Advances in Computational Mathematics

, Volume 42, Issue 2, pp 313–332 | Cite as

Geometric integration of non-autonomous linear Hamiltonian problems



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.


Non-autonomous mechanics Hamiltonian problems Canonical transformations Geometric integration 

Mathematics Subject Classification (2010)

65P10 37Jxx 37M15 53D22 70H15 


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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorwegian University of Science and Technology (NTNU)TrondheimNorway

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