非线性方程的保辛近似算法

保守体系的微分方程可用Hamilton体系的方法描述,其特点是保辛。两个辛矩阵之和不能保辛,两个辛矩阵的乘积仍是辛矩阵。最常用的小参数摄动法用的是加法,因此对辛矩阵不能保辛。从保辛的角度,要用正则变换。本文针对非线性微分方程,运用自变量坐标变换,对原系统进行变换。由此推导出变换后系统的变分原理。引入Hamilton对偶变量,通过数学变换,得到变系数非线性方程。针对该方程,本文提出了保辛摄动算法。通过数值算例,对不同步长下,保辛摄动法、多尺度摄动法、龙格库塔法和精确解的结果做了比较。数值例题表明,对于非线性方程,本文提出的保辛摄动算法有良好的精度。在步长增大的情况下,保辛摄动保持了良好的稳定性。

Symplectic Conservative Integration for Nonlinear Differential Equation

A differential equation of conservative system can be described in Hamilton system.The distinguishing feature of conservative system is symplectic conservation.The sum of two symplectic matrixes is not symplectic conservation.The product of two symplectic matrixes is symplectic conservation.Addition is used in method of small parameter perturbation.It is not symplectic conservation.The canonical transformation is symplectic conservation.The nonlinear differential equation problem was discussed.The method of coordinate transformation was applied. The variation of principle was derived.The dual variables of Hamilton were introduced,and a variable coefficient nonlinear equation was derived through transformation.A symplectic conservative integration method was presented.The results of symplectic conservative integration,method of multiple scales,fourthorder Runge-Kutta and analytic solution were given.Numerical results demonstrate the effectiveness of the present method for nonlinear different equation.