Order Barriers for Symplectic Multi-Value Methods
E. Hairer
and
P. Leone
Abstract.
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.
Key Words.
Hamiltonian systems, multistep methods, one-leg methods,
general linear methods, symplecticity