{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,21]],"date-time":"2026-05-21T14:19:48Z","timestamp":1779373188977,"version":"3.53.1"},"reference-count":37,"publisher":"Wiley","license":[{"start":{"date-parts":[[2012,10,17]],"date-time":"2012-10-17T00:00:00Z","timestamp":1350432000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11201239"],"award-info":[{"award-number":["11201239"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11126292"],"award-info":[{"award-number":["11126292"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11201239"],"award-info":[{"award-number":["11201239"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11126292"],"award-info":[{"award-number":["11126292"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Advances in Numerical Analysis"],"published-print":{"date-parts":[[2012,10,17]]},"abstract":"<jats:p>A new compact difference scheme is proposed for solving the nonlinear Schr\u00f6dinger equation. The scheme is proved to conserve the total mass and the total energy and the optimal convergent rate, without any restriction on the grid ratio, at the order of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mi>O<\/mml:mi><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mrow><mml:msup><mml:mi>h<\/mml:mi><mml:mn>8<\/mml:mn><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mi>\u03c4<\/mml:mi><mml:mn>2<\/mml:mn><\/mml:msup><\/mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math> in the discrete <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:msup><mml:mi>L<\/mml:mi><mml:mi>\u221e<\/mml:mi><\/mml:msup><\/mml:mrow><\/mml:math>-norm with time step <jats:italic>\u03c4<\/jats:italic> and mesh size <jats:italic>h<\/jats:italic>. In numerical analysis, beside the standard techniques of the energy method, a new technique named \u201cregression of compactness\u201d and some lemmas are proposed to prove the high-order convergence. For computing the nonlinear algebraical systems generated by the nonlinear compact scheme, an efficient iterative algorithm is constructed. Numerical examples are given to support the theoretical analysis.<\/jats:p>","DOI":"10.1155\/2012\/913429","type":"journal-article","created":{"date-parts":[[2012,10,19]],"date-time":"2012-10-19T01:12:07Z","timestamp":1350609127000},"page":"1-24","source":"Crossref","is-referenced-by-count":17,"title":["Convergence of an Eighth-Order Compact Difference Scheme for the Nonlinear Schr\u00f6dinger Equation"],"prefix":"10.1155","volume":"2012","author":[{"given":"Tingchun","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","reference":[{"key":"1","year":"1995"},{"key":"2","first-page":"392","volume":"5","year":"1998","journal-title":"Journal of the Optical Society of America B"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1143\/JPSJ.61.2241"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/13.1.115"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1137\/0724025"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1998.6120"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1137\/0729011"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1016\/j.cpc.2009.08.015"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142997319728"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1996.5589"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.2307\/2007397"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2005.09.011"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142994270636"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/6.1.25"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1016\/0096-3003(94)00152-T"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1137\/0732083"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1007\/BF01385769"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.1137\/0730018"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.1002\/num.1690100609"},{"key":"20","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142902413391"},{"issue":"5","key":"21","doi-asserted-by":"crossref","first-page":"4684","DOI":"10.1103\/PhysRevE.49.4684","volume":"49","year":"1994","journal-title":"Physical Review E"},{"key":"22","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/0021-9991(83)90015-3","volume":"52","year":"1983","journal-title":"Journal of Computational Physics"},{"key":"23","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142996302396"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050067"},{"key":"25","doi-asserted-by":"publisher","DOI":"10.1137\/050622833"},{"key":"26","year":"2002"},{"key":"27","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827502410797"},{"key":"28","doi-asserted-by":"publisher","DOI":"10.1137\/030602459"},{"key":"29","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827503425900"},{"key":"30","doi-asserted-by":"publisher","DOI":"10.1002\/fld.263"},{"key":"31","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(92)90324-R"},{"key":"32","doi-asserted-by":"publisher","DOI":"10.1002\/num.20414"},{"key":"33","doi-asserted-by":"publisher","DOI":"10.1137\/080714907"},{"key":"34","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2008.11.011"},{"key":"35","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827599350320"},{"key":"37","year":"1985"},{"key":"36","series-title":"Proceedings of Symposia in Applied Mathematics","first-page":"24","volume-title":"Existence and uniqueness theorems for solutions of nonlinear boundary value problems","year":"1965"}],"container-title":["Advances in Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/archive\/2012\/913429.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/archive\/2012\/913429.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/archive\/2012\/913429.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,12,10]],"date-time":"2020-12-10T05:24:00Z","timestamp":1607577840000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/ana\/2012\/913429\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,10,17]]},"references-count":37,"alternative-id":["913429","913429"],"URL":"https:\/\/doi.org\/10.1155\/2012\/913429","relation":{},"ISSN":["1687-9562","1687-9570"],"issn-type":[{"value":"1687-9562","type":"print"},{"value":"1687-9570","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,10,17]]}}}