三级四阶显式辛R-K-N格式的完全构造

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方法所满足的条件方程有冗余. 数值实验结果显示出新的辛格式在一定的条件下有着较好的误差精度.

THE COMPLETE CONSTRUCTION OF FOURTH ORDER SYMPLECTIC EXPLICIT R-K-N METHODS OF THREE STAGES

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.