One of the most classical pairs of symplectic and conjugate-symplectic schemes is given by the Midpoint method (the Gauss-Runge-Kutta method of order 2) and the Trapezoidal rule. These can be interpreted as compositions of the Implicit and Explicit Euler methods, taken in direct and reverse order, respectively. This naturally raises the question of whether a similar composition structure exists for higher-order Gauss-Legendre methods. In this paper, we provide a positive answer by first examining the fourth-order case and then outlining a generalization to higher orders.

翻译：最经典的辛格式与共轭辛格式对之一由中点法（二阶高斯-龙格-库塔方法）和梯形法则给出。这两种方法可分别解释为隐式欧拉方法与显式欧拉方法按正向及反向顺序的组合。这自然引出一个问题：高阶高斯-勒让德方法是否具有类似的组合结构？本文通过先分析四阶情形，进而概述向更高阶的推广，对此问题给出了肯定回答。