{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T15:11:43Z","timestamp":1772291503171,"version":"3.50.1"},"reference-count":13,"publisher":"Wiley","license":[{"start":{"date-parts":[[2010,2,1]],"date-time":"2010-02-01T00:00:00Z","timestamp":1264982400000},"content-version":"unspecified","delay-in-days":2588,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"published-print":{"date-parts":[[2003]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The Alekseev\u2013Gr\u00f6bner lemma is combined with the theory of modified equations to obtain an a <jats:italic>priori<\/jats:italic> estimate for the global error of numerical integrators. This estimate is correct up to a remainder term of order <jats:italic>h<\/jats:italic><jats:sup>2<jats:italic>p<\/jats:italic><\/jats:sup>, where <jats:italic>h<\/jats:italic> denotes the step size and <jats:italic>p<\/jats:italic> the order of the method. It is applied to nonlinear oscillators whose behaviour is described by the Emden\u2013Fowler equation <jats:italic>y<\/jats:italic>\u2033+<jats:italic>t<\/jats:italic>\u03bd<jats:italic>y<jats:sup>n<\/jats:sup><\/jats:italic>=0. The result shows explicitly that later terms sometimes blow up faster than the leading term of order <jats:italic>h<jats:sup>p<\/jats:sup><\/jats:italic>, necessitating the whole computation. This is supported by numerical experiments.<\/jats:p>","DOI":"10.1112\/s1461157000000358","type":"journal-article","created":{"date-parts":[[2013,8,6]],"date-time":"2013-08-06T07:41:45Z","timestamp":1375774905000},"page":"18-28","source":"Crossref","is-referenced-by-count":2,"title":["A Priori Estimates for the Global Error Committed by Runge-Kutta Methods for a Nonlinear Oscillator"],"prefix":"10.1112","volume":"6","author":[{"given":"Jitse","family":"Niesen","sequence":"first","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2010,2,1]]},"reference":[{"key":"S1461157000000358_ref005","unstructured":"5 Gragg W. , \u2018Repeated extrapolation to the limit in the numerical solution of ordinary differential equations\u2019, Ph.D. thesis, University of California, Los Angeles, 1964."},{"key":"S1461157000000358_ref003","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/18.1.57"},{"key":"S1461157000000358_ref004","volume-title":"An introduction to the study of stellar structure","author":"Chandrasekhar","year":"1939"},{"key":"S1461157000000358_ref011","volume-title":"Numerical methods for ordinary differential equations: the initial value problem","author":"Lambert","year":"1991"},{"key":"S1461157000000358_ref010","doi-asserted-by":"publisher","DOI":"10.1023\/A:1022049814688"},{"key":"S1461157000000358_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF02188219"},{"key":"S1461157000000358_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BF01391413"},{"key":"S1461157000000358_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1274-4_5"},{"key":"S1461157000000358_ref012","volume-title":"Jacobian elliptic functions","author":"Neville","year":"1944"},{"key":"S1461157000000358_ref008","volume-title":"Structure-preserving algorithms for ordinary differential equations","volume":"31","author":"Hairer","year":"2002"},{"key":"S1461157000000358_ref013","doi-asserted-by":"publisher","DOI":"10.1137\/1017036"},{"key":"S1461157000000358_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/0168-9274(95)00046-W"},{"key":"S1461157000000358_ref009","volume-title":"Nonstiff problems","author":"Hairer","year":"1993"}],"container-title":["LMS Journal of Computation and Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1461157000000358","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,7]],"date-time":"2019-06-07T19:49:16Z","timestamp":1559936956000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1461157000000358\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003]]},"references-count":13,"alternative-id":["S1461157000000358"],"URL":"https:\/\/doi.org\/10.1112\/s1461157000000358","relation":{},"ISSN":["1461-1570"],"issn-type":[{"value":"1461-1570","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003]]}}}