Symplectic Properties of multistep Runge-Kutta methods

We investigate the symplecticity of multistep Runge-Kutta methods (MRKMs) as general linear methods (GLMs) for Hamiltonian systems in accordance with the definition due to Bochev and Scovel [1], Eirola and Sanz-Serna [2], and Hairer and Leone [3,4]. We present a necessary and sufficient condition for an MRKM to be symplectic, and show that many typical high-order MRKMs cannot be symplectic unless they degenerate into one-step Runge-Kutta methods (RKMs). We also show that the order of any symplectic two-step RKM is at most 2. We conjecture that there exist order barriers for symplectic MRKMs, and more generally, for symplectic GLMs.