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Math. Softw."],"published-print":{"date-parts":[[2009,8]]},"abstract":"<jats:p>\n            Highly accurate long-term numerical integration of nearly oscillatory systems of ordinary differential equations (ODEs) is a common problem in astrodynamics. Scheifele\u2019s algorithm is one of 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\n            <jats:italic>G<\/jats:italic>\n            -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\u2019s method capable of integrating exactly harmonic oscillations in two known frequencies is developed here, after introducing a two parametric family of analytical\n            <jats:italic>\u03c6<\/jats:italic>\n            -functions. Moreover, the local error contains the perturbation parameter as a factor when the algorithm is applied to perturbed problems. The good behavior 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 behavior in the long-term integration of the Kepler problem and the other oscillatory systems is well stated in recent literature.\n          <\/jats:p>","DOI":"10.1145\/1555386.1555390","type":"journal-article","created":{"date-parts":[[2009,9,1]],"date-time":"2009-09-01T17:52:59Z","timestamp":1251827579000},"page":"1-34","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":5,"title":["Accurate Numerical Integration of Perturbed Oscillatory Systems in Two Frequencies"],"prefix":"10.1145","volume":"36","author":[{"given":"Fernando","family":"Garc\u00eda-Alonso","sequence":"first","affiliation":[{"name":"Alicante University"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jos\u00e9 A.","family":"Reyes","sequence":"additional","affiliation":[{"name":"Alicante University"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jos\u00e9 M.","family":"Ferr\u00e1ndiz","sequence":"additional","affiliation":[{"name":"Alicante University"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jes\u00fas","family":"Vigo-Aguiar","sequence":"additional","affiliation":[{"name":"Salamanca University"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2009,8]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02163028"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01235122"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1145\/62038.69650"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(90)90045-3"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/16.2.179"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1016\/0898-1221(91)90096-M"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1016\/0168-9274(93)90131-A"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01601932"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/7.4.423"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/11.2.297"},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1063\/1.168504"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01238770"},{"key":"e_1_2_1_13_1","doi-asserted-by":"crossref","unstructured":"Ferr\u00e1ndiz J. M. and Novo S. 1991. Improved Bettis methods for long-term prediction. In Predictability stability and chaos in N-body dynamical systems. Plenum Publishing Corporation NATO ASI Series C. Plenum New York 515--521.  Ferr\u00e1ndiz J. M. and Novo S. 1991. Improved Bettis methods for long-term prediction. In Predictability stability and chaos in N-body dynamical systems . Plenum Publishing Corporation NATO ASI Series C. Plenum New York 515--521.","DOI":"10.1007\/978-1-4684-5997-5_46"},{"key":"e_1_2_1_14_1","unstructured":"Ferr\u00e1ndiz J. M. Vigo J. and Mart\u00edn P. 1991. Reducing the error growth in the numerical propagation of satellite orbits. ESA SP-326. European Space Agency Paris France 49--54.  Ferr\u00e1ndiz J. M. Vigo J. and Mart\u00edn P. 1991. Reducing the error growth in the numerical propagation of satellite orbits. ESA SP-326. European Space Agency Paris France 49--54."},{"key":"e_1_2_1_15_1","first-page":"193","article-title":"M\u00e9todos adaptados de tipo St\u00f6rmer-Cowell de orden elevado","volume":"7","author":"Franco J. M.","year":"1991","unstructured":"Franco , J. M. , Correas , J. M. , and Petriz , F. 1991 . M\u00e9todos adaptados de tipo St\u00f6rmer-Cowell de orden elevado . Rev. Internac. M\u00e9tod. Num\u00e9r. C\u00e1lc. Dise\u00f1. Ing. 7 , 193 -- 216 . Franco, J. M., Correas, J. M., and Petriz, F. 1991. M\u00e9todos adaptados de tipo St\u00f6rmer-Cowell de orden elevado. Rev. Internac. M\u00e9tod. Num\u00e9r. C\u00e1lc. Dise\u00f1. Ing. 7, 193--216.","journal-title":"Rev. Internac. M\u00e9tod. Num\u00e9r. C\u00e1lc. Dise\u00f1. Ing."},{"key":"e_1_2_1_16_1","doi-asserted-by":"publisher","DOI":"10.1086\/376476"},{"key":"e_1_2_1_17_1","doi-asserted-by":"publisher","DOI":"10.1086\/423039"},{"key":"e_1_2_1_19_1","doi-asserted-by":"crossref","unstructured":"Hairer E. and Hairer M. 2002. 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Gr. , Vanden Berghe , G. , and De Meyer , H. 2001 . Exponentially fitted variable two-step BDF algorithm for first order ODEs . Comput. Phys. Commun. 140 , 346 -- 357 . Ixaru, L. Gr., Vanden Berghe, G., and De Meyer, H. 2001. Exponentially fitted variable two-step BDF algorithm for first order ODEs. Comput. Phys. Commun. 140, 346--357.","journal-title":"Comput. Phys. Commun."},{"volume-title":"Numerical Solution of Differential Equations","author":"Jain M. K.","key":"e_1_2_1_22_1","unstructured":"Jain , M. K. 1984. Numerical Solution of Differential Equations . Wiley Eastern , New York, NY . Jain, M. K. 1984. Numerical Solution of Differential Equations. Wiley Eastern, New York, NY."},{"volume-title":"Numerical Methods for Ordinary Differential Systems","author":"Lambert J. D.","key":"e_1_2_1_23_1","unstructured":"Lambert , J. D. 1991. Numerical Methods for Ordinary Differential Systems . John Wiley and Sons, New York , NY. Lambert, J. D. 1991. 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In Proceedings of the Conference on the Numerical Solution of Differential Equations. A. Dold and B. Eckmann, Eds. Lecture Notes in Mathematics, vol. 109. Springer-Verlag, Berl\u00edn, Germany. 214--219."},{"volume-title":"Proceedings of the AAS\/AIAA Conference on Spaceflight Mechanics, (Victoria, BC, Canada).","author":"Richardson L.","key":"e_1_2_1_29_1","unstructured":"Richardson , L. and Panovskyj , A . 1993. A family of implicit Chebyshev methods for the numerical integration of first-order differential equations . In Proceedings of the AAS\/AIAA Conference on Spaceflight Mechanics, (Victoria, BC, Canada). Richardson, L. and Panovskyj, A. 1993. A family of implicit Chebyshev methods for the numerical integration of first-order differential equations. 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