Variable time step integration with symplectic methods
E. Hairer
Abstract.
Symplectic methods for Hamiltonian systems are known to have
favourable pro\-per\-ties 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 favourable properties. We theoretically
justify the algorithm
by a backward error analysis, and illustrate its performance by
numerical experiments.
Key Words.
Hamiltonian systems, symplectic integration, variable step sizes,
backward error analysis, Kepler's problem, Verlet scheme