{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:44:38Z","timestamp":1776836678956,"version":"3.51.2"},"reference-count":52,"publisher":"American Mathematical Society (AMS)","issue":"341","license":[{"start":{"date-parts":[[2023,11,29]],"date-time":"2023-11-29T00:00:00Z","timestamp":1701216000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We establish improved uniform error bounds for the time-splitting methods for the long-time dynamics of the Schr\u00f6dinger equation with small potential and the nonlinear Schr\u00f6dinger equation (NLSE) with weak nonlinearity. For the Schr\u00f6dinger equation with small potential characterized by a dimensionless parameter\n \n \n \n \n \n \u03b5\n \n <\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n \\varepsilon \\in (0, 1]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , we employ the unitary flow property of the (second-order) time-splitting Fourier pseudospectral (TSFP) method in\n \n \n \n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n L^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -norm to prove a uniform error bound at time\n \n \n \n \n \n t<\/mml:mi>\n \n \u03b5\n \n <\/mml:mi>\n <\/mml:msub>\n =<\/mml:mo>\n t<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n \u03b5\n \n <\/mml:mi>\n <\/mml:mrow>\n t_\\varepsilon =t\/\\varepsilon<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n as\n \n \n \n \n C<\/mml:mi>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n \n \n C<\/mml:mi>\n \n ~\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n (<\/mml:mo>\n T<\/mml:mi>\n )<\/mml:mo>\n (<\/mml:mo>\n \n h<\/mml:mi>\n m<\/mml:mi>\n <\/mml:msup>\n +<\/mml:mo>\n \n \n \u03c4\n \n <\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n C(t)\\widetilde {C}(T)(h^m +\\tau ^2)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n up to\n \n \n \n \n \n t<\/mml:mi>\n \n \u03b5\n \n <\/mml:mi>\n <\/mml:msub>\n \n \u2264\n \n <\/mml:mo>\n \n T<\/mml:mi>\n \n \u03b5\n \n <\/mml:mi>\n <\/mml:msub>\n =<\/mml:mo>\n T<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n \u03b5\n \n <\/mml:mi>\n <\/mml:mrow>\n t_\\varepsilon \\leq T_\\varepsilon = T\/\\varepsilon<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for any\n \n \n \n \n T<\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n T>0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and uniformly for\n \n \n \n \n \n \u03b5\n \n <\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n \\varepsilon \\in (0,1]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , while\n \n \n \n h<\/mml:mi>\n h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the mesh size,\n \n \n \n \n \u03c4\n \n <\/mml:mi>\n \\tau<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the time step,\n \n \n \n \n m<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n m \\ge 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n \n C<\/mml:mi>\n \n ~\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n (<\/mml:mo>\n T<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\tilde {C}(T)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n (the local error bound) depend on the regularity of the exact solution, and\n \n \n \n \n C<\/mml:mi>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n \n C<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n +<\/mml:mo>\n \n C<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n t<\/mml:mi>\n <\/mml:mrow>\n C(t) =C_0+C_1t<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n grows at most linearly with respect to\n \n \n \n t<\/mml:mi>\n t<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with\n \n \n \n \n C<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n C_0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n C<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n C_1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n two positive constants independent of\n \n \n \n T<\/mml:mi>\n T<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n \u03b5\n \n <\/mml:mi>\n \\varepsilon<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n h<\/mml:mi>\n h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \u03c4\n \n <\/mml:mi>\n \\tau<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Then by introducing a new technique of\n regularity compensation oscillation<\/italic>\n (RCO) in which the high frequency modes are controlled by regularity and the low frequency modes are analyzed by phase cancellation and energy method, an improved uniform (w.r.t\n \n \n \n \n \u03b5\n \n <\/mml:mi>\n \\varepsilon<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ) error bound at\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n h<\/mml:mi>\n \n m<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n +<\/mml:mo>\n \n \u03b5\n \n <\/mml:mi>\n \n \n \u03c4\n \n <\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n O(h^{m-1} + \\varepsilon \\tau ^2)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is established in\n \n \n \n \n H<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msup>\n H^1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -norm for the long-time dynamics up to the time at\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n \u03b5\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n O(1\/\\varepsilon )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of the Schr\u00f6dinger equation with\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n \u03b5\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n O(\\varepsilon )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -potential with\n \n \n \n \n m<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n m \\geq 3<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Moreover, the RCO technique is extended to prove an improved uniform error bound at\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n h<\/mml:mi>\n \n m<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n +<\/mml:mo>\n \n \n \u03b5\n \n <\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \n \n \u03c4\n \n <\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n O(h^{m-1} + \\varepsilon ^2\\tau ^2)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in\n \n \n \n \n H<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msup>\n H^1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -norm for the long-time dynamics up to the time at\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n \n \u03b5\n \n <\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n O(1\/\\varepsilon ^2)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of the cubic NLSE with\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n \n \u03b5\n \n <\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n O(\\varepsilon ^2)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -nonlinearity strength. Extensions to the first-order and fourth-order time-splitting methods are discussed. Numerical results are reported to validate our error estimates and to demonstrate that they are sharp.\n <\/p>","DOI":"10.1090\/mcom\/3801","type":"journal-article","created":{"date-parts":[[2022,11,29]],"date-time":"2022-11-29T11:01:17Z","timestamp":1669719677000},"page":"1109-1139","source":"Crossref","is-referenced-by-count":27,"title":["Improved uniform error bounds of the time-splitting methods for the long-time (nonlinear) Schr\u00f6dinger equation"],"prefix":"10.1090","volume":"92","author":[{"given":"Weizhu","family":"Bao","sequence":"first","affiliation":[]},{"given":"Yongyong","family":"Cai","sequence":"additional","affiliation":[]},{"given":"Yue","family":"Feng","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2022,11,29]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1093\/imanum\/13.1.115","article-title":"Finite difference discretization of the cubic Schr\u00f6dinger equation","volume":"13","author":"Akrivis, Georgios D.","year":"1993","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"12","key":"2","doi-asserted-by":"publisher","first-page":"2621","DOI":"10.1016\/j.cpc.2013.07.012","article-title":"Computational methods for the dynamics of the nonlinear Schr\u00f6dinger\/Gross-Pitaevskii equations","volume":"184","author":"Antoine, Xavier","year":"2013","journal-title":"Comput. Phys. Commun.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-4655","issn-type":"print"},{"issue":"1","key":"3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.3934\/krm.2013.6.1","article-title":"Mathematical theory and numerical methods for Bose-Einstein condensation","volume":"6","author":"Bao, Weizhu","year":"2013","journal-title":"Kinet. Relat. Models","ISSN":"https:\/\/id.crossref.org\/issn\/1937-5093","issn-type":"print"},{"issue":"3","key":"4","doi-asserted-by":"publisher","first-page":"1103","DOI":"10.1137\/120866890","article-title":"Uniform and optimal error estimates of an exponential wave integrator sine pseudospectral method for the nonlinear Schr\u00f6dinger equation with wave operator","volume":"52","author":"Bao, Weizhu","year":"2014","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"5","doi-asserted-by":"publisher","first-page":"1962","DOI":"10.1137\/21M1449774","article-title":"Improved uniform error bounds on time-splitting methods for long-time dynamics of the nonlinear Klein-Gordon equation with weak nonlinearity","volume":"60","author":"Bao, Weizhu","year":"2022","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"6","unstructured":"W. Bao, Y. Cai, and Y. Feng, Improved uniform error bounds on time-splitting methods for the long-time dynamics of the weakly nonlinear Dirac equation, arXiv:2203.05886."},{"issue":"325","key":"7","doi-asserted-by":"publisher","first-page":"2141","DOI":"10.1090\/mcom\/3536","article-title":"Super-resolution of time-splitting methods for the Dirac equation in the nonrelativistic regime","volume":"89","author":"Bao, Weizhu","year":"2020","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"8","doi-asserted-by":"publisher","first-page":"1040","DOI":"10.1137\/19M1271828","article-title":"Uniform error bounds of time-splitting methods for the nonlinear Dirac equation in the nonrelativistic regime without magnetic potential","volume":"59","author":"Bao, Weizhu","year":"2021","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"9","doi-asserted-by":"publisher","first-page":"1040","DOI":"10.1137\/22M146995X","article-title":"Improved uniform error bounds on time-splitting methods for the long-time dynamics of the Dirac equation with small potentials","volume":"20","author":"Bao, Weizhu","year":"2022","journal-title":"Multiscale Model. Simul.","ISSN":"https:\/\/id.crossref.org\/issn\/1540-3459","issn-type":"print"},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"318","DOI":"10.1016\/S0021-9991(03)00102-5","article-title":"Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation","volume":"187","author":"Bao, Weizhu","year":"2003","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"2","key":"11","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1006\/jcph.2001.6956","article-title":"On time-splitting spectral approximations for the Schr\u00f6dinger equation in the semiclassical regime","volume":"175","author":"Bao, Weizhu","year":"2002","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"6","key":"12","doi-asserted-by":"publisher","first-page":"2010","DOI":"10.1137\/030601211","article-title":"A fourth-order time-splitting Laguerre-Hermite pseudospectral method for Bose-Einstein condensates","volume":"26","author":"Bao, Weizhu","year":"2005","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"1","key":"13","doi-asserted-by":"publisher","first-page":"26","DOI":"10.1137\/S0036142900381497","article-title":"Order estimates in time of splitting methods for the nonlinear Schr\u00f6dinger equation","volume":"40","author":"Besse, Christophe","year":"2002","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"2","key":"14","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1007\/BF01896020","article-title":"Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schr\u00f6dinger equations","volume":"3","author":"Bourgain, J.","year":"1993","journal-title":"Geom. Funct. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1016-443X","issn-type":"print"},{"issue":"1","key":"15","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1007\/s002200050644","article-title":"Growth of Sobolev norms in linear Schr\u00f6dinger equations with quasi-periodic potential","volume":"204","author":"Bourgain, J.","year":"1999","journal-title":"Comm. Math. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3616","issn-type":"print"},{"issue":"7","key":"16","doi-asserted-by":"publisher","first-page":"1407","DOI":"10.1002\/cpa.21749","article-title":"Effective dynamics of the nonlinear Schr\u00f6dinger equation on large domains","volume":"71","author":"Buckmaster, T.","year":"2018","journal-title":"Comm. Pure Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3640","issn-type":"print"},{"issue":"3","key":"17","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1353\/ajm.2004.0016","article-title":"Strichartz inequalities and the nonlinear Schr\u00f6dinger equation on compact manifolds","volume":"126","author":"Burq, N.","year":"2004","journal-title":"Amer. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9327","issn-type":"print"},{"issue":"6","key":"18","doi-asserted-by":"publisher","first-page":"3232","DOI":"10.1137\/120892416","article-title":"On Fourier time-splitting methods for nonlinear Schr\u00f6dinger equations in the semiclassical limit","volume":"51","author":"Carles, R\u00e9mi","year":"2013","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"2","key":"19","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"},{"key":"20","series-title":"Courant Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/cln\/010","volume-title":"Semilinear Schr\\\"{o}dinger equations","volume":"10","author":"Cazenave, Thierry","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/0821833995"},{"issue":"3","key":"21","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1007\/s10208-007-9016-7","article-title":"Symmetric exponential integrators with an application to the cubic Schr\u00f6dinger equation","volume":"8","author":"Celledoni, Elena","year":"2008","journal-title":"Found. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1615-3375","issn-type":"print"},{"issue":"302","key":"22","doi-asserted-by":"publisher","first-page":"2863","DOI":"10.1090\/mcom\/3088","article-title":"Improved error estimates for splitting methods applied to highly-oscillatory nonlinear Schr\u00f6dinger equations","volume":"85","author":"Chartier, Philippe","year":"2016","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"23","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1007\/s00211-007-0123-9","article-title":"Energy-conserved splitting FDTD methods for Maxwell\u2019s equations","volume":"108","author":"Chen, Wenbin","year":"2008","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"4","key":"24","doi-asserted-by":"publisher","first-page":"1530","DOI":"10.1137\/090765857","article-title":"Energy-conserved splitting finite-difference time-domain methods for Maxwell\u2019s equations in three dimensions","volume":"48","author":"Chen, Wenbin","year":"2010","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"25","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1007\/s10208-002-0062-x","article-title":"Modulated Fourier expansions of highly oscillatory differential equations","volume":"3","author":"Cohen, David","year":"2003","journal-title":"Found. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1615-3375","issn-type":"print"},{"issue":"5-6","key":"26","doi-asserted-by":"publisher","first-page":"659","DOI":"10.4310\/MRL.2002.v9.n5.a9","article-title":"Almost conservation laws and global rough solutions to a nonlinear Schr\u00f6dinger equation","volume":"9","author":"Colliander, J.","year":"2002","journal-title":"Math. Res. Lett.","ISSN":"https:\/\/id.crossref.org\/issn\/1073-2780","issn-type":"print"},{"issue":"5","key":"27","doi-asserted-by":"publisher","first-page":"3705","DOI":"10.1137\/080744578","article-title":"Modified energy for split-step methods applied to the linear Schr\u00f6dinger equation","volume":"47","author":"Debussche, Arnaud","year":"2009","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"2","key":"28","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1016\/0021-9991(81)90052-8","article-title":"Finite-difference solutions of a nonlinear Schr\u00f6dinger equation","volume":"44","author":"Delfour, M.","year":"1981","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"8","key":"29","doi-asserted-by":"publisher","first-page":"1839","DOI":"10.1016\/j.apnum.2009.02.002","article-title":"Exponential Runge-Kutta methods for the Schr\u00f6dinger equation","volume":"59","author":"Dujardin, Guillaume","year":"2009","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"issue":"2","key":"30","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1007\/s00211-007-0119-5","article-title":"Normal form and long time analysis of splitting schemes for the linear Schr\u00f6dinger equation with small potential","volume":"108","author":"Dujardin, Guillaume","year":"2007","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"3","key":"31","doi-asserted-by":"publisher","first-page":"515","DOI":"10.1007\/s00222-006-0022-1","article-title":"Derivation of the cubic non-linear Schr\u00f6dinger equation from quantum dynamics of many-body systems","volume":"167","author":"Erd\u0151s, L\u00e1szl\u00f3","year":"2007","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"key":"32","series-title":"Zurich Lectures in Advanced Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.4171\/100","volume-title":"Geometric numerical integration and Schr\\\"{o}dinger equations","author":"Faou, Erwan","year":"2012","ISBN":"https:\/\/id.crossref.org\/isbn\/9783037191002"},{"issue":"7","key":"33","doi-asserted-by":"publisher","first-page":"1123","DOI":"10.1080\/03605302.2013.785562","article-title":"Sobolev stability of plane wave solutions to the cubic nonlinear Schr\u00f6dinger equation on a torus","volume":"38","author":"Faou, Erwan","year":"2013","journal-title":"Comm. Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0360-5302","issn-type":"print"},{"issue":"4","key":"34","doi-asserted-by":"publisher","first-page":"915","DOI":"10.1090\/jams\/845","article-title":"The weakly nonlinear large-box limit of the 2D cubic nonlinear Schr\u00f6dinger equation","volume":"29","author":"Faou, Erwan","year":"2016","journal-title":"J. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0894-0347","issn-type":"print"},{"issue":"3","key":"35","doi-asserted-by":"publisher","first-page":"429","DOI":"10.1007\/s00211-009-0258-y","article-title":"Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. I. Finite-dimensional discretization","volume":"114","author":"Faou, Erwan","year":"2010","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"2","key":"36","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1007\/s10208-010-9059-z","article-title":"Nonlinear Schr\u00f6dinger equations and their spectral semi-discretizations over long times","volume":"10","author":"Gauckler, Ludwig","year":"2010","journal-title":"Found. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1615-3375","issn-type":"print"},{"issue":"3","key":"37","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1007\/s10208-010-9063-3","article-title":"Splitting integrators for nonlinear Schr\u00f6dinger equations over long times","volume":"10","author":"Gauckler, Ludwig","year":"2010","journal-title":"Found. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1615-3375","issn-type":"print"},{"key":"38","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","volume-title":"Geometric numerical integration","volume":"31","author":"Hairer, Ernst","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9783642051579"},{"key":"39","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1515\/crelle-2012-0013","article-title":"Strichartz estimates for partially periodic solutions to Schr\u00f6dinger equations in 4\ud835\udc51 and applications","volume":"690","author":"Herr, Sebastian","year":"2014","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"key":"40","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1017\/S0962492910000048","article-title":"Exponential integrators","volume":"19","author":"Hochbruck, Marlis","year":"2010","journal-title":"Acta Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0962-4929","issn-type":"print"},{"issue":"4","key":"41","doi-asserted-by":"publisher","first-page":"735","DOI":"10.1023\/A:1022396519656","article-title":"Error bounds for exponential operator splittings","volume":"40","author":"Jahnke, Tobias","year":"2000","journal-title":"BIT","ISSN":"https:\/\/id.crossref.org\/issn\/0006-3835","issn-type":"print"},{"issue":"264","key":"42","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":"43","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1017\/S0962492902000053","article-title":"Splitting methods","volume":"11","author":"McLachlan, Robert I.","year":"2002","journal-title":"Acta Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0962-4929","issn-type":"print"},{"issue":"1","key":"44","doi-asserted-by":"publisher","first-page":"299","DOI":"10.1088\/0951-7715\/13\/1\/314","article-title":"Resonant and Diophantine step sizes in computing invariant tori of Hamiltonian systems","volume":"13","author":"Shang, Zai-jiu","year":"2000","journal-title":"Nonlinearity","ISSN":"https:\/\/id.crossref.org\/issn\/0951-7715","issn-type":"print"},{"key":"45","doi-asserted-by":"crossref","unstructured":"J. Shen, T. Tang, and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer-Verlag Berlin Heidelberg, 2011.","DOI":"10.1007\/978-3-540-71041-7"},{"key":"46","doi-asserted-by":"publisher","first-page":"506","DOI":"10.1137\/0705041","article-title":"On the construction and comparison of difference schemes","volume":"5","author":"Strang, Gilbert","year":"1968","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"47","series-title":"Applied Mathematical Sciences","isbn-type":"print","volume-title":"The nonlinear Schr\\\"{o}dinger equation","volume":"139","author":"Sulem, Catherine","year":"1999","ISBN":"https:\/\/id.crossref.org\/isbn\/0387986111"},{"key":"48","series-title":"CBMS Regional Conference Series in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/cbms\/106","volume-title":"Nonlinear dispersive equations","volume":"106","author":"Tao, Terence","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/0821841432"},{"issue":"4","key":"49","doi-asserted-by":"publisher","first-page":"2022","DOI":"10.1137\/060674636","article-title":"High-order exponential operator splitting methods for time-dependent Schr\u00f6dinger equations","volume":"46","author":"Thalhammer, Mechthild","year":"2008","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"6","key":"50","doi-asserted-by":"publisher","first-page":"3231","DOI":"10.1137\/120866373","article-title":"Convergence analysis of high-order time-splitting pseudospectral methods for nonlinear Schr\u00f6dinger equations","volume":"50","author":"Thalhammer, Mechthild","year":"2012","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"11","key":"51","doi-asserted-by":"publisher","first-page":"2926","DOI":"10.1016\/j.jfa.2007.11.012","article-title":"Bounded Sobolev norms for linear Schr\u00f6dinger equations under resonant perturbations","volume":"254","author":"Wang, W.-M.","year":"2008","journal-title":"J. Funct. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-1236","issn-type":"print"},{"issue":"3","key":"52","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"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-341\/S0025-5718-2022-03801-1\/S0025-5718-2022-03801-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:53:15Z","timestamp":1776833595000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-341\/S0025-5718-2022-03801-1\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,29]]},"references-count":52,"journal-issue":{"issue":"341","published-print":{"date-parts":[[2023,5]]}},"alternative-id":["S0025-5718-2022-03801-1"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3801","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":[[2022,11,29]]}}}