Asymptotic Error Analysis of the Adaptive Verlet Method
S. Cirilli, E. Hairer and B. Leimkuhler
Abstract.
The Adaptive Verlet method and variants are
time-reversible schemes for treating Hamiltonian systems
subject to a Sundman time transformation. These methods have been
observed in computer experiments to exhibit superior numerical stability
when implemented in a counterintuitive
``reciprocal'' formulation.
Here we give a theoretical explanation
of this behavior by examining the leading terms of the
modified equation (backward error analysis) and those of the asymptotic
error expansion. With this insight we are able to
improve the algorithm by simply
correcting the starting stepsize.
Key Words.
Adaptive Verlet method,
time-reversible variable stepsizes,
Hamiltonian systems,
Sundman time-transformations,
backward error analysis,
asymptotic expansions.