Geometric integration of non-autonomous Hamiltonian problems
For researchers in geometric numerical integration, this provides a principled framework for symplectic methods in non-autonomous systems, though the extension is incremental.
The paper extends symplectic integration methods from autonomous to non-autonomous Hamiltonian systems using canonical transformations, showing that exponential integrators with canonical and symmetric properties achieve superior long-time accuracy compared to general ODE schemes.
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.