@Article{GarciaAlonso:2009:ANI,
  author =       "Fernando Garcia-Alonso and Jos\'{e} A. Reyes and 
                  Jos\'{e} M. Ferr\'{a}ndiz and Jes\'{u}s Vigo-Aguiar",
  title =        "Accurate Numerical Integration of Perturbed Oscillatory Systems
                  in Two Frequencies",
  journal =      "{ACM} Transactions on Mathematical Software",
  volume =       "36",
  number =       "4",
  month =        aug,
  pages =        "21:1--21:34",
  year =         "2009",
  URL =          "http://doi.acm.org/10.1145/1555386.1555390",
  accepted =     "18 February 2009",
  abstract =     "
                 Highly accurate long-term numerical
                 integration of nearly oscillatory systems
                 of ODE's is a common problem in
                 Astrodynamics. Scheifele's algorithm
                 is one the excellent integrators
                 developed in the past years to take
                 advantage of special transformations
                 of variables such as the K-S set. It is
                 based on using expansions in series of
                 the so-called G-functions, and generalizes
                 the Taylor series integrators but with the
                 remarkable property of integrating without
                 truncation error oscillations in one
                 basic known frequency. A generalization
                 of Scheifele's method capable to
                 integrate exactly harmonic oscillations in
                 two known frequencies is developed here,
                 after introducing a two parametric family
                 of analytical $\varphi$-functions. Moreover,
                 the local error contains the perturbation
                 parameter as a factor when the algorithm
                 is applied to perturbed problems. The good
                 behaviour and the long-term accuracy of
                 the new method are shown through several
                 examples, including systems with low
                 and high frequency constituents and a
                 perturbed satellite orbit. The new methods
                 provide significantly higher accuracy
                 and efficiency than a selection of
                 well-reputed general-purpose integrators
                 and even recent symplectic or symmetric
                 integrators, whose good behaviour in
                 the long-term integration of the Kepler
                 problem and the other oscillatory systems
                 is well stated in recent literature.",
}