{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,8]],"date-time":"2026-08-08T06:07:36Z","timestamp":1786169256214,"version":"3.56.0"},"reference-count":7,"publisher":"American Mathematical Society (AMS)","issue":"302","license":[{"start":{"date-parts":[[2017,2,16]],"date-time":"2017-02-16T00:00:00Z","timestamp":1487203200000},"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>\n                    In this work, the error behavior of operator splitting methods is analyzed for highly-oscillatory differential equations. The scope of applications includes time-dependent nonlinear Schr\u00f6dinger equations, where the evolution operator associated with the principal linear part is highly-oscillatory and periodic in time. In a first step, a known convergence result for the second-order Strang splitting method applied to the cubic Schr\u00f6dinger equation is adapted to a wider class of nonlinearities. In a second step, the dependence of the global error on the decisive parameter\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"0 greater-than epsilon greater-than greater-than 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>\n                              \u03b5\n                              \n                            <\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mspace width=\"negativethinmathspace\"\/>\n                            <mml:mspace width=\"negativethinmathspace\"\/>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">0 &gt; \\varepsilon &gt;\\!\\!&gt; 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , defining the length of the period, is examined. The main result states that, compared to established error estimates, the Strang splitting method is more accurate by a factor\u00a0\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"epsilon\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03b5\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\varepsilon<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , provided that the time stepsize is chosen as an integer fraction of the period. This improved error behavior over a time interval of fixed length, which is independent of the period, is due to an averaging effect. The extension of the convergence result to higher-order splitting methods and numerical illustrations complement the investigations.\n                  <\/p>","DOI":"10.1090\/mcom\/3088","type":"journal-article","created":{"date-parts":[[2015,12,30]],"date-time":"2015-12-30T09:46:25Z","timestamp":1451468785000},"page":"2863-2885","source":"Crossref","is-referenced-by-count":37,"title":["Improved error estimates for splitting methods applied to highly-oscillatory nonlinear Schr\u00f6dinger equations"],"prefix":"10.1090","volume":"85","author":[{"given":"Philippe","family":"Chartier","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Florian","family":"M\u00e9hats","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mechthild","family":"Thalhammer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yong","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2016,2,16]]},"reference":[{"key":"1","series-title":"Graduate Studies in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/082","volume-title":"Pseudo-differential operators and the Nash-Moser theorem","volume":"82","author":"Alinhac, Serge","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9780821834541"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"519","DOI":"10.1007\/s10208-014-9235-7","article-title":"Stroboscopic averaging for the nonlinear Schr\u00f6dinger equation","volume":"15","author":"Castella, F.","year":"2015","journal-title":"Found. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1615-3375","issn-type":"print"},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"396","DOI":"10.1093\/imanum\/drp041","article-title":"Convergence of a split-step Hermite method for the Gross-Pitaevskii equation","volume":"31","author":"Gauckler, Ludwig","year":"2011","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"264","key":"4","doi-asserted-by":"publisher","first-page":"2141","DOI":"10.1090\/S0025-5718-08-02101-7","article-title":"On splitting methods for Schr\u00f6dinger-Poisson and cubic nonlinear Schr\u00f6dinger equations","volume":"77","author":"Lubich, Christian","year":"2008","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"5","series-title":"Cambridge Studies in Advanced Mathematics","isbn-type":"print","volume-title":"Classical and multilinear harmonic analysis. Vol. II","volume":"138","author":"Muscalu, Camil","year":"2013","ISBN":"https:\/\/id.crossref.org\/isbn\/9781107031821"},{"issue":"3","key":"6","doi-asserted-by":"publisher","first-page":"485","DOI":"10.1137\/0723033","article-title":"Split-step methods for the solution of the nonlinear Schr\u00f6dinger equation","volume":"23","author":"Weideman, J. A. C.","year":"1986","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"5-7","key":"7","doi-asserted-by":"publisher","first-page":"262","DOI":"10.1016\/0375-9601(90)90092-3","article-title":"Construction of higher order symplectic integrators","volume":"150","author":"Yoshida, Haruo","year":"1990","journal-title":"Phys. Lett. A","ISSN":"https:\/\/id.crossref.org\/issn\/0375-9601","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2016-85-302\/S0025-5718-2016-03088-4\/S0025-5718-2016-03088-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-302\/S0025-5718-2016-03088-4\/S0025-5718-2016-03088-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:56:41Z","timestamp":1776797801000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-302\/S0025-5718-2016-03088-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,2,16]]},"references-count":7,"journal-issue":{"issue":"302","published-print":{"date-parts":[[2016,11]]}},"alternative-id":["S0025-5718-2016-03088-4"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3088","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":[[2016,2,16]]}}}