高振荡微分方程的辛几何算法
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摘要
高振荡微分方程是指其解具有高振荡性的一类微分方程,在分子动力学、天体力学、量子化学以及原子物理等方面有着广泛的应用。因此,研究其数值解法具有重要意义。
设计数值计算格式的一个基本想法是数值解法保持原问题的基本特征。根据这种指导思想,构造Hamilton系统的算法,就应该在Hamilton系统的同一框架中进行。辛几何就是Hamilton系统的数学框架,由此产生辛几何算法。
本文介绍了Hamilton方程、高振荡微分方程、辛几何算法、对称的数值解法。主要讨论了形如(x｜¨)+Ω~2x=g(x)的一类高振荡Hamilton微分方程。详细地研究了一些辛算法,同时给出了一个新的辛格式。FPU问题的数值实验结果显示,与其他辛算法相比较,这个解法具有较好的能量保守性。
Highly-oscillatory differential equations are a kind of equations whose solutions are highly-oscillatory,which are extensively applied in molecular dynamics,celestial mechanics,quantum chemistry,atomic physics and so on. Therefore,it is significant to study its numerical methods.
A basic idea behind the design of the numerical schemes is that they can preserve the properties of the problems as much as possible.According to this guiding ideology,in order to construct algorithms for Hamiltonian system,it should be in the same framework of the Hamiltonian systems.Symplectic geometry is the mathematical framework of the Hamiltonian systems,resulting symplectic geometric algorithms.
In this paper,we introduce the properties of Hamilton equation,symmetric numerical methods,highly-oscillatory differential equations,symplectic geometric algorithms and.We mainly discuss a kind of high-oscillatory Hamilton differential equations which take form(?)+Ω~2x=g(x).We study some symplectic methods in full length,and give a new symplectic scheme.Compared to other symplectic methods, the numerical experiment results of FPU problems show that symplectic methods have better behavior of energy conservation.
设计数值计算格式的一个基本想法是数值解法保持原问题的基本特征。根据这种指导思想,构造Hamilton系统的算法,就应该在Hamilton系统的同一框架中进行。辛几何就是Hamilton系统的数学框架,由此产生辛几何算法。
本文介绍了Hamilton方程、高振荡微分方程、辛几何算法、对称的数值解法。主要讨论了形如(x｜¨)+Ω~2x=g(x)的一类高振荡Hamilton微分方程。详细地研究了一些辛算法,同时给出了一个新的辛格式。FPU问题的数值实验结果显示,与其他辛算法相比较,这个解法具有较好的能量保守性。
Highly-oscillatory differential equations are a kind of equations whose solutions are highly-oscillatory,which are extensively applied in molecular dynamics,celestial mechanics,quantum chemistry,atomic physics and so on. Therefore,it is significant to study its numerical methods.
A basic idea behind the design of the numerical schemes is that they can preserve the properties of the problems as much as possible.According to this guiding ideology,in order to construct algorithms for Hamiltonian system,it should be in the same framework of the Hamiltonian systems.Symplectic geometry is the mathematical framework of the Hamiltonian systems,resulting symplectic geometric algorithms.
In this paper,we introduce the properties of Hamilton equation,symmetric numerical methods,highly-oscillatory differential equations,symplectic geometric algorithms and.We mainly discuss a kind of high-oscillatory Hamilton differential equations which take form(?)+Ω~2x=g(x).We study some symplectic methods in full length,and give a new symplectic scheme.Compared to other symplectic methods, the numerical experiment results of FPU problems show that symplectic methods have better behavior of energy conservation.
引文
[1]A.Iserles.On the global error of discretization methods for highly-Oscillatory Ordinary Differential Equations.BIT.42.2002,561-599.
[2]A.Iserles.On the numerical quadrature of highly-oscillating integrals I:Fourier transforms.SIMA J.Num.Anal.24,2004,365-391.
[3]A.Iserles.Think globally,act locally:Solving highly-oscillatory ordinary differential equations.Applied Num.Anal.43,2002,145-160.
[4]A.Iserles,H.Munthe-Kaas,A.Zanna.Lie-group methods.Acta Numerical,2000.9,215-365.
[5]A.Iserles,S.P.Nφrsett,A.F.Rasmussen.Time-symmetry and high-order Magnus methods.Applied Num maths.39,2001,379-401.
[6]B.Garcia,J.M.Sanz-Serna,R.Skeel.Long-time-step methods for Oscillatory Differential Equations,SIAM J.Sci.Comput,20(1999),pp.930-963.
[7]E.Hairer,Christian Lubich,Gerhard Wanner.Geometric Numerical integration.Structure-Preserving Algorithms for Ordinary Differential Equations,Springer Series in Computational Mathematics31,Berlin:Springer-Verlag,2002.
[8]Ernst Hairer,Christian Lubich.Long-Time Energy Conservation of Numerical Methods for Oscillatory Differential Equations,SIAM J.Numer.Anal.vol.38.NO.2(2001).pp.414-441
[9]冯康、秦孟兆著.《哈密尔顿系统的辛几何算法》,浙江科学技术出版社,2003.12
[10]Feng Kang.Difference schemes for Hamiltonian formalism and symplectic geometry.J Comp Math,1986 4(3):279-289.
[11]Feng Kang.On difference schemes and symplectic geometry,In Feng K,ed,Proc 1984Beijing Symp Dif Geometry and Dif Equations,Beijing:Science Press,1985,42-58.
[12]Feng Kang,M.Z.Qin.Hamiltonian algorithms for Hamiltonian dynamical systems.Progr.Nature.Sci,1991,1(2):105-116.
[13]廖新浩,刘林.Hamilton系统数值计算的新方法.天文学进展,1996.14(1):3-11.
[14]赵平福.Kdv方程保结构计算方法的研究[博士论文],2000.5.
[15]A.Iserles.On Cayley-transform methods for the discretization of Lie-group equations.Found.Comput.Math,2001,129-160.
[16]V.I.Arnold.Mathematical methods of classical mechanics.In:GTM 60,Second edition.Springer-Verlag,1989.
[17]Jerrold E.Mamden,Tudor S.Ratiu.王丽瑾等译.力学和对称性导论:经典力学系统初探.北京:清华大学出版社,2006.
[18]J.C.Butcher.Implicit Runge-Kutta process.Math,Comp.,1964(18):50-64
[19]陈钊.高振荡微分方程的对称解法[硕士论文].2008.6
[20]李庆扬,王能超,易大义.数值分析.北京:清华大学出版社,2001.8.
[21]J.C.Butcher.Implicit Runge-Kutta processes.Math,Comp,1964(18):50-64.
[22]E.Hairer,S.P.Norsett,G.Wanner.Solving Ordinary Differential Equations Ⅱ.Stiff and Differential- Algebraic Problems,2nd edition,Springer Series in Computational Mathematics 14,Springer-Verlag Berlin,1996.
[2]A.Iserles.On the numerical quadrature of highly-oscillating integrals I:Fourier transforms.SIMA J.Num.Anal.24,2004,365-391.
[3]A.Iserles.Think globally,act locally:Solving highly-oscillatory ordinary differential equations.Applied Num.Anal.43,2002,145-160.
[4]A.Iserles,H.Munthe-Kaas,A.Zanna.Lie-group methods.Acta Numerical,2000.9,215-365.
[5]A.Iserles,S.P.Nφrsett,A.F.Rasmussen.Time-symmetry and high-order Magnus methods.Applied Num maths.39,2001,379-401.
[6]B.Garcia,J.M.Sanz-Serna,R.Skeel.Long-time-step methods for Oscillatory Differential Equations,SIAM J.Sci.Comput,20(1999),pp.930-963.
[7]E.Hairer,Christian Lubich,Gerhard Wanner.Geometric Numerical integration.Structure-Preserving Algorithms for Ordinary Differential Equations,Springer Series in Computational Mathematics31,Berlin:Springer-Verlag,2002.
[8]Ernst Hairer,Christian Lubich.Long-Time Energy Conservation of Numerical Methods for Oscillatory Differential Equations,SIAM J.Numer.Anal.vol.38.NO.2(2001).pp.414-441
[9]冯康、秦孟兆著.《哈密尔顿系统的辛几何算法》,浙江科学技术出版社,2003.12
[10]Feng Kang.Difference schemes for Hamiltonian formalism and symplectic geometry.J Comp Math,1986 4(3):279-289.
[11]Feng Kang.On difference schemes and symplectic geometry,In Feng K,ed,Proc 1984Beijing Symp Dif Geometry and Dif Equations,Beijing:Science Press,1985,42-58.
[12]Feng Kang,M.Z.Qin.Hamiltonian algorithms for Hamiltonian dynamical systems.Progr.Nature.Sci,1991,1(2):105-116.
[13]廖新浩,刘林.Hamilton系统数值计算的新方法.天文学进展,1996.14(1):3-11.
[14]赵平福.Kdv方程保结构计算方法的研究[博士论文],2000.5.
[15]A.Iserles.On Cayley-transform methods for the discretization of Lie-group equations.Found.Comput.Math,2001,129-160.
[16]V.I.Arnold.Mathematical methods of classical mechanics.In:GTM 60,Second edition.Springer-Verlag,1989.
[17]Jerrold E.Mamden,Tudor S.Ratiu.王丽瑾等译.力学和对称性导论:经典力学系统初探.北京:清华大学出版社,2006.
[18]J.C.Butcher.Implicit Runge-Kutta process.Math,Comp.,1964(18):50-64
[19]陈钊.高振荡微分方程的对称解法[硕士论文].2008.6
[20]李庆扬,王能超,易大义.数值分析.北京:清华大学出版社,2001.8.
[21]J.C.Butcher.Implicit Runge-Kutta processes.Math,Comp,1964(18):50-64.
[22]E.Hairer,S.P.Norsett,G.Wanner.Solving Ordinary Differential Equations Ⅱ.Stiff and Differential- Algebraic Problems,2nd edition,Springer Series in Computational Mathematics 14,Springer-Verlag Berlin,1996.