Long-Term Stability of Symmetric Partitioned Linear Multistep Methods

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.