刚性延迟系统组合RK-Rosenbrock方法的稳定性分析
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摘要
刚性延迟微分方程广泛地出现于自动控制、电力工程、生态学等领域。目前,国内外文献大都集中对求解这类问题的Runge-Kutta方法、Rosenbrock方法、线性多步法及更一般的线性方法进行研究。在实际问题中,一些刚性延迟系统往往可分解为两个耦合的子系统:刚性延迟子系统和非刚性延迟子系统,自然可采用组合方法进行求解,且往往更为高效。因此,针对求解刚性延迟微分方程的组合方法的研究具有重要的理论和实际意义。
本文第二章介绍了三种求解刚性微分方程和刚性延迟微分方程的Rosenbrock型方法;第三章对求解刚性延迟微分方程的组合RK-Rosenbrock方法的稳定性进行了分析,给出了求解该方程的组合RK-Rosenbrock方法GP-稳定的充要条件;第四章对求解刚性延迟微分方程的组合连续RK-Rosenbrock方法的稳定性进行了分析,给出了求解该方程的组合连续RK-Rosenbrock方法GP-稳定的充要条件;第五章对求解刚性延迟微分方程的连续Rosenbrock方法和连续并行Rosenbrock方法的稳定性进行了分析,利用B-级数理论得到了方法的阶条件方程,且构造了一类方法,并证明了该方法具有GP-稳定性;第六章通过数值试验验证了Rosenbrock型方法对刚性延迟系统的有效性。
Delay differential equations(DDEs) arise widely in many fields such as control engineering,power engineering,ecology etc.In recent years,much discussion about Runge-Kutta methods,Rosenbrock methods,linear multistep methods and general linear methods for DDEs is found in majority relevant literatures.In the practical application,stiff delay systems could be often partitioned into two coupling subsystems:stiff delay subsystems and non-stiff subsystems,thus we can use combined methods to solve these systems,and they are often more efficient. Therefore,it is significant in theory and practice to study the combined methods for stiff DDEs.
In Chapter 2,we present three kinds of Rosenbrock methods for stiff differential equations and stiff delay differential equations.
In Chapter 3,we analyze the stability properties of combined RK-Rosenbrock methods for stiff delay systems,and give the sufficient and necessary conditions under which the combined RK-Rosenbrock methods are GP-stable.
In Chapter 4,we analyze the stability properties of combined continuous RK-Rosenbrock methods for stiff delay systems,get the methods' order condition equations,and give the sufficient and necessary conditions under which the combined continuous RK-Rosenbrock methods are GP-stable.
In Chapter 5,we analyze the stability properties of continuous Rosenbrock methods and continuous parallel Rosenbrock methods for stiff delay differential equations,use B-series to get the order conditions,and give the sufficient and necessary conditions under which the methods are GP-stable.Numerical experiments show that the methods are efficient.
本文第二章介绍了三种求解刚性微分方程和刚性延迟微分方程的Rosenbrock型方法;第三章对求解刚性延迟微分方程的组合RK-Rosenbrock方法的稳定性进行了分析,给出了求解该方程的组合RK-Rosenbrock方法GP-稳定的充要条件;第四章对求解刚性延迟微分方程的组合连续RK-Rosenbrock方法的稳定性进行了分析,给出了求解该方程的组合连续RK-Rosenbrock方法GP-稳定的充要条件;第五章对求解刚性延迟微分方程的连续Rosenbrock方法和连续并行Rosenbrock方法的稳定性进行了分析,利用B-级数理论得到了方法的阶条件方程,且构造了一类方法,并证明了该方法具有GP-稳定性;第六章通过数值试验验证了Rosenbrock型方法对刚性延迟系统的有效性。
Delay differential equations(DDEs) arise widely in many fields such as control engineering,power engineering,ecology etc.In recent years,much discussion about Runge-Kutta methods,Rosenbrock methods,linear multistep methods and general linear methods for DDEs is found in majority relevant literatures.In the practical application,stiff delay systems could be often partitioned into two coupling subsystems:stiff delay subsystems and non-stiff subsystems,thus we can use combined methods to solve these systems,and they are often more efficient. Therefore,it is significant in theory and practice to study the combined methods for stiff DDEs.
In Chapter 2,we present three kinds of Rosenbrock methods for stiff differential equations and stiff delay differential equations.
In Chapter 3,we analyze the stability properties of combined RK-Rosenbrock methods for stiff delay systems,and give the sufficient and necessary conditions under which the combined RK-Rosenbrock methods are GP-stable.
In Chapter 4,we analyze the stability properties of combined continuous RK-Rosenbrock methods for stiff delay systems,get the methods' order condition equations,and give the sufficient and necessary conditions under which the combined continuous RK-Rosenbrock methods are GP-stable.
In Chapter 5,we analyze the stability properties of continuous Rosenbrock methods and continuous parallel Rosenbrock methods for stiff delay differential equations,use B-series to get the order conditions,and give the sufficient and necessary conditions under which the methods are GP-stable.Numerical experiments show that the methods are efficient.
引文
[1]刘德贵,费景高.动力学系统数字仿真算法[M].科学出版社,2000.
[2]C.W.Gear,D.H.Wang.Real-time integration formulas with off-step inputs and their stability[C].Dept.of Comput.Sci.,University of Illinois at Urbana-Champaign,Report No.UIUCDCS-R,1986:86-1246.
[3]K.C.Lin,R.M.Howe.Simulation using staggered integration steps part I:Intermediate-step predictor methods[J].Trans.Soc.Comput.Simul.,1993,10:153-164.
[4]朱珍民,刘德贵,李寿佛.一类实时仿真算法的稳定性和收敛性[J].系统仿真学报,2000,12:70-74.
[5]陈丽容.刚性大系统数值仿真算法[D].中国航天工业总公司第二研究院 204所博士研究生毕业论文,1996.
[6]曹学年.刚性微分方程的并行Rosenbrock方法[D].中国工程物理研究院博士研究生毕业论文,2001.
[7]X.N.Cao,D.G.Liu,S.F.Li,A class of parallel algorithms of real-time numerical simulation for stiff dynamic system[J].J.Syst.Engin.Elect.,2000,11:51-58.
[8]K.Strehmel,R.Weiner,Partitioned adaptive Runge-Kutta methods and their stability[J].Numer.Math.,1984,45:283-300.
[9]M.Buttner,B.A.Schmitt and R.Weiner,Automatic partitioning in linearlyimplicit Runge-Kutta methods[J],Appl.Numer.Math.,1993,13:41-55.
[10]费景高.常微分方程向前步组合离散方法[J].计算数学,1991,3:209-250.
[11]刘德贵,费景高,韩天敏.刚性大系统数值仿真方法[M].河南科技出版社,1996.
[12]甘特马赫尔.《矩阵论》(上卷,柯召译).高教出版社,1955.
[13]陈丽容,刘德贵.一类分解刚性大系统的组合方法[J].计算物理.1997,14:509-512.
[14]陈丽容,刘德贵.一类刚性动力学系统实时数值仿真组合方法[J].J.Syst.En- gin.Elect.1999.2:64-69.
[15]陈丽容,刘德贵.一类刚性大系统实时数值仿真的并行组合方法[J].系统仿真学报.1999,1l:53.57
[16]K.J.In't Hout.Stability analysis of Runge-Kutta methods for systems of delay differential equations[J].IMA Mumer.Anal.1997,17:17-27.
[17]V.K.Barwell.Specials stability problems for functional equations[J].BIT,1975,15:130-135.
[18]M.Zennaro.P-stability properties of Runge-Kutta methods for delay differential equations[J].Numer.Math,1986,49:305-318.
[19]A.Bellen,Z.Jackiewicz,M.Zennaro.Stability analysis of one step methods for neutral delay differential equations[J].Numberische Mathematic,1988,52(3):605-619.
[20]L.Torelli.Stability of numerical methods for delay differential equations[J].J.Comput.Appl.Math.,1989,25:15-26.
[21]M.Z.Liu,M Spijker.The stability of θ methods in the numerical solution of delay differential equations[J].IMA Journal of Numerical Analysis,1990,10(1):31-48.
[22]K.J.In't Hout.The stability of θ methods for systems of delay differential equations[J].Applied Numerical Mathematics,1994,1(3):323-334.
[23]T.Koto.A stability property of P-stable natural Runge-Kutta methods for systems for systems of delay differential equations[J].BIT,1994,34(2):262-267.
[24]E.Hairer,Order conditons for numberical methods for partitioned ordinary differential equations,Numer.Math.1981,36:431-445.
[25]E.Hairer,S.N φ rsett,G.Wanner,Solving ordinary differential equations I:Nonstiff Problems,Berlin,Heidelberg,Springer 1987.
[26]J.C.Butcher,The numberical analysis of ordinary differential equations:Rungle-Kutta and General Linear Methods,[M].John Wiley Sons,1987.
[27] 李寿佛.刚性微分方程算法理论[M].湖南科学技术出版社, 1997.
[28] L.R. Chen, D.G. Liu, Parallel rosenbrock methods for solving stiff systems in real-time simulation.[J]. J. Comput. Math., 2000,18:375-386.
[29] Kuang Jiaoxun, Tian Hongjiong. The asymptotic behaviour of theoretical and numberical solutions for nonlinear differential systems with several delay terms.[J]. Journal of Shanghai Teachers University, 1995,24(1):1-7.
[30] Cheng-jian Zhang, Geng Sun. Boundedness and asymptotic stability of mul-tistep methods for generalized pantograph equations[J]. Journal of Computational Mathematics,2004,22(3):447-456.
[31] Alfredo Bellen, Marino Zennaro. Numerical Methods for Delay Differential Equations[M]. Oxford: Clarendon Press,2003.
[32] H.H. Rosenbrock. Some general implicit processes for the numerical solution of differential equations[J], Comput. J., 1963,5:329-331.
[33] K. Burrage. Parallel and sequential methods for ordinary differential equations [M]. Clarendon Press, Oxford, 1995.
[34] D.A. Calahan. A-stable accurate method of numerical integration for nonlinear systems[J]. Proc. IEEE, 1968,56:944.
[35] P.J. van der Houwen. One-step methods with adaptive stability functions for the integration of differential equations [C]. Lecture Notes in Mathematics No.333, Springer-Verlag, Berlin, 1973:164-174.
[36] J.R. Cash. Semi-implicit Runge-Kutta prcedures with error estimates for the numerical integration of stiff systems of ordinary differential equtions [J].JACM, 1976,23:455-460.
[37] S.P Norsett.Runge-Kutta methods coefficients depending on the Jacobian [C].Report No.l/75,Dept. of Math.,Univ. of Trondheim, Norway,1975.
[38] S.P. N(?)rsett, A. Wolfbrand. Order conditions for Rosenbrock type methods [J]. Numer. Math., 1979,32:1-15.
[39]P.Kaps,P.Rentrop.Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations[J].Numer.Math.,1979,33:55-68.
[40]P.Kaps,G,Wanner.A study of Rosenbrock type methods of high order[J].Numer.Math,1981,38:279-298.
[41]L.F.Shampine.Implementation of Rosenbrock methods[J].ACM Trans.on Math.Soft.,1982,8:93-113.
[42]P.Kaps,S.W.H.Pooh,T.D.Bui.Rossenbrock methods for stiff ordinary differential equations:acomparison of Richardson eatrapolation andembedding and technique[J].Coputing,1985,34:17-40.
[43]P.Kaps,A.Ostermann.Rosenbrock methods using few LU-decompositions [J].IMA J.Numer.Anal.,1989,9:15-27.
[44]P.Kaps,A.Ostermann.L(α)-stable variable order Rosenbrock-methods[C].Proceedings of the Halle Seminar 1989,Teubner Texte zur Mathmatics,1990.
[45]E.Hairer,S.P.Norsett.Solving Ordinary Differential Equations Ⅱ:Stiff and Differential-Algebraic Problem[M].Springer-Verlag,1996.
[46]E.Hairer,S.P.Norsett,G.Wanner.Solving ordinary differential equations .[M].New York:Spinger Verlag,2000,103-117.
[47]曹学年,刘德贵,李寿佛.求解延迟微分方程的Rosenbrock方法的渐进稳定性[J].系统仿真学报,2004,14(3):290-292.
[48]丛玉豪,才佳宁,项家祥.求解时滞微分方程组的Rosenbrock方法的GP-稳定性[J].应用数学与力学,2004,12(25):1285-1291.
[49]冷欣,刘德贵,宋晓秋,陈丽容.刚性延迟微分方程数值仿真的两步连续Rosenbrock方法[J],系统仿真学报,2006,18(7):1758-1762.
[50]Liu Jianguo,Asymptotic stability of Rosenbrock methods for pantograph equations[J].Mathematical Theory and Application.2006,1(26):61-64.
[51]刘建国.求解延迟微分方程组的Rosenbrock方法的渐近稳定性[J].系统仿真学报,2007,14(3):290-292。
[52]K.Burrage.Parallel methods for initial value problem[J].Appl.Numer.Math.,1993,11:5-25.
[53]H.Podhaisky,B.A.Schmitt,R.Weiner.Design analysis and testing of some parallel two-step W-methods for stiff systems[J].Appl.Numer.Math.,2002,42:381-395.
[54]冷欣,刘德贵,宋晓秋,陈丽容.一类组合两步连续RK-Rosenbrock方法[J].系统仿真学报,2007,19(17):3945-3948.
[55]Qiao Zhu,Aiguo Xiao.Parallel two-step ROW-methods for stiff delay differential equations[J].Appl.Numer.Math.2009,59:I768-1778.
[56]袁兆鼎,费景高,刘德贵.刚性常微分方程初值问题数值解法[M].科学出版社,1987年.
[57]Jiaoxun Kuang,Yuhao Cong.Stability of numerical methods for delay differential equations[M].Sinece Press,2005.
[2]C.W.Gear,D.H.Wang.Real-time integration formulas with off-step inputs and their stability[C].Dept.of Comput.Sci.,University of Illinois at Urbana-Champaign,Report No.UIUCDCS-R,1986:86-1246.
[3]K.C.Lin,R.M.Howe.Simulation using staggered integration steps part I:Intermediate-step predictor methods[J].Trans.Soc.Comput.Simul.,1993,10:153-164.
[4]朱珍民,刘德贵,李寿佛.一类实时仿真算法的稳定性和收敛性[J].系统仿真学报,2000,12:70-74.
[5]陈丽容.刚性大系统数值仿真算法[D].中国航天工业总公司第二研究院 204所博士研究生毕业论文,1996.
[6]曹学年.刚性微分方程的并行Rosenbrock方法[D].中国工程物理研究院博士研究生毕业论文,2001.
[7]X.N.Cao,D.G.Liu,S.F.Li,A class of parallel algorithms of real-time numerical simulation for stiff dynamic system[J].J.Syst.Engin.Elect.,2000,11:51-58.
[8]K.Strehmel,R.Weiner,Partitioned adaptive Runge-Kutta methods and their stability[J].Numer.Math.,1984,45:283-300.
[9]M.Buttner,B.A.Schmitt and R.Weiner,Automatic partitioning in linearlyimplicit Runge-Kutta methods[J],Appl.Numer.Math.,1993,13:41-55.
[10]费景高.常微分方程向前步组合离散方法[J].计算数学,1991,3:209-250.
[11]刘德贵,费景高,韩天敏.刚性大系统数值仿真方法[M].河南科技出版社,1996.
[12]甘特马赫尔.《矩阵论》(上卷,柯召译).高教出版社,1955.
[13]陈丽容,刘德贵.一类分解刚性大系统的组合方法[J].计算物理.1997,14:509-512.
[14]陈丽容,刘德贵.一类刚性动力学系统实时数值仿真组合方法[J].J.Syst.En- gin.Elect.1999.2:64-69.
[15]陈丽容,刘德贵.一类刚性大系统实时数值仿真的并行组合方法[J].系统仿真学报.1999,1l:53.57
[16]K.J.In't Hout.Stability analysis of Runge-Kutta methods for systems of delay differential equations[J].IMA Mumer.Anal.1997,17:17-27.
[17]V.K.Barwell.Specials stability problems for functional equations[J].BIT,1975,15:130-135.
[18]M.Zennaro.P-stability properties of Runge-Kutta methods for delay differential equations[J].Numer.Math,1986,49:305-318.
[19]A.Bellen,Z.Jackiewicz,M.Zennaro.Stability analysis of one step methods for neutral delay differential equations[J].Numberische Mathematic,1988,52(3):605-619.
[20]L.Torelli.Stability of numerical methods for delay differential equations[J].J.Comput.Appl.Math.,1989,25:15-26.
[21]M.Z.Liu,M Spijker.The stability of θ methods in the numerical solution of delay differential equations[J].IMA Journal of Numerical Analysis,1990,10(1):31-48.
[22]K.J.In't Hout.The stability of θ methods for systems of delay differential equations[J].Applied Numerical Mathematics,1994,1(3):323-334.
[23]T.Koto.A stability property of P-stable natural Runge-Kutta methods for systems for systems of delay differential equations[J].BIT,1994,34(2):262-267.
[24]E.Hairer,Order conditons for numberical methods for partitioned ordinary differential equations,Numer.Math.1981,36:431-445.
[25]E.Hairer,S.N φ rsett,G.Wanner,Solving ordinary differential equations I:Nonstiff Problems,Berlin,Heidelberg,Springer 1987.
[26]J.C.Butcher,The numberical analysis of ordinary differential equations:Rungle-Kutta and General Linear Methods,[M].John Wiley Sons,1987.
[27] 李寿佛.刚性微分方程算法理论[M].湖南科学技术出版社, 1997.
[28] L.R. Chen, D.G. Liu, Parallel rosenbrock methods for solving stiff systems in real-time simulation.[J]. J. Comput. Math., 2000,18:375-386.
[29] Kuang Jiaoxun, Tian Hongjiong. The asymptotic behaviour of theoretical and numberical solutions for nonlinear differential systems with several delay terms.[J]. Journal of Shanghai Teachers University, 1995,24(1):1-7.
[30] Cheng-jian Zhang, Geng Sun. Boundedness and asymptotic stability of mul-tistep methods for generalized pantograph equations[J]. Journal of Computational Mathematics,2004,22(3):447-456.
[31] Alfredo Bellen, Marino Zennaro. Numerical Methods for Delay Differential Equations[M]. Oxford: Clarendon Press,2003.
[32] H.H. Rosenbrock. Some general implicit processes for the numerical solution of differential equations[J], Comput. J., 1963,5:329-331.
[33] K. Burrage. Parallel and sequential methods for ordinary differential equations [M]. Clarendon Press, Oxford, 1995.
[34] D.A. Calahan. A-stable accurate method of numerical integration for nonlinear systems[J]. Proc. IEEE, 1968,56:944.
[35] P.J. van der Houwen. One-step methods with adaptive stability functions for the integration of differential equations [C]. Lecture Notes in Mathematics No.333, Springer-Verlag, Berlin, 1973:164-174.
[36] J.R. Cash. Semi-implicit Runge-Kutta prcedures with error estimates for the numerical integration of stiff systems of ordinary differential equtions [J].JACM, 1976,23:455-460.
[37] S.P Norsett.Runge-Kutta methods coefficients depending on the Jacobian [C].Report No.l/75,Dept. of Math.,Univ. of Trondheim, Norway,1975.
[38] S.P. N(?)rsett, A. Wolfbrand. Order conditions for Rosenbrock type methods [J]. Numer. Math., 1979,32:1-15.
[39]P.Kaps,P.Rentrop.Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations[J].Numer.Math.,1979,33:55-68.
[40]P.Kaps,G,Wanner.A study of Rosenbrock type methods of high order[J].Numer.Math,1981,38:279-298.
[41]L.F.Shampine.Implementation of Rosenbrock methods[J].ACM Trans.on Math.Soft.,1982,8:93-113.
[42]P.Kaps,S.W.H.Pooh,T.D.Bui.Rossenbrock methods for stiff ordinary differential equations:acomparison of Richardson eatrapolation andembedding and technique[J].Coputing,1985,34:17-40.
[43]P.Kaps,A.Ostermann.Rosenbrock methods using few LU-decompositions [J].IMA J.Numer.Anal.,1989,9:15-27.
[44]P.Kaps,A.Ostermann.L(α)-stable variable order Rosenbrock-methods[C].Proceedings of the Halle Seminar 1989,Teubner Texte zur Mathmatics,1990.
[45]E.Hairer,S.P.Norsett.Solving Ordinary Differential Equations Ⅱ:Stiff and Differential-Algebraic Problem[M].Springer-Verlag,1996.
[46]E.Hairer,S.P.Norsett,G.Wanner.Solving ordinary differential equations .[M].New York:Spinger Verlag,2000,103-117.
[47]曹学年,刘德贵,李寿佛.求解延迟微分方程的Rosenbrock方法的渐进稳定性[J].系统仿真学报,2004,14(3):290-292.
[48]丛玉豪,才佳宁,项家祥.求解时滞微分方程组的Rosenbrock方法的GP-稳定性[J].应用数学与力学,2004,12(25):1285-1291.
[49]冷欣,刘德贵,宋晓秋,陈丽容.刚性延迟微分方程数值仿真的两步连续Rosenbrock方法[J],系统仿真学报,2006,18(7):1758-1762.
[50]Liu Jianguo,Asymptotic stability of Rosenbrock methods for pantograph equations[J].Mathematical Theory and Application.2006,1(26):61-64.
[51]刘建国.求解延迟微分方程组的Rosenbrock方法的渐近稳定性[J].系统仿真学报,2007,14(3):290-292。
[52]K.Burrage.Parallel methods for initial value problem[J].Appl.Numer.Math.,1993,11:5-25.
[53]H.Podhaisky,B.A.Schmitt,R.Weiner.Design analysis and testing of some parallel two-step W-methods for stiff systems[J].Appl.Numer.Math.,2002,42:381-395.
[54]冷欣,刘德贵,宋晓秋,陈丽容.一类组合两步连续RK-Rosenbrock方法[J].系统仿真学报,2007,19(17):3945-3948.
[55]Qiao Zhu,Aiguo Xiao.Parallel two-step ROW-methods for stiff delay differential equations[J].Appl.Numer.Math.2009,59:I768-1778.
[56]袁兆鼎,费景高,刘德贵.刚性常微分方程初值问题数值解法[M].科学出版社,1987年.
[57]Jiaoxun Kuang,Yuhao Cong.Stability of numerical methods for delay differential equations[M].Sinece Press,2005.