Reversible Long-Term Integration with Variable Stepsizes

The numerical integration of reversible dynamical systems is considered. A backward analysis for variable stepsize one-step methods is developed, and it is shown that the numerical solution of a symmetric one-step method, implemented with a reversible stepsize 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 nonsymmetric integrators (quadratic error growth).