Generalized Additive Runge-Kutta schemes have shown to be a suitable tool for solving ordinary differential equations with additively partitioned right-hand sides. This work develops symplectic GARK schemes for additively partitioned Hamiltonian systems. In a general setting, we derive conditions for symplecticness, as well as symmetry and time-reversibility. We show how symplectic and symmetric schemes can be constructed based on schemes which are only symplectic, or only symmetric. Special attention is given to the special case of partitioned schemes for Hamiltonians split into multiple potential and kinetic energies. Finally we show how symplectic GARK schemes can leverage different time scales and evaluation costs for different potentials, and provide efficient numerical solutions by using different order for these parts.

翻译：广义加性龙格-库塔方法已被证明是求解右端项具有加性分部的常微分方程的有效工具。本文针对加性分部的哈密顿系统，发展了辛GARK格式。在一般框架下，我们推导了辛性、对称性及时间可逆性的条件。展示了如何基于仅具有辛性或仅具有对称性的格式构造辛对称格式。特别关注了哈密顿量分裂为多个势能与动能分量的分部格式特例。最后，我们论证了辛GARK格式如何利用不同势能分量的时间尺度差异与评估成本差异，并通过对这些分量采用不同阶数提供高效数值解。