The purpose of the present work is to construct a new Runge–Kutta pair of orders five and four to outperform the state-of-the-art in these kind of methods when addressing problems with periodic solutions. We consider the family of such pairs that the celebrated Dormand–Prince pair also belongs. The chosen family comes with coefficients that all depend on five free parameters. These latter parameters are tuned in a way to furnish a new method that performs best on a couple of oscillators. Then, we observe that this trained pair outperforms other well known methods in the relevant literature in a standard set of problems with periodic solutions. This is remarkable since no special property holds such as high phase-lag order or an extended interval of periodicity.

中文翻译：

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Runge-Kutta 阶数对的演化推导 5(4) 专门针对周期解问题进行了调整

当前工作的目的是构建一个新的 Runge-Kutta 对 5 阶和 4 阶，以在解决周期性解问题时优于这些方法的最新技术。我们考虑著名的 Dormand-Prince 对也属于此类对的家族。所选系列的系数都取决于五个自由参数。这些后面的参数以某种方式进行调整，以提供一种在几个振荡器上表现最佳的新方法。然后，我们观察到，在具有周期性解决方案的一组标准问题中，该训练对优于相关文献中的其他众所周知的方法。这是显着的，因为没有特殊属性，例如高相位滞后阶数或延长的周期性间隔。