LongTime Behaviour of Numerically Computed Orbits: Small and Intermediate Timestep Analysis of OneDimensional Systems
Abstract
The longtime behaviour of numerically computed orbits in onedimensional systems is studied by deriving a continuoustime "pseudodynamics" equivalent to the discretetime numerical dynamics. The derivation applies to any numerical algorithm which conserves phasespace volume. A conservation law of the continuoustime system (conservation of the "pseudoHamiltonian") guarantees that the numerical orbits are close to the exact orbits, even after an unlimited number of timesteps. The equivalence between the discretetime and continuoustime 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 "superadiabatic" 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 highorder 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.
 Publication:

Journal of Computational Physics
 Pub Date:
 March 1991
 DOI:
 10.1016/00219991(91)90079Z
 Bibcode:
 1991JCoPh..93..189A