Linear multistep methods for integrating reversible differential equations

This paper studies multistep methods for the integration of reversible dyna mical systems, with particular emphasis on the planar Kepler problem. It ha s previously been shown by Cane & Sanz-Serna that reversible linear multist eps for first-order differential equations are generally unstable. Here we report on a subset of these methods-the zero-growth methods-that evade thes e instabilities. We provide an algorithm for identifying these rare methods . We find and study all zero-growth, reversible multisteps with six or fewe r steps. This select group includes two well-known second-order multisteps (the trapezoidal and explicit midpoint methods), as well as three new fourt h-order multisteps-one of which is explicit. Variable time steps can be rea dily implemented without spoiling the reversibility. Tests on Keplerian orb its show that these new reversible multisteps work. well on orbits with low or moderate eccentricity, although at least 100 steps per radian are requi red for stability.