The paper presents fourth order Runge–Kutta methods derived from symmetric Hermite–Obreshkov schemes by suitably approximating the involved higher derivatives. In particular, starting from the multi-derivative extension of the midpoint method we have obtained a new symmetric implicit Runge–Kutta method of order four, for the numerical solution of first-order differential equations. The new method is symplectic and is suitable for the solution of both initial and boundary value Hamiltonian problems. Moreover, starting from the conjugate class of multi-derivative trapezoidal schemes, we have derived a new method that is conjugate to the new symplectic method.

中文翻译：

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中点和梯形方法的四阶辛和共轭辛扩展

通过适当地近似所涉及的较高导数，本文介绍了从对称Hermite-Obreshkov方案派生的四阶Runge-Kutta方法。特别是，从中点方法的多导数展开开始，我们获得了一种新的四阶对称隐式Runge-Kutta方法，用于一阶微分方程的数值解。新方法是辛的，适用于初值和边值哈密顿问题的求解。此外，从多导数梯形方案的共轭类开始，我们得出了与新辛方法共轭的新方法。