The generalized additive Runge-Kutta (GARK) framework provides a powerful approach for solving additively partitioned ordinary differential equations. This work combines the ideas of symplectic GARK schemes and multirate GARK schemes to efficiently solve additively partitioned Hamiltonian systems with multiple time scales. Order conditions, as well as conditions for symplecticity and time-reversibility, are derived in the general setting of non-separable Hamiltonian systems. Investigations of the special case of separable Hamiltonian systems are also carried out. We show that particular partitions may introduce stability issues, and discuss partitions that enable an implicit-explicit integration leading to improved stability properties. Higher-order symplectic multirate GARK schemes based on advanced composition techniques are discussed. The performance of the schemes is demonstrated by means of the Fermi-Pasta-Ulam problem.

翻译：广义加性龙格-库塔（GARK）框架为解决加性分解常微分方程提供了一种强有力的方法。本文结合辛GARK格式与多速率GARK格式的思想，高效求解具有多时间尺度的加性分解哈密顿系统。在非可分哈密顿系统的一般框架下，推导了阶条件以及辛性与时间可逆性条件，同时对可分哈密顿系统的特例进行了研究。我们表明特定的分解可能引入稳定性问题，并讨论了能够实现隐式-显式积分从而改善稳定性的分解方法。基于高级组合技术的高阶辛多速率GARK格式也得到了探讨。通过费米-帕斯塔-乌拉姆问题验证了这些格式的性能。