{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T04:25:19Z","timestamp":1787372719969,"version":"build-2736575974"},"reference-count":12,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>In this paper several Nystr\u00f6m methods for the van der Pol equation are considered. In an earlier study by Cai, Aoyagi, and Abe it was shown that the second order Nystr\u00f6m, or leapfrog, method fails to approximate the limit cycle of the van der Pol equation, exhibiting a periodic modulation of the amplitude and sporadic high frequency noise instead. Cai et al. did a linear analysis and concluded that the spurious behavior was due to the interaction of the main part of the solution with a high frequency computational mode. In this paper we also apply a third and fourth order Nystr\u00f6m method to the van der Pol equation. Numerical experiments show that in these cases the high frequency mode causes blowup after some time. The onset of the instability can be delayed by decreasing the time step. We also improve on their analysis of the second order scheme by doing a nonlinear analysis, to wit a discrete multiple scales analysis. By this means we are able to explain the spurious behavior of this system completely.<\/jats:p>","DOI":"10.1137\/050644148","type":"journal-article","created":{"date-parts":[[2008,3,7]],"date-time":"2008-03-07T18:06:03Z","timestamp":1204913163000},"page":"1147-1165","source":"Crossref","is-referenced-by-count":1,"title":["High Frequency Induced Instability in Nystr\u00f6m Methods for the van der Pol Equation"],"prefix":"10.1137","volume":"46","author":[{"given":"S. W.","family":"Schoombie","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"E.","family":"Mar\u00e9","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,3,7]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1993.1131"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"E. Hairer, S. P. N\u00f8rsett, and G. Wanner,\n                      Solving Ordinary Differential Equations\n                      I, Springer-Verlag, Berlin, 1988.","DOI":"10.1007\/978-3-662-12607-3"},{"key":"R3","doi-asserted-by":"crossref","unstructured":"D. W. Jordan and P. Smith,\n                      Nonlinear Ordinary Differential Equations\n                      , 3rd ed., Oxford University Press, Oxford, UK, 1999.","DOI":"10.1093\/oso\/9780198565635.001.0001"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1080\/16073606.1994.9631770"},{"key":"R5","unstructured":"J. D. Murray,\n                      Mathematical Biology.\n                      I, 3rd ed., Springer-Verlag, New York, 2002."},{"key":"R6","unstructured":"A. H. Nayfeh,\n                      Perturbation Methods\n                      , Wiley, New York, 1973."},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/0906062"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/0147005"},{"key":"R9","unstructured":"S. W. Schoombie,\n                      A Discrete Multiple Scales Analysis of a Discrete Version of the Korteweg\u2013de Vries Equation\n                      , Report NA\/109, University of Dundee, 1987."},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(92)90042-W"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(86)90269-X"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1080\/14786442208633932"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/050644148","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:13:12Z","timestamp":1787325192000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/050644148"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":12,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/050644148"],"URL":"https:\/\/doi.org\/10.1137\/050644148","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}