We introduce the definition of state-dependent symplecticity as a useful tool of investigation to discover nearby symplecticity in symmetric non-symplectic one-step methods applied to two-dimensional Hamiltonian systems. We first relate this property to Poisson systems and to the trapezoidal method, and then investigate Runge-Kutta and discrete gradient symmetric methods. © 2006 Elsevier B.V. All rights reserved.
Iavernaro, F., & Trigiante, D. (2007). State-dependent symplecticity and area preserving numerical methods. Journal of Computational and Applied Mathematics, 205(2), 814–825. https://doi.org/10.1016/j.cam.2006.02.058