  The recently-introduced relaxation approach for Runge-Kutta methods can be
used to enforce conservation of energy in the integration of Hamiltonian
systems. We study the behavior of implicit and explicit relaxation Runge-Kutta
methods in this context. We find that, in addition to their useful conservation
property, the relaxation methods yield other improvements. Experiments show
that their solutions bear stronger qualitative similarity to the true solution
and that the error grows more slowly in time. We also prove that these methods
are superconvergent for a certain class of Hamiltonian systems.
