求解变系数奇异摄动问题的精细积分法
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摘要
将高阶乘法摄动法与子区段消元法结合,提出一种求解一端有边界层的变系数奇异摄动2点边值问题的精细积分方法.首先用一个不大的步长将求解区域均匀离散,然后采用高阶乘法摄动方法求解出每个子区段内的传递矩阵.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,该方程可通过递推消元法高效求解.由于每个子区段内的传递矩阵为一系列指数矩阵之积,可利用精细积分法精确计算,因此该方法具有很高的精度和效率.数值算例证明了方法的有效性.
A precise integration method was presented,which was based on high order perturbation multiplication method and reduction method for variable coefficient singularly perturbed two point boundary value problems.Firstly,the interval was evenly divided by a small step.Secondly,the transfer matrix in every sub-interval was worked out using the high order multiplication perturbation method.Thirdly,through the precise integration relationship of the state parameter of adjacent node,a set of algebraic equations was given and can be efficiently worked out using the reduction method.Since the transfer matrix in every sub-interval is the product of a series of exponential matrices,which can be solved accurately through the precise integration method.Therefore,the method of this paper is of high precision and efficiency.Numerical examples show the validity of the present method.
A precise integration method was presented,which was based on high order perturbation multiplication method and reduction method for variable coefficient singularly perturbed two point boundary value problems.Firstly,the interval was evenly divided by a small step.Secondly,the transfer matrix in every sub-interval was worked out using the high order multiplication perturbation method.Thirdly,through the precise integration relationship of the state parameter of adjacent node,a set of algebraic equations was given and can be efficiently worked out using the reduction method.Since the transfer matrix in every sub-interval is the product of a series of exponential matrices,which can be solved accurately through the precise integration method.Therefore,the method of this paper is of high precision and efficiency.Numerical examples show the validity of the present method.
引文
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[10]向宇,袁丽芸,邹时智,等.求解非线性动力方程的一种齐次扩容精细积分法[J].华中科技大学学报:自然科学版,2007,35(11):109-111.
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[14]谭述君,钟万勰.变系数微分Riccati方程的保辛摄动近似求解[J].大连理工大学学报,2006,46(S1):7-13.
[15]富明慧,蓝林华,陆克浪,等.时变动力系统的高阶乘法摄动方法[J].中国科学:物理学力学天文学,2012,42:185-191.
[16]Fu M H,Cheung M C,Sheshenin S.Precise inte-gration method for solving singular perturbationproblems[J].Applied Mathematics and Mechanics,2010,31(11):1463-1472.
[2]Andargie A,Reddy Y N.Fitted fourth-order tridiago-nal finite differencemethod for singular perturbationproblems[J].Appl Maths Comput,2007,192:90-100.
[3]Stynes M,O′Riordan E.A uniformly accurate finite-element method for a singular-perturbation problem inconservative form[J].SIAM J Numer Anal,1986,23:369-375.
[4]Vigo-Aguiar J,Natesan S.A parallel boundary valuetechniques for singularly perturbed two-point bounda-ry value problems[J].The J Supercomput,2004,27:195-206.
[5]Kadalbajoo M K,Kumar D.Initial value technique forsingularly perturbed two point boundary value prob-lems using an exponentially fitted finite differencescheme[J].Comput Maths Appl,2009,57:1147-1156.
[6]Tirmizi I A,Fazal-i-Haq,Siraj-ul-Islam.Non-polyno-mial spline solution of singularly perturbed boundary-value problems[J].Appl Maths Comput,2008,196:6-16.
[7]Zhong W X,Williams F W.A precise time step inte-gration method[J].J Mech Engin Sci,1994,208(C6):427-430.
[8]Lin J H,Sun D K,Zhong W X,et al.High efficiencycomputation of the variances of structural evolutionaryrandom responses[J].Shock Vibr,2000,7(4):209-216.
[9]Gu Y X,Chen B S,Zhang H W,et al.Precise timeintegration method with dimensional expanding forstructural dynamic equations[J].AIAA Journal,2001,39(12):2394-2399.
[10]向宇,袁丽芸,邹时智,等.求解非线性动力方程的一种齐次扩容精细积分法[J].华中科技大学学报:自然科学版,2007,35(11):109-111.
[11]Gu Y X,Chen B S,Zhang H W,et al.A sensitivityanalysis method for linear and nonlinear transientheat conduction with precise time integration[J].Struct Multidisc Optim,2002,24:23-37.
[12]Zhong W X.Combined method for the solution ofasymmetric Riccati differential equations[J].ComputMeth Appl Mech Engin,2001,191:93-102.
[13]Chen B S,Tong L Y,Gu Y X.Precise time integra-tion for linear two-point boundary value problems[J].Appl Math and Comput,2006,175:182-211.
[14]谭述君,钟万勰.变系数微分Riccati方程的保辛摄动近似求解[J].大连理工大学学报,2006,46(S1):7-13.
[15]富明慧,蓝林华,陆克浪,等.时变动力系统的高阶乘法摄动方法[J].中国科学:物理学力学天文学,2012,42:185-191.
[16]Fu M H,Cheung M C,Sheshenin S.Precise inte-gration method for solving singular perturbationproblems[J].Applied Mathematics and Mechanics,2010,31(11):1463-1472.