Long-term stability of symmetric partitioned linear multistep methods
P. Console and E. Hairer
Abstract.Long-time integration of Hamiltonian systems is an important
issue in many applications -- for example the planetary motion in astronomy or
simulations in molecular dynamics. Symplectic and symmetric
one-step methods are known to have favorable numerical
features like near energy preservation over long times and at most linear
error growth for nearly integrable systems. This work studies the suitability
of linear multistep methods for this kind of problems. It turns out that
the symmetry of the method is essential for good conservation properties,
and the more general class of partitioned linear multistep methods permits
to obtain more favorable long-term stability of the integration. Insight into the
long-time behavior is obtained by a
backward error analysis, where the underlying one-step method and also
parasitic solution components are investigated. In this way one approaches
a classification of problems, for which multistep methods are an
interesting class of integrators when long-time integration is important.
Numerical experiments confirm the theoretical findings.
Key Words.Partitioned linear multistep methods, Hamiltonian systems,
long-term integration, energy preservation.