A Variable-Step Double-Integration Multi-Step Integrator
Berry, Matthew M.
Healy, Liam M.
in Coffey, Mazzoleni, Luu, Glover, eds., Spaceflight Mechanics 2004: Proceedings of the 14th AAS/AIAA Space Flight Mechanics Meeting held Feb. 8-12, 2004, Maui, Hawai'i, Advances in the Astronautical Sciences, Volume 119, pp. 2097-2116. Paper AAS 04-238.
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A variable-step double-integration multi-step integrator is derived using divided differences. The derivation is based upon the derivation of Shampine-Gordon, a single-integration method. Variable-step integrators are useful for propagating elliptical orbits, because larger steps can be taken near apogee. As a double-integration method, the integrator performs only one function evaluation per step, whereas Shampine-Gordon requires two evaluations per step, giving the integrator a significant speed advantage over Shampine-Gordon. Though several implementation issues remain, preliminary results show the integrator to be effective.
See also the dissertation of Matt Berry at http://scholar.lib.vt.edu/theses/available/etd-04282004-071227/