  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.
