Accurate Long-Term Integration of
Dynamical Systems
M.P. Calvo
and
E. Hairer
Abstract.
Symplectic or symmetric integration methods do not only
preserve geometrical structures of the flow of the
differential equation, but they have also favourable
properties concerning their global error when the integration
is performed over a very long time. The subject of this paper
is to provide new insight into this phenomenon. For problems
with periodic solution and for integrable systems we prove that
the error growth is only linear for symplectic and symmetric
methods, compared to a quadratic error growth in the general case.
Furthermore, for symmetric collocation methods we explain a
variable step size implementation which does not distroy
the above mentioned properties.
Key Words.
Symmetric Runge-Kutta methods, symplectic methods,
long-term integration, Hamiltonian problems, reversible systems.