Geometric integration of non-autonomous Hamiltonian problems

TL;DR

This paper extends symplectic integration methods to non-autonomous Hamiltonian systems using canonical transformations, demonstrating their effectiveness and superior accuracy for exponential integrators through theoretical and numerical analysis.

Contribution

It introduces a framework for symplectic integration of non-autonomous Hamiltonian problems via canonical transformations, unifying existing methods and highlighting their advantages.

Findings

Canonical transformations extend symplectic methods to non-autonomous systems.

Exponential integrators with canonical and symmetric properties perform well long-term.

Exponential integrators outperform general ODE schemes in accuracy for non-autonomous linear problems.

Abstract

Symplectic integration of autonomous Hamiltonian systems is a well-known field of study in geometric numerical integration, but for non-autonomous systems the situation is less clear, since symplectic structure requires an even number of dimensions. We show that one possible extension of symplectic methods in the autonomous setting to the non-autonomous setting is obtained by using canonical transformations. Many existing methods fit into this framework. We also perform experiments which indicate that for exponential integrators, the canonical and symmetric properties are important for good long time behaviour. In particular, the theoretical and numerical results support the well documented fact from the literature that exponential integrators for non-autonomous linear problems have superior accuracy compared to general ODE schemes.