Other
Scientific paper
Dec 1999
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1999baas...31.1590l&link_type=abstract
American Astronomical Society, DPS Meeting #31, late abstracts, #59.33; Bulletin of the American Astronomical Society, Vol. 31,
Other
Scientific paper
Since Lagrange, dynamicists have invented integration methods that can explicitly take into account the near-integrable character of problems in celestial mechanics. Any scheme that returns the exact Kepler orbit of a two-body problem when perturbations are removed describes the system much better than a `blind' conventional N-body method, the foremost advantage being a longer timestep. Traditional schemes modelled in cumbersome forms and variables have been replaced by symplectic integrators (SIs); development in this field has been rapid in recent years, and some of the disadvantages of early SIs have been alleviated by, e.g., symplectic correctors and limited adjustability of stepsize. However, SIs cannot by definition tackle non-Hamiltonian forces, and there is as yet no proper way of using multistep information to build inexpensive high-order schemes. We present a simple near-integrable non-symplectic but SI-like formulation that is suitable for almost any numerical integrator; thus, multistep schemes are easy to build, stepsize can be adjusted, and dissipative forces are allowed. The formulation is based on the choice of variables: we use the phase-space coordinates the object would have at a given point in its (Keplerian) orbit if the perturbing forces were removed. In one choice of frame, low-order methods resemble SIs with similar `kicks' from perturbations and `drifts' via Gauss´ f- and g-functions. A fourth-order scheme needs (asymptotically) only one and a half force evaluations per step. What is more, the variational equations for the Liapunov exponent are especially simple to integrate in this approach. In another frame, any high-order multistep scheme can be efficiently applied. Our formulation is complementary to SIs, offering an increase in speed/accuracy in problems of celestial mechanics where SIs cannot be employed.
Kaasalainen Mikko
Laakso T.
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