The recently-introduced relaxation approach for Runge–Kutta methods can be used to enforce conservation of energy in the integration of Hamiltonian systems. We study the behavior of implicit and explicit relaxation Runge–Kutta methods in this context. We find that, in addition to their useful conservation property, the relaxation methods yield other improvements. Experiments show that their solutions bear stronger qualitative similarity to the true solution and that the error grows more slowly in time. We also prove that these methods are superconvergent for a certain class of Hamiltonian systems.

中文翻译：

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哈密​​顿问题的松弛Runge–Kutta方法

最近为Runge-Kutta方法引入的松弛方法可用于在哈密顿系统的集成中强制执行能量守恒。在这种情况下，我们研究了隐式和显式松弛Runge-Kutta方法的行为。我们发现，松弛方法除了具有有用的保存特性外，还产生了其他改进。实验表明，他们的解决方案与真实解决方案具有更强的定性相似性，并且误差随时间的增长更慢。我们还证明了这些方法对于特定类别的哈密顿系统是超收敛的。