{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T12:48:12Z","timestamp":1774615692434,"version":"3.50.1"},"reference-count":30,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T00:00:00Z","timestamp":1627603200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T00:00:00Z","timestamp":1627603200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/100000086","name":"Directorate for Mathematical and Physical Sciences","doi-asserted-by":"publisher","award":["DMS-2012585"],"award-info":[{"award-number":["DMS-2012585"]}],"id":[{"id":"10.13039\/100000086","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2021,9]]},"DOI":"10.1007\/s10915-021-01576-y","type":"journal-article","created":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T05:02:36Z","timestamp":1627621356000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Second-Order SAV Schemes for the Nonlinear Schr\u00f6dinger Equation and Their Error Analysis"],"prefix":"10.1007","volume":"88","author":[{"given":"Beichuan","family":"Deng","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4885-5732","authenticated-orcid":false,"given":"Jie","family":"Shen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qingqu","family":"Zhuang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,7,30]]},"reference":[{"key":"1576_CR1","volume-title":"Optical Solitons","author":"F Abdullaev","year":"2014","unstructured":"Abdullaev, F., Darmanyan, S., Khabibullaev, P., Engelbrecht, J.: Optical Solitons. Springer, Berlin (2014)"},{"issue":"1","key":"1576_CR2","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1007\/BF01385769","volume":"59","author":"G Akrivis","year":"1991","unstructured":"Akrivis, G., Dougalis, V., Karakashian, O.: On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear schr\u00f6dinger equation. Numer. Math. 59(1), 31\u201353 (1991)","journal-title":"Numer. Math."},{"issue":"4","key":"1576_CR3","first-page":"729","volume":"4","author":"X Antoine","year":"2008","unstructured":"Antoine, X., Arnold, A., Besse, C., Ehrhardt, M., Schadle, A.: A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrodinger equations. Commun. Comput. Phys. 4(4), 729\u2013796 (2008)","journal-title":"Commun. Comput. Phys."},{"issue":"12","key":"1576_CR4","doi-asserted-by":"publisher","first-page":"2621","DOI":"10.1016\/j.cpc.2013.07.012","volume":"184","author":"X Antoine","year":"2013","unstructured":"Antoine, X., Bao, W., Besse, C.: Computational methods for the dynamics of the nonlinear Schr\u00f6dinger\/Gross\u2013Pitaevskii equations. Comput. Phys. Commun. 184(12), 2621\u20132633 (2013)","journal-title":"Comput. Phys. Commun."},{"issue":"2","key":"1576_CR5","doi-asserted-by":"publisher","first-page":"1008","DOI":"10.1137\/090780535","volume":"33","author":"X Antoine","year":"2011","unstructured":"Antoine, X., Besse, C., Klein, P.: Absorbing boundary conditions for general nonlinear Schr\u00f6dinger equations. SIAM J. Sci. Comput. 33(2), 1008\u20131033 (2011)","journal-title":"SIAM J. Sci. Comput."},{"issue":"8","key":"1576_CR6","doi-asserted-by":"publisher","first-page":"1491","DOI":"10.1134\/S1054660X11150011","volume":"21","author":"X Antoine","year":"2011","unstructured":"Antoine, X., Besse, C., Klein, P.: Numerical solution of time-dependent nonlinear Schr\u00f6dinger equations using domain truncation techniques coupled with relaxation scheme. Laser Phys. 21(8), 1491\u20131502 (2011)","journal-title":"Laser Phys."},{"key":"1576_CR7","doi-asserted-by":"crossref","unstructured":"Antoine, X., Shen, J., Tang, Q.: Scalar auxiliary variable\/lagrange multiplier based pseudospectral schemes for the dynamics of nonlinear Schr\u00f6dinger\/Gross\u2013Pitaevskii equations (2020). https:\/\/hal.archives-ouvertes.fr\/hal-02940080\/document","DOI":"10.1016\/j.jcp.2021.110328"},{"issue":"2","key":"1576_CR8","doi-asserted-by":"publisher","first-page":"492","DOI":"10.1137\/110830800","volume":"50","author":"W Bao","year":"2012","unstructured":"Bao, W., Cai, Y.: Uniform error estimates of finite difference methods for the nonlinear Schr\u00f6dinger equation with wave operator. SIAM J. Numer. Anal. 50(2), 492\u2013521 (2012)","journal-title":"SIAM J. Numer. Anal."},{"issue":"6","key":"1576_CR9","doi-asserted-by":"publisher","first-page":"2010","DOI":"10.1137\/030601211","volume":"26","author":"W Bao","year":"2005","unstructured":"Bao, W., Shen, J.: A fourth-order time-splitting laguerre-hermite pseudospectral method for Bose\u2013Einstein condensates. SIAM J. Sci. Comput. 26(6), 2010\u20132028 (2005)","journal-title":"SIAM J. Sci. Comput."},{"issue":"3","key":"1576_CR10","doi-asserted-by":"publisher","first-page":"934","DOI":"10.1137\/S0036142901396521","volume":"42","author":"C Besse","year":"2004","unstructured":"Besse, C.: A relaxation scheme for the nonlinear Schr\u00f6dinger equation. SIAM J. Numer. Anal. 42(3), 934\u2013952 (2004)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"1576_CR11","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1016\/j.cam.2005.04.066","volume":"193","author":"A Borz\u00ec","year":"2006","unstructured":"Borz\u00ec, A., Decker, E.: Analysis of a leap-frog pseudospectral scheme for the Schr\u00f6dinger equation. J. Comput. Appl. Math. 193(1), 65\u201388 (2006)","journal-title":"J. Comput. Appl. Math."},{"key":"1576_CR12","doi-asserted-by":"crossref","unstructured":"Cazenave, T.: Semilinear Schr\u00f6dinger Equations, volume\u00a010 of Courant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI (2003)","DOI":"10.1090\/cln\/010"},{"issue":"4","key":"1576_CR13","doi-asserted-by":"publisher","first-page":"A2489","DOI":"10.1137\/19M1306221","volume":"42","author":"Q Cheng","year":"2020","unstructured":"Cheng, Q., Shen, J.: Global constraints preserving scalar auxiliary variable schemes for gradient flows. SIAM J. Sci. Comput. 42(4), A2489\u2013A2513 (2020)","journal-title":"SIAM J. Sci. Comput."},{"issue":"4","key":"1576_CR14","doi-asserted-by":"publisher","first-page":"979","DOI":"10.1002\/num.20468","volume":"26","author":"M Dehghan","year":"2010","unstructured":"Dehghan, M., Taleei, A.: Numerical solution of nonlinear Schr\u00f6dinger equation by using time\u2013space pseudo-spectral method. Numer. Methods Partial Differ. Equ. 26(4), 979\u2013992 (2010)","journal-title":"Numer. Methods Partial Differ. Equ."},{"issue":"3\u20134","key":"1576_CR15","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1016\/0045-7825(93)90007-K","volume":"108","author":"LRT Gardner","year":"1993","unstructured":"Gardner, L.R.T., Gardner, G.A., Zaki, S.I., El-Sahrawi, Z.: B-spline finite element studies of the non-linear schr\u00f6dinger equation. Comput. Methods Appl. Mech. Eng. 108(3\u20134), 303\u2013318 (1993)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1576_CR16","unstructured":"Guo, B., Shen, J., Xu, C.: Spectral and Pseudospectral Approximations Using Hermite Functions: Application to the Dirac Equation, vol.\u00a019, pp. 35\u201355 (2003). Challenges in Computational Mathematics, Pohang (2001)"},{"issue":"2","key":"1576_CR17","doi-asserted-by":"publisher","first-page":"1366","DOI":"10.1137\/070683787","volume":"47","author":"LI Ignat","year":"2009","unstructured":"Ignat, L.I., Zuazua, E.: Numerical dispersive schemes for the nonlinear Schr\u00f6dinger equation. SIAM J. Numer. Anal. 47(2), 1366\u20131390 (2009)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1\u20132","key":"1576_CR18","doi-asserted-by":"publisher","first-page":"588","DOI":"10.1016\/j.cam.2007.09.025","volume":"220","author":"T Lu","year":"2008","unstructured":"Lu, T., Cai, W.: Fourier spectral-discontinuous Galerkin method for time-dependent 3-d Schrodinger\u2013Poisson equations with discontinuous potentials. J. Comput. Appl. Math. 220(1\u20132), 588\u2013614 (2008)","journal-title":"J. Comput. Appl. Math."},{"issue":"7","key":"1576_CR19","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1016\/S0898-1221(97)00042-4","volume":"33","author":"MP Robinson","year":"1997","unstructured":"Robinson, M.P.: The solution of nonlinear Schr\u00f6dinger equations using orthogonal spline collocation. Comput. Math. Appl. 33(7), 39\u201357 (1997)","journal-title":"Comput. Math. Appl."},{"issue":"6","key":"1576_CR20","doi-asserted-by":"publisher","first-page":"1049","DOI":"10.1103\/PhysRev.28.1049","volume":"28","author":"E Schr\u00f6dinger","year":"1926","unstructured":"Schr\u00f6dinger, E.: An undulatory theory of the mechanics of atoms and molecules. Phys. Rev. 28(6), 1049 (1926)","journal-title":"Phys. Rev."},{"issue":"5","key":"1576_CR21","doi-asserted-by":"publisher","first-page":"2895","DOI":"10.1137\/17M1159968","volume":"56","author":"J Shen","year":"2018","unstructured":"Shen, J., Xu, J.: Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows. SIAM J. Numer. Anal. 56(5), 2895\u20132912 (2018)","journal-title":"SIAM J. Numer. Anal."},{"key":"1576_CR22","doi-asserted-by":"publisher","first-page":"407","DOI":"10.1016\/j.jcp.2017.10.021","volume":"353","author":"J Shen","year":"2018","unstructured":"Shen, J., Xu, J., Yang, J.: The scalar auxiliary variable (SAV) approach for gradient fluids. J. Comput. Phys. 353, 407\u2013416 (2018)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"1576_CR23","doi-asserted-by":"publisher","first-page":"474","DOI":"10.1137\/17M1150153","volume":"61","author":"J Shen","year":"2019","unstructured":"Shen, J., Xu, J., Yang, J.: A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61(3), 474\u2013506 (2019)","journal-title":"SIAM Rev."},{"key":"1576_CR24","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-71041-7","volume-title":"Spectral Methods: Algorithms, Analysis and Applications","author":"J Shen","year":"2011","unstructured":"Shen, J., Tang, T., Wang, L.-L.: Spectral Methods: Algorithms, Analysis and Applications, vol. 41. Springer, Berlin (2011)"},{"key":"1576_CR25","volume-title":"The Nonlinear Schr\u00f6dinger Equation: Self-focusing and Wave Collapse","author":"C Sulem","year":"2007","unstructured":"Sulem, C., Sulem, P.-L.: The Nonlinear Schr\u00f6dinger Equation: Self-focusing and Wave Collapse, vol. 139. Springer, Berlin (2007)"},{"issue":"4","key":"1576_CR26","doi-asserted-by":"publisher","first-page":"2022","DOI":"10.1137\/060674636","volume":"46","author":"M Thalhammer","year":"2008","unstructured":"Thalhammer, M.: High-order exponential operator splitting methods for time-dependent Schr\u00f6dinger equations. SIAM J. Numer. Anal. 46(4), 2022\u20132038 (2008)","journal-title":"SIAM J. Numer. Anal."},{"key":"1576_CR27","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1016\/j.cpc.2009.10.007","volume":"181","author":"H Wang","year":"2010","unstructured":"Wang, H.: An efficient Chebyshev\u2013Tau spectral method for Ginzburg\u2013Landau\u2013Schr\u00f6dinger equations. Comput. Phys. Commun. 181, 325\u2013340 (2010)","journal-title":"Comput. Phys. Commun."},{"issue":"2","key":"1576_CR28","doi-asserted-by":"publisher","first-page":"390","DOI":"10.1007\/s10915-013-9799-4","volume":"60","author":"J Wang","year":"2014","unstructured":"Wang, J.: A new error analysis of Crank-Nicolson Galerkin fems for a generalized nonlinear Schr\u00f6dinger equation. J. Sci. Comput. 60(2), 390\u2013407 (2014)","journal-title":"J. Sci. Comput."},{"issue":"2","key":"1576_CR29","doi-asserted-by":"publisher","first-page":"479","DOI":"10.1007\/s00211-017-0944-0","volume":"139","author":"J Wang","year":"2018","unstructured":"Wang, J.: Unconditional stability and convergence of Crank\u2013Nicolson Galerkin FEMs for a nonlinear Schr\u00f6dinger\u2013Helmholtz system. Numer. Math. 139(2), 479\u2013503 (2018)","journal-title":"Numer. Math."},{"issue":"3","key":"1576_CR30","doi-asserted-by":"publisher","first-page":"389","DOI":"10.1051\/m2an:2001121","volume":"35","author":"G Zouraris","year":"2001","unstructured":"Zouraris, G.: On the convergence of a linear two-step finite element method for the nonlinear Schrodinger equation. ESAIM Math. Model. Numer. Anal. Model. Math. 35(3), 389\u2013405 (2001)","journal-title":"ESAIM Math. Model. Numer. Anal. Model. Math."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-021-01576-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-021-01576-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-021-01576-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,9,3]],"date-time":"2021-09-03T22:48:51Z","timestamp":1630709331000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-021-01576-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,30]]},"references-count":30,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,9]]}},"alternative-id":["1576"],"URL":"https:\/\/doi.org\/10.1007\/s10915-021-01576-y","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,30]]},"assertion":[{"value":"3 October 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 May 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 July 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 July 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"69"}}