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We conduct a systematic comparison of spectral methods with some
symplectic methods in solving Hamiltonian dynamical systems. Our
main emphasis is on the non-linear problems. Numerical evidence has
demonstrated that the proposed spectral collocation method preserves
both energy and symplectic structure up to the machine error in each
time (large) step, and therefore has a better long time behavior.
Kanyamee, Nairat, "Spectral Methods For The Hamiltonian Systems" (2010). Wayne State University Dissertations. Paper 133.