The Newton-Störmer/Verlet-leapfrog method (S/V) is a symplectic and symmetric one of order two, which, when applied to separable Hamiltonian dynamical systems, becomes explicit and conserves quadratic first integrals, e.g., the angular momentum in the N-body problem.
As for high accuracy long-time computations required by dynamical systems coming from astronomy, the order two is too low, we consider composed S/V methods up to order 10. Beyond all these remarkable qualities, our numerical experiments on the outer as well as inner Solar system (N-body problem for N larger than 5) show that, in practice, the conservation of the Hamiltonian of the flow, up to a prescribed order, still remains an open task. Specifically, the order at which the Hamiltonian is conserved seems to be quasi independent of the order of the composed method in spite of the fact that the compensated summation, as a specific technique to reduce the round-off errors, was used. In order to clear up this aspect we perform a round-off error analysis of S/V method as a one-step method applied to an arbitrary vector field.
Despite of the fact that we succeeded in improving the classical round-off error estimates, numerical experiments lead to the opinion that the accumulation of round-off errors bears the main responsibility for the lack of accuracy in long time energy conservation.
See the expanding block below.
C.I. Gheorghiu, A.C. Mureşan, On the cccuracy of Stoermer/Verlet method as the numerical integrator of the n-body problem – Application to solar system, Rom. Astron. J., 16 (2006) 93-102.
Romanian Astronomical Journal
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