Long-Time Behaviour of Numerically Computed Orbits: Small and Intermediate Timestep Analysis of One-Dimensional Systems
The long-time behaviour of numerically computed orbits in one-dimensional systems is studied by deriving a continuous-time "pseudo-dynamics" equivalent to the discrete-time numerical dynamics. The derivation applies to any numerical algorithm which conserves phase-space volume. A conservation law of the continuous-time system (conservation of the "pseudo-Hamiltonian") guarantees that the numerical orbits are close to the exact orbits, even after an unlimited number of timesteps. The equivalence between the discrete-time and continuous-time dynamics holds only for sufficiently small of the timestep Δ. For intermediate values of Δ (sufficiently large that the conservation law does not hold, but sufficiently small that the numerical orbits are not chaotic) a new "super-adiabatic" invariant Δ is derived, and it is shown that conservation of Δ forces the numerical orbits to lie on smooth closed curves. If the potential energy varies rapidly over a small region, it is shown that very high-order resonances between the timestep and the orbital period T, (i.e., T/ Δ = n, where n is a large integer) produce large deviations of these closed curves from the exact orbit. Such resonances also cause extreme sensitivity of the numerical orbit to the timestep.