求解奇异摄动边值问题的精细积分法

富明慧; 张文志; S·V· 薛申宁

doi:10.3879/j.issn.1000-0887.2010.11.011

求解奇异摄动边值问题的精细积分法

doi: 10.3879/j.issn.1000-0887.2010.11.011

中山大学应用力学与工程系,广州 510275;2.莫斯科大学力学数学系,莫斯科 119992, 俄罗斯

基金项目: 国家自然科学基金资助项目(10672194);中俄NSFC-RFBR资助项目(10811120012)

详细信息

作者简介:
富明慧(1966- ),男,黑龙江人,满族,教授,博士,博士生导师(联系人.E-mail:stsfmh@mail.sysu.edu.cn).

中图分类号: O175.8; O241.81
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- 被引次数: 0
出版历程
- 收稿日期: 1900-01-01
- 修回日期: 2010-09-15
- 刊出日期: 2010-11-15

Precise Integration Method for Solving Singular Perturbation Problems

Department of Applied Mechanics and Engineering, Sun Yat-sen University, Guangzhou 510275, P. R. China;

摘要

摘要: 提出了一种求解一端有边界层的奇异摄动边值问题的精细方法．首先将求解区域均匀离散，由状态参量在相邻节点间的精细积分关系式确定一组代数方程，并将其写成矩阵形式．代入边界条件后，该代数方程组的系数矩阵可化为块三对角形式，针对这一特性，给出了一种高效递推消元方法．由于在离散过程中，精细积分关系式不会引入离散误差，故所提出的方法具有极高的精度．数值算例充分证明了所提出方法的有效性．
- 奇异摄动问题 /
- 一阶常微分方程组 /
- 两点边值问题 /
- 精细积分法 /
- 递推方法
Abstract: A precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end was presented. Firstly,the interval was divided evenly,then a set of algebraic equations in the form of matrix by the precise integration relationship of each segment was given. Substituting the boundary conditions into the algebraic equations,the coefficient matrix could be transformed to the form of block tridiagonal matrix. Combining the special nature of the problem,an efficient reduction method for singular perturbation problems was given. Since the precise integration relationship gives no discrete error in the discrete process,the present method has very high precision. Numerical examples show the validity of the present method.
- singular perturbation problems /
- first-order ordinary differential equations /
- two point boundary value problems /
- precise integration method /
- reduction method

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参考文献(22)

[1]	苏煜城. 奇异摄动中的边界层校正法[M]. 上海: 上海科学技术出版社, 1983.
[2]	Kadalbajoo M K, Reddy Y N. Asymptotic and numerical analysis of singular perturbation problems: a survey[J]. Applied Mathematics and Computation, 1989, 30(3): 223-259. doi: 10.1016/0096-3003(89)90054-4
[3]	Chawla M M. A fourth-order tridiagonal finite difference method for general non-linear two-point boundary value problems with mixed boundary conditions[J]. IMA J Appl Math, 1978, 21(1): 83-93. doi: 10.1093/imamat/21.1.83
[4]	苏煜城, 吴启光. 奇异摄动问题数值方法引论[M]. 重庆: 重庆出版社, 1991.
[5]	Andargie A, Reddy Y N. Fitted fourth-order tridiagonal finite difference method for singular perturbation problems[J]. Applied Mathematics and Computation, 2007, 192(1): 90-100. doi: 10.1016/j.amc.2007.02.123
[6]	Stynes M, O’Riordan E. A uniformly accurate finite-element method for a singular-perturbation problem in conservative form[J]. SIAM Journal on Numerical Analysis, 1986, 23(2): 369-375. doi: 10.1137/0723024
[7]	Vigo-Aguiar J, Natesan S. A parallel boundary value technique for singularly perturbed two-point boundary value problems[J]. The Journal of Supercomputing, 2004, 27(2): 195-206. doi: 10.1023/B:SUPE.0000009322.23950.53
[8]	Reddy Y N, Chakravarthy P P. An initial-value approach solving for singularly perturbed two-point boundary value problems[J]. Applied Mathematics and Computation, 2004, 155(1): 95-110. doi: 10.1016/S0096-3003(03)00763-X
[9]	Kadalbajoo M K, Kumar D. Initial value technique for singularly perturbed two point boundary value problems using an exponentially fitted finite difference scheme[J]. Computers and Mathematics With Applications, 2009, 57(7): 1147-1156. doi: 10.1016/j.camwa.2009.01.010
[10]	Aziz T, Khan A. A spline method for second-order singularly perturbed boundary-value problems[J]. Journal of Computational and Applied Mathematics, 2002, 147(2): 445-452. doi: 10.1016/S0377-0427(02)00479-X
[11]	Tirmizi I A, Fazal-i-Haq, Siraj-ul-Islam. Non-polynomial spline solution of singularly perturbed boundary-value problems[J]. Applied Mathematics and Computation, 2008, 196(1): 6-16. doi: 10.1016/j.amc.2007.05.029
[12]	Zhong W X, Williams F W. A precise time step integration method[J]. Proceedings of the Institute of Mechanical Engineers Part C, Journal of Mechanical Engineering Science, 1994, 208(C6): 427-430. doi: 10.1243/PIME_PROC_1994_208_148_02
[13]	Lin J H, Sun D K, Zhong W X, Zhang W S. High efficiency computation of the variances of structural evolutionary random responses[J]. Shock and Vibration, 2000, 7(4): 209-216.
[14]	Gu Y X, Chen B S, Zhang H W, Grandhi R V. A sensitivity analysis method for linear and nonlinear transient heat conduction with precise time integration[J]. Structural and Multidisciplinary Optimization, 2002, 24(1): 23-37. doi: 10.1007/s00158-002-0211-5
[15]	Zhang H W, Zhang X W, Chen J S. A new algorithm for numerical solution of dynamic elastic-plastic hardening and softening problems[J]. Computers and Structures, 2003, 81(17): 1739-1749. doi: 10.1016/S0045-7949(03)00167-6
[16]	Zhong W X. Combined method for the solution of asymmetric Riccati differential equations[J]. Computer Methods in Applied Mechanics and Engineering, 2001, 191(1/2): 93-102. doi: 10.1016/S0045-7825(01)00246-8
[17]	Chen B S, Tong L Y, Gu Y X. Precise time integration for linear two-point boundary value problems[J]. Applied Mathematics and Computation, 2006, 175(1): 182-211. doi: 10.1016/j.amc.2005.08.001
[18]	富明慧, 林敬华. 一类指数矩阵函数及其应用[J]. 力学学报, 2009, 41(5): 808-814.
[19]	谭述君, 钟万勰. 非齐次动力方程Duhamel项的精细积分法[J]. 力学学报, 2007, 39(3): 374-381.
[20]	Wang M F, Zhou X Y. Renewal precise time step integration method of structural dynamic analysis[J]. Acta Mechanica Sinica, 2004, 36(2): 191-195.
[21]	任传波, 贺光宗, 李忠芳. 结构动力学精细积分的一种高精度通用计算格式[J]. 机械科学与技术, 2005, 24(12): 1507-1509.
[22]	富明慧，梁华力. 一种改进的精细-龙格库塔法[J]. 中山大学学报(自然科学版), 2009, 48(5): 1-5.