERNEL[J]. Mathematica Numerica Sinica, 2023, 45(1): 39-56.

ALTERNATING DIRECTION IMPLICIT EULER METHOD FOR THE HIGH-DIMENSIONAL EVOLUTION EQUATIONS WITH WEAKLY SINGULAR KERNEL

Yang Xuehua, Liu Yanling, Zhang Haixiang

College of Sciences, Hunan University of Technology, Zhuzhou 412007, China

Received:2021-05-22 Online:2023-02-14 Published:2023-02-13
Supported by:
The project was supported by the National Key Research and Development Program of China (2019YFC1905301);National Natural Science Foundation of China (22078115,21776108,21690083,22008078).

本文主要研究高维带弱奇异核的发展型方程的交替方向隐式 (ADI) 差分方法. 向后欧拉 (Euler) 方法联立一阶卷积求积公式处理时间方向的离散, 有限差分方法处理空间方向的离散, 并进一步构造了 ADI 全离散差分格式. 然后将二维问题延伸到三维问题, 构造三维空间问题的 ADI 差分格式. 基于离散能量法, 详细证明了全离散格式的稳定性和误差分析. 随后给出了 2 个数值算例, 数值结果进一步验证了时间方向的收敛阶为一阶, 空间方向的收敛阶为二阶, 和理论分析结果一致.

关键词： 发展型方程}, 交替方向隐式方法, 向后Euler法, 离散能量法, 有限差分方法

In this paper, an alternating direction implicit (ADI) difference method is considered for the high dimensional evolution equations with the weak singular kernel. The backward Euler method and the first-order convolution quadrature formula are used in time direction, finite difference method is used in space direction, and then the ADI fully discrete scheme is obtained. The stability and error analysis are proved by the discrete energy method. We extend the two-dimensional problem to the three-dimensional problem and construct the ADI difference scheme for the three-dimensional space problem. Finally, two numerical examples are given. The numerical results show that the convergence order of our scheme is order one in time direction and order two in space direction, which is consistent with the theoretical analysis results.

Key words: Evolution equations, Alternating direction implicit method, Backward Euler method, Discrete energy method, Finite difference methods

MR(2010)主题分类:

65M12
65M06

分享此文:

[1] Friedman A, Shinbrot M. Volterra integral equations in Banach space[J]. Transactions of the American Mathematical Society, 1967, 126(1):131-179.
[2] Heard M L. An Abstract Parabolic Volterra Integrodifferential Equation[J]. SIAM Journal on Mathematical Analysis, 1982, 13(1):81-105.
[3] López-Marcos J C. A difference scheme for a nonlinear partial integro-differential equation[J]. SIAM Journal on Numerical Analysis, 1990, 27(1):20-31.
[4] Tang T. A finite difference scheme for a partial integro-differential equation with a weakly singular kernel[J]. Applied Numerical Mathematics, 1993, 11(4):309-319.
[5] Yanik E G, Fairweather G. Finite element methods for parabolic and hyperbolic partial integrodifferential equations[J]. Nonlinear Analysis, 1988, 12(8):785-809.
[6] Mustapha K, Mustapha H. A second-order accurate numerical method for a semilinear integrodifferential equation with a weakly singular kernel[J]. IMA Journal of Numerical Analysis, 2010, 30(2):555-578.
[7] Chen H B, Xu D. A compact difference scheme for an evolution equation with a weakly singular kernel[J]. Numerical Mathematics-Theory, Methods and Applications, 2012, 5(4):559-572.
[8] Chen H B, Xu D, Cao J L, Zhou J. A formally second order BDF ADI difference scheme for the three-dimensional time-fractional heat equation[J]. International Journal of Computer Mathematics, 2020, 97(5):1100-1117.
[9] 张海湘, 杨雪花, 汤琼. 多项复合型黏弹性波问题的离散奇异卷积方法[J]. 湖南工业大学学报, 2019, 33(3):1-5.
[10] 罗曼, 徐大, 吴珍珍. Crank-Nicolson/sinc方法求解带弱奇异核的偏积分微分方程[J]. 工业技术创新, 2020, 07(5):81-84.
[11] 刘轩宇, 罗鲲, 王皓. 抛物型积分微分方程的新型全离散弱Galerkin有限元法[J]. 四川大学学报(自然科学版), 2020, 57(5):830-840.
[12] 申雪, 王怡昕, 朱爱玲. 二阶线性抛物型积分微分方程的$(r, r-1, r-1)$ 阶弱Galerkin有限元数值模拟[J]. 山东师范大学学报(自然科学版), 2018, 33(3):271-277.
[13] Hu S F, Qiu W L, Chen H B. A backward Euler difference scheme for the integro-differential equations with the multi-term kernels[J]. International Journal of Computer Mathematics, 2020, 97(6):1254-1267.
[14] Qiu W L, Chen H B, Zheng X. An implicit difference scheme and algorithm implementation for the one-dimensional time-fractional Burgers equations[J]. Mathematics and Computers in Simulation, 2019, 166:298-314.
[15] Peaceman D W, Rachford H H. The numerical solution of parabolic and elliptic differential equations[J]. Journal of the Society for industrial and Applied Mthematics, 1955, 3(1):28-41.
[16] Cheng H, Lin P, Sheng Q, Tan R C E. Solving degenerate reaction-diffusion equations via variable step Peaceman-Rachford splitting[J]. SIAM Journal on Scientific Computing, 2004, 25(4):1273-1292.
[17] Karaa S. A high-order compact ADI method for parabolic problems with variable coefficients[J]. Numerical Methods for Partial Differential Equations, 2006, 22:983-993.
[18] 王文兵, 周辉, 马良, 程引会, 刘逸飞. 共形单步交替方向隐式时域有限差分方法及其改进[J]. 强激光与粒子束, 2018, 30(7):110-117.
[19] 高林, 谢国大, 黄志祥, 许杰, 吴先良. 3维交替方向隐式时域有限差分算法及其应用[J]. 安徽大学学报(自然科学版), 2020, 44(4):52-59.
[20] Zhang Y N, Sun Z Z. Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation[J]. Journal of Computational Physics, 2011, 230(24):8713-8728.
[21] Zhang Y N, Sun Z Z, Zhao X. Compact alternating direction implicit scheme for the twodimensional fractional diffusion-wave equation[J]. SIAM Journal on Numerical Analysis, 2012, 50:1535-1555.
[22] Li L M, Xu D. Alternating direction implicit-Euler method for the two-dimensional fractional evolution equation[J]. Journal of Computational Physics, 2013, 236:157-168.
[23] Lin Y P, Thomée V, Wahlbin L B. Ritz-Volterra projections to finite element spaces and applications to integro differential and related equations[J]. SIAM Journal on Numerical Analysis, 1991, 28:1047-1070.
[24] McLean W, Thomée V. Numerical solution of an evolution equation with a positive-type memory term[J]. The ANZIAM Journal, 1993, 35(1):23-70.
[25] Sloan I H, Thomée V. Time discretization of an integro-differential equation of parabolic type[J]. SIAM Journal on Numerical Analysis, 1986, 23(5):1052-1061.
[26] Xu D. The global behavior of time discretization for an abstract Volterra equation in Hilbert space[J]. Calcolo, 1997, 34(1):93-116.

[1]	北京应用物理与计算数学研究所. 周毓麟先生在计算数学领域的成就与贡献——纪念周毓麟院士百年诞辰[J]. 计算数学, 2023, 45(1): 3-7.
[2]	付姚姚, 曹礼群. 矩阵形式二次修正Maxwell-Dirac系统的多尺度算法[J]. 计算数学, 2019, 41(4): 419-439.
[3]	王坤, 张扬, 郭瑞. Helmholtz方程有限差分方法概述[J]. 计算数学, 2018, 40(2): 171-190.
[4]	汤华中,邬华谟. 离散速度动力学方程组的数值方法研究 Ⅰ.半隐式差分格式[J]. 计算数学, 2000, 22(2): 183-190.

阅读次数

全文

摘要

计算高维带弱奇异核发展型方程的交替方向隐式欧拉方法

ALTERNATING DIRECTION IMPLICIT EULER METHOD FOR THE HIGH-DIMENSIONAL EVOLUTION EQUATIONS WITH WEAKLY SINGULAR KERNEL

扫码分享