High order starting iterates for implicit Runge-Kutta methods: An improvement for variable-step symplectic integrators (Q2783768)

Symplectic mappings are considered in connection with Hamiltonian systems. A special differential form is preserved by symplectic mappings. In particular the flow of a canonical system is symplectic.NEWLINENEWLINENEWLINEWith numerical integration methods symplecticity is usually destroyed. Special implicit Runge-Kutta methods are developed which preserve symplecticity. In the paper starting values for the iterative solution of the implicit equations are proposed. The values in each stage of the Runge-Kutta method are considered as the realization of a time-series. A stochastic model is proposed to represent the time-series. The Kepler-problem in two dimensions is investigated as a test problem. The proposed starting procedure is compared with two other methods. In addition results for a second test problem the so-called outer solar system are given.