Collocation Integrator Lobbie in Orbital Dynamics Problems

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Abstract
The paper investigates the efficiency of the new collocation integrator Lobbie, presented in (Avdyushev, 2020), in comparison with other integrators widely used in practice, namely, collocation Runge–Kutta, extrapolation Gragg–Bulirsch–Stoer, multistep Adams–Multon–Bashforth integrators, and also with the Everhart integrator, well known in celestial mechanics. The integrators are tested in orbital dynamics problems. In particular, a comparative analysis of efficiency shows that when simulating a complex orbital motion (strongly elliptical or with gravity assist maneuvers), Lobbie excels the other integrators (except Everhart) by several times in performance, and by several orders of magnitude in accuracy. A correct comparison of the efficiency of the Everhart integrator and Lobbie is not possible, since they have no common orders: the former has only odd orders on the Radau spacings, while the latter has only even orders on the Lobatto spacings. Nevertheless, if we compare the efficiency of integrators of adjacent orders, then in the strongly elliptic case the Everhart integrator (with a higher order) is one order of magnitude inferior to Lobbie in accuracy. Another advantage of Lobbie is that it allows solving mixed systems of differential equations of the second and first orders, which, for example, are used in celestial mechanics to study dynamic chaos, as well as to linearize, regularize, and stabilize the equations of motion. To use the Everhart integrator to solve such systems, all second-order equations must be reduced to first-order ones. However, as applied to systems of first-order equations, the efficiency of the Everhart integrator becomes noticeably worse.

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