Adaptive geometric integrators for Hamiltonian problems with approximate scale invariance

We consider adaptive geometric integrators for the numerical integration of Hamiltonian systems with greatly varying time scales. A time regularization is considered using either the Sundman or the Poincaré transformation. In the latter case, this gives a new Hamiltonian which is usually separable, but with one of the parts not always exactly solvable. This system can be numerically integrated with a splitting scheme where each part can be computed using a symplectic implicit or explicit method, preserving the qualitative properties of the exact solution. In this case, a backward error analysis for the numerical integration 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 monitor function g, which is also observed when using other symmetric nonsymplectic integrators. We also show how this dependence greatly increases with the order of the numerical integrator used. The optimal choice corresponds to the function g, which nearly preserves the scaling invariance of the system. Numerical examples supporting this result are presented. In some cases a canonical transformation can also be considered, making the system more regular or easy to compute, and this is also illustrated with some examples.