线性Hamilton系统边值问题的保辛数值方法

A Symplectic Approach for Boundary-Value Problems of Linear Hamiltonian Systems

以Hamilton系统的正则变换和生成函数为基础研究线性时变Hamilton系统边值问题的保辛数值求解算法.根据第二类生成函数系数矩阵与状态传递矩阵的关系,构造了生成函数系数矩阵的区段合并递推算法,并进一步将递推算法推广到线性非齐次边值问题中;然后利用生成函数的性质将边值问题转化为初值问题,最后采用初值问题的保辛算法求解以达到整个Hamilton系统保辛的目的.数值算例表明该方法能够有效地求解线性齐次与非齐次问题,并能很好地保持Hamilton系统的固有特性.

A symplectic approach based on canonical transformation and generating functions was proposed to solve boundary-value problems of linear Hamiltonian systems. According to the relationship between the generating function and the state-transition matrix,an interval merge recursive algorithm was constructed to calculate the coefficient matrices of the 2nd-type generating function for linear homogeneous Hamiltonian systems,which was further extended to nonhomogeneous cases. Then the properties of the generating function were used to transform the boundary-value problems to initialvalue problems. Finally,the general initial-value problems were solved with the symplectic numerical method to preserve the geometric structure of the Hamiltonian system. Numerical simulations show the validity of the presented approach for linear homogeneous and nonhomogeneous problems,and the advantages of the symplectic numerical method to preserve the intrinsic properties of Hamiltonian systems.