Reversible long-term integration with
variable step sizes
E. Hairer
and
D. Stoffer
Abstract.
The numerical integration of reversible dynamical
systems is considered.
A backward analysis for variable step size one-step methods is developed
and it is shown that the numerical solution of a symmetric one-step method,
implemented with a reversible step size strategy, is formally equal to
the exact solution of a perturbed differential equation, which again
is reversible. This explains geometrical properties of the numerical flow,
such as the nearby preservation of invariants. In a second part, the
efficiency of symmetric implicit Runge-Kutta methods (linear error growth
when applied to integrable systems) is compared with explicit non-symmetric
integrators (quadratic error growth).
Key Words.
symmetric Runge-Kutta methods, extrapolation methods,
long-term integration, Hamiltonian problems, reversible systems