Variable time step integration with symplectic methods

Abstract

Symplectic methods for Hamiltonian systems are known to have favorable properties concerning long-time integrations (no secular terms in the error of the energy integral, linear error growth in the angle variables instead of quadratic growth, correct qualitative behaviour) if they are applied with constant step sizes, while all of these properties are lost in a standard variable step size implementation. In this article we present a “meta-algorithm” which allows us to combine the use of variable steps with symplectic integrators, without destroying the above mentioned favorable properties. We theoretically justify the algorithm by a backward error analysis, and illustrate its performance by numerical experiments.