• 论文 •

### 三级四阶显式辛R-K-N格式的完全构造

1. (1)电子科技大学自动化工程学院, 成都 610054;宁波大学理学院, 宁波 315211;(2)电子科技大学自动化工程学院, 成都 610054.
• 收稿日期:2009-05-31 修回日期:1900-01-01 出版日期:2009-09-25 发布日期:2009-09-25

XU Song;HOU Xiaorong. The Complete Construction of Fourth Order Symplectic Explicit R-K-N Methods of Three Stages[J]. Journal of Systems Science and Mathematical Sciences, 2009, 29(9): 1211-1221.

### The Complete Construction of Fourth Order Symplectic Explicit R-K-N Methods of Three Stages

XU Song(1), HOU Xiaorong(2)

1. (1)College of Automation, University of Electronic Science and Technology of China Chengdu 610054;Department of Mathematics, Ningbo University,Ningbo 315211;(2)College of Automation, University of Electronic Science and Technology of China, Chengdu 610054.
• Received:2009-05-31 Revised:1900-01-01 Online:2009-09-25 Published:2009-09-25
s级p阶辛Runge-Kutta-Nystr\"om(R-K-N)方法的一种充要条件是用关于参数的非线性方程组来表示的,辛R-K-N格式的构造问题因而转化为该方程组的求解问题. 在一些特殊的限定条件下, 已有该方程组在s=3,p=4时的两组解,即得到了两个三级四阶显式辛格式. 对于s=3,p=4情形,基于吴方法,利用计算机代数系统Maple及软件包wsolve给出了对应的非线性方程组的全部解, 这样就构造了所有的三级四阶显式辛R-K-N格式, 并证明了三级四阶显式辛R-K-N方法所满足的

The sufficient and necessary conditions for a $p$-th order symmetric Runge-Kutta-Nystr\"om (R-K-N) method of s stages were expressed by a system of nonlinear equations. Thus the construction problem of symmetric schemes
is transformed into the problem of solving the system. Under some special conditions, two solutions of the system for s=3 and p=4 were presented, so two fourth order symmetric explicit schemes of three stages were given. Based on Wu's method, we obtain all the solutions of the system for s=3 and p=4 by using Maple software and the software package wsolve in this paper, that is, we construct all the fourth order symplectic explicit R-K-N schemes of three stages. Moreover, we prove that the conditions for a fourth order symmetric explicit R-K-N method of three stages are redundant. Numerical results on the two-body problem indicate the higher precision of the new schemes compared to some existing schemes.

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