Order Barriers for Symplectic Multi-value Methods

We study the question whether multistep methods or general multi-value methods can be symplectic. Already a precise understanding of symplecticity of such methods is a nontrivial task, because they advance the numerical solution in a higher dimensional space (in contrast to one-step methods). The essential ingredient for the present study is the formal existence of an invariant manifold, on which the multi-value method is equivalent to a one-step method. Assuming this underlying one-step method to be symplectic, we prove that the order of the multi-value method has to be at least twice its stage order. As special cases we obtain: multistep methods can never be symplectic; the only symplectic one-leg method is the implicit mid-point rule, and the Gauss methods are the only symplectic Runge-Kutta collocation methods.