Variable time step integration with symplectic methods (Q1372703)

It is a fundamental problem in the symplectic integration of Hamiltonian systems that the favorable properties of such methods get lost, if one uses variable stepsizes. The author proposes to perform a time reparametrization in the Hamiltonian system instead of changing the stepsize during the numerical integration. A naive transformation of the time variable will not preserve the Hamiltonian character of the system. But the author shows how one can define for given initial data a new Hamiltonian with a transformed time variable such that the solutions of the two Hamiltonian systems for these special initial data coincide. A backward error analysis of the approach is given. The author discusses several concrete choices of time reparametrization and demonstrates their performances in numerical experiments with the Kepler problem.