Explicit, Adaptive, Symplectic(EASY) Integrators using Scale Invariant Regularisations and Canonical Transformations

We present explicit, adaptive symplectic (EASY) integrators for the numerical integration of Hamiltonian systems with greatly varying time-scales. A time regularisation is considered using the Poincar\'e transformation. This gives a new Hamiltonian which is usually not separable, and to recover the original separability a canonical transformation is considered. A backward error analysis for the numerical integration with a splitting symplectic integrator is presented. For a one-dimensional near singular problem, this analysis reveals a strong dependence of the performance of the method with the choice of the regularisation function, $g$, and the order of the method. The optimal choice corresponds to the function $g$ which nearly preserves the scaling invariance of the system. Numerical examples supporting this result are presented. Finally, an EASY method for the two- and three-dimensional Lennard-Jones problem, nearly preserving scaling invariance is also presented.