{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:47:56Z","timestamp":1776728876276,"version":"3.51.2"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"244","license":[{"start":{"date-parts":[[2004,5,29]],"date-time":"2004-05-29T00:00:00Z","timestamp":1085788800000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>In this paper we deal with several issues concerning variable-stepsize linear multistep methods. First, we prove their stability when their fixed-stepsize counterparts are stable and under mild conditions on the stepsizes and the variable coefficients. Then we prove asymptotic expansions on the considered tolerance for the global error committed. Using them, we study the growth of error with time when integrating periodic orbits. We consider strongly and weakly stable linear multistep methods for the integration of first-order differential systems as well as those designed to integrate special second-order ones. We place special emphasis on the latter which are also symmetric because of their suitability when integrating moderately eccentric orbits of reversible systems. For these types of methods, we give a characterization for symmetry of the coefficients, which allows their construction, and provide some numerical results for them.<\/p>","DOI":"10.1090\/s0025-5718-03-01538-2","type":"journal-article","created":{"date-parts":[[2003,6,20]],"date-time":"2003-06-20T10:23:27Z","timestamp":1056104607000},"page":"1769-1801","source":"Crossref","is-referenced-by-count":5,"title":["Analysis of variable-stepsize linear multistep methods with special emphasis on symmetric ones"],"prefix":"10.1090","volume":"72","author":[{"given":"B.","family":"Cano","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"A.","family":"Dur\u00e1n","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2003,5,29]]},"reference":[{"issue":"1-3","key":"1","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1016\/0168-9274(95)00046-W","article-title":"Accurate long-term integration of dynamical systems","volume":"18","author":"Calvo, M. P.","year":"1995","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0168-9274(98)00035-X","article-title":"Variable step implementation of geometric integrators","volume":"28","author":"Calvo, M. P.","year":"1998","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"issue":"4","key":"3","doi-asserted-by":"publisher","first-page":"401","DOI":"10.1016\/0168-9274(95)00107-7","article-title":"A generalization to variable stepsizes of St\u00f6rmer methods for second-order differential equations","volume":"19","author":"Cano, Bego\u00f1a","year":"1996","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"key":"4","unstructured":"Cano, B. and Dur\u00e1n, A., A technique to construct symmetric variable-stepsize lines multistep methods for second-order systems, Math. Comp., posted on May 29, 2003, PII S 0025-5718(03)01546-1 (to appear in print)."},{"issue":"4","key":"5","doi-asserted-by":"publisher","first-page":"1391","DOI":"10.1137\/S0036142995281152","article-title":"Error growth in the numerical integration of periodic orbits, with application to Hamiltonian and reversible systems","volume":"34","author":"Cano, B.","year":"1997","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"6","unstructured":"Cano, B., Integraci\u00f3n num\u00e9rica de \u00f3rbitas peri\u00f3dicas con m\u00e9todos multipaso, PhD Thesis, Universidad de Valladolid, 1996."},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1093\/imanum\/18.1.57","article-title":"Error growth in the numerical integration of periodic orbits by multistep methods, with application to reversible systems","volume":"18","author":"Cano, B.","year":"1998","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"3","key":"8","doi-asserted-by":"publisher","first-page":"512","DOI":"10.1137\/0721037","article-title":"The convergence of variable-stepsize, variable-formula, multistep methods","volume":"21","author":"Crouzeix, M.","year":"1984","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"9","doi-asserted-by":"crossref","unstructured":"Evans, N. W. and Tremaine, S., Linear multistep methods for integrating reversible differential equations, Astron. J 118 1888 (1999).","DOI":"10.1086\/301057"},{"key":"10","doi-asserted-by":"publisher","first-page":"1025","DOI":"10.1137\/0711079","article-title":"The effect of variable mesh size on the stability of multistep methods","volume":"11","author":"Gear, C. W.","year":"1974","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"11","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","volume-title":"Solving ordinary differential equations. I","volume":"8","author":"Hairer, E.","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/3540566708","edition":"2"},{"issue":"2-3","key":"12","doi-asserted-by":"publisher","first-page":"219","DOI":"10.1016\/S0168-9274(97)00061-5","article-title":"Variable time step integration with symplectic methods","volume":"25","author":"Hairer, E.","year":"1997","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"issue":"1","key":"13","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1137\/S1064827595285494","article-title":"Reversible long-term integration with variable stepsizes","volume":"18","author":"Hairer, Ernst","year":"1997","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"key":"14","volume-title":"Discrete variable methods in ordinary differential equations","author":"Henrici, Peter","year":"1962"},{"issue":"1","key":"15","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1137\/S1064827595284658","article-title":"The adaptive Verlet method","volume":"18","author":"Huang, Weizhang","year":"1997","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"key":"16","doi-asserted-by":"crossref","unstructured":"Hut, P., Makino, J. and McMillan, S., Building a better leapfrog, Astrophys. J., 443 (1995), pp. L93\u2013L96.","DOI":"10.1086\/187844"},{"issue":"5","key":"17","doi-asserted-by":"publisher","first-page":"1549","DOI":"10.1137\/S0036142997329797","article-title":"Backward error analysis for numerical integrators","volume":"36","author":"Reich, Sebastian","year":"1999","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"18","series-title":"Springer Tracts in Natural Philosophy, Vol. 23","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-65471-8","volume-title":"Analysis of discretization methods for ordinary differential equations","author":"Stetter, Hans J.","year":"1973"},{"issue":"1","key":"19","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF02238234","article-title":"Variable steps for reversible integration methods","volume":"55","author":"Stoffer, D.","year":"1995","journal-title":"Computing","ISSN":"https:\/\/id.crossref.org\/issn\/0010-485X","issn-type":"print"},{"key":"20","unstructured":"Stoffer, D., On reversible and canonical integration methods. Res. Rep. 88\u201305, Applied Mathematics, Eidgen\u00f6ssische Technische Hochschule (ETH), Z\u00fcrich, 1988."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2003-72-244\/S0025-5718-03-01538-2\/S0025-5718-03-01538-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-244\/S0025-5718-03-01538-2\/S0025-5718-03-01538-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:25:29Z","timestamp":1776727529000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-244\/S0025-5718-03-01538-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,5,29]]},"references-count":20,"journal-issue":{"issue":"244","published-print":{"date-parts":[[2003,10]]}},"alternative-id":["S0025-5718-03-01538-2"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-03-01538-2","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2003,5,29]]}}}