{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T15:43:27Z","timestamp":1786981407277,"version":"build-2736575974"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2013,11,8]],"date-time":"2013-11-08T00:00:00Z","timestamp":1383868800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2014,8]]},"DOI":"10.1007\/s10915-013-9799-4","type":"journal-article","created":{"date-parts":[[2013,11,7]],"date-time":"2013-11-07T06:50:06Z","timestamp":1383807006000},"page":"390-407","source":"Crossref","is-referenced-by-count":172,"title":["A New Error Analysis of Crank\u2013Nicolson Galerkin FEMs for a Generalized Nonlinear Schr\u00f6dinger Equation"],"prefix":"10.1007","volume":"60","author":[{"given":"Jilu","family":"Wang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2013,11,8]]},"reference":[{"key":"9799_CR1","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/BF01385769","volume":"59","author":"GD Akrivis","year":"1991","unstructured":"Akrivis, G.D., Dougalis, V.A., Karakashian, O.A.: On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schr\u00f6dinger equation. Numer. Math. 59, 31\u201353 (1991)","journal-title":"Numer. Math."},{"key":"9799_CR2","doi-asserted-by":"crossref","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, 1008\u20131033 (2011)","journal-title":"SIAM J. Sci. Comput."},{"key":"9799_CR3","doi-asserted-by":"crossref","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, 492\u2013521 (2012)","journal-title":"SIAM J. Numer. Anal."},{"key":"9799_CR4","unstructured":"Borzi, A., Decker, E.: Analysis of a leap-frog pseudospectral scheme for the Schr\u00f6dinger equation. J. Comput. Appl. Math. 193, 65\u201388 (2006)"},{"key":"9799_CR5","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1002\/num.20541","volume":"27","author":"AG Bratsos","year":"2011","unstructured":"Bratsos, A.G.: A modified numerical scheme for the cubic Schr\u00f6dinger equation. Numer. Methods Partial Differ. Equ. 27, 608\u2013620 (2011)","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"9799_CR6","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/BF02575943","volume":"25","author":"JR Cannon","year":"1988","unstructured":"Cannon, J.R., Lin, Y.: Nonclassical $$H^1$$ H 1 projection and Galerkin methods for nonlinear parabolic integro-differential equations. Calcolo 25, 187\u2013201 (1988)","journal-title":"Calcolo"},{"key":"9799_CR7","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1006\/jcph.1998.6120","volume":"148","author":"Q Chang","year":"1999","unstructured":"Chang, Q., Jia, E., Sun, W.: Difference schemes for solving the generalized nonlinear Schr\u00f6dinger equation. J. Comput. Phys. 148, 397\u2013415 (1999)","journal-title":"J. Comput. Phys."},{"key":"9799_CR8","doi-asserted-by":"crossref","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-space pseudo-spectral method. Numer. Methods Partial Differ. Equ. 26, 979\u2013990 (2010)","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"9799_CR9","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1051\/m2an\/1983170302491","volume":"17","author":"J Douglas Jr","year":"1983","unstructured":"Douglas Jr, J., Ewing, R.E., Wheeler, M.F.: A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media. RAIRO Anal. Numer. 17, 249\u2013265 (1983)","journal-title":"RAIRO Anal. Numer."},{"key":"9799_CR10","doi-asserted-by":"crossref","first-page":"392","DOI":"10.1137\/0711034","volume":"11","author":"T Dupont","year":"1974","unstructured":"Dupont, T., Fairweather, G., Johnson, J.P.: Three-level Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 11, 392\u2013410 (1974)","journal-title":"SIAM J. Numer. Anal."},{"key":"9799_CR11","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1137\/0717029","volume":"17","author":"RE Ewing","year":"1980","unstructured":"Ewing, R.E., Wheeler, M.F.: Galerkin methods for miscible displacement problems in porous media. SIAM J. Numer. Anal. 17, 351\u2013365 (1980)","journal-title":"SIAM J. Numer. Anal."},{"key":"9799_CR12","doi-asserted-by":"crossref","first-page":"2097","DOI":"10.1090\/S0025-5718-08-02127-3","volume":"77","author":"Y He","year":"2008","unstructured":"He, Y.: The Euler implicit\/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data. Math. Comput. 77, 2097\u20132124 (2008)","journal-title":"Math. Comput."},{"key":"9799_CR13","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1137\/110854813","volume":"51","author":"Y Hou","year":"2013","unstructured":"Hou, Y., Li, B., Sun, W.: Error analysis of splitting Galerkin methods for heat and sweat transport in textile materials. SIAM J. Numer. Anal. 51, 88\u2013111 (2013)","journal-title":"SIAM J. Numer. Anal."},{"key":"9799_CR14","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1016\/j.cam.2007.08.006","volume":"220","author":"J Jin","year":"2008","unstructured":"Jin, J., Wu, X.: Analysis of finite element method for one-dimensional time-dependent Schr\u00f6dinger equation on unbounded domain. J. Comput. Appl. Math. 220, 240\u2013256 (2008)","journal-title":"J. Comput. Appl. Math."},{"key":"9799_CR15","unstructured":"Li, B.: Mathematical modeling, analysis and computation for some complex and nonlinear flow problems. PhD Thesis, City University of Hong Kong, Hong Kong, July (2012)"},{"key":"9799_CR16","first-page":"622","volume":"10","author":"B Li","year":"2013","unstructured":"Li, B., Sun, W.: Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations. Int. J. Numer. Anal. Model. 10, 622\u2013633 (2013)","journal-title":"Int. J. Numer. Anal. Model."},{"key":"9799_CR17","doi-asserted-by":"crossref","first-page":"1959","DOI":"10.1137\/120871821","volume":"51","author":"B Li","year":"2013","unstructured":"Li, B., Sun, W.: Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media. SIAM J. Numer. Anal. 51, 1959\u20131977 (2013)","journal-title":"SIAM J. Numer. Anal."},{"key":"9799_CR18","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1137\/S0036142996302396","volume":"35","author":"BK Li","year":"1998","unstructured":"Li, B.K., Fairweather, G., Bialecki, B.: Discrete-time orthogonal spline collocation methods for Schr\u00f6dinger equations in two space variables. SIAM J. Numer. Anal. 35, 453\u2013477 (1998)","journal-title":"SIAM J. Numer. Anal."},{"key":"9799_CR19","doi-asserted-by":"crossref","first-page":"4381","DOI":"10.1137\/080714907","volume":"47","author":"H Liao","year":"2010","unstructured":"Liao, H., Sun, Z., Shi, H.: Error estimate of fourth-order compact scheme for linear Schr\u00f6dinger equations. SIAM J. Numer. Anal. 47, 4381\u20134401 (2010)","journal-title":"SIAM J. Numer. Anal."},{"key":"9799_CR20","unstructured":"L\u00f3pez-Marcos, J.C., Sanz-Serna, J.M.: A definition of stability for nonlinear problems. In: Numerical Treatment of Differential Equations, Teubner-Texte zur Mathematik, Band 104, Leipzig, pp. 216\u2013226 (1988)"},{"key":"9799_CR21","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1088\/0031-8949\/39\/6\/001","volume":"39","author":"D Pathria","year":"1989","unstructured":"Pathria, D.: Exact solutions for a generalized nonlinear Schr\u00f6dinger equation. Physica Scripta 39, 673\u2013679 (1989)","journal-title":"Physica Scripta"},{"key":"9799_CR22","doi-asserted-by":"crossref","first-page":"1940","DOI":"10.1103\/PhysRevE.53.1940","volume":"53","author":"DE Pelinovsky","year":"1996","unstructured":"Pelinovsky, D.E., Afanasjev, V.V., Kivshar, Y.S.: Nonlinear theory of oscillating, decaying, and collapsing solitons in the generalized nonlinear Schr\u00f6dinger equation. Phys. Rev. E 53, 1940\u20131953 (1996)","journal-title":"Phys. Rev. E"},{"key":"9799_CR23","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1016\/j.camwa.2007.04.038","volume":"55","author":"B Reichel","year":"2008","unstructured":"Reichel, B., Leble, S.: On convergence and stability of a numerical scheme of coupled nonlinear Schr\u00f6dinger equations. Comput. Math. Appl. 55, 745\u2013759 (2008)","journal-title":"Comput. Math. Appl."},{"key":"9799_CR24","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1090\/S0025-5718-1984-0744922-X","volume":"43","author":"JM Sanz-Serna","year":"1984","unstructured":"Sanz-Serna, J.M.: Methods for the numerical solution of nonlinear Schr\u00f6dinger equation. Math. Comput. 43, 21\u201327 (1984)","journal-title":"Math. Comput."},{"key":"9799_CR25","doi-asserted-by":"crossref","first-page":"4312","DOI":"10.1103\/PhysRevE.54.4312","volume":"54","author":"HW Sch\u00fcrmann","year":"1996","unstructured":"Sch\u00fcrmann, H.W.: Traveling-wave solutions of the cubic-quintic nonlinear Schr\u00f6dinger equation. Phys. Rev. E 54, 4312\u20134320 (1996)","journal-title":"Phys. Rev. E"},{"key":"9799_CR26","first-page":"276","volume":"8","author":"J Shen","year":"1990","unstructured":"Shen, J.: On an unconditionally stable scheme for the unsteady Navier-Stokes equations. J. Comput. Math. 8, 276\u2013288 (1990)","journal-title":"J. Comput. Math."},{"key":"9799_CR27","doi-asserted-by":"crossref","first-page":"3286","DOI":"10.1016\/j.camwa.2010.03.012","volume":"59","author":"Z Sun","year":"2010","unstructured":"Sun, Z., Zhao, D.: On the $$L_{\\infty }$$ L \u221e convergence of a difference scheme for coupled nonlinear Schr\u00f6dinger equations. Comput. Math. Appl. 59, 3286\u20133300 (2010)","journal-title":"Comput. Math. Appl."},{"key":"9799_CR28","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-03359-3","volume-title":"Galerkin Finite Element Methods for Parabolic Problems","author":"V Thom\u00e9e","year":"1997","unstructured":"Thom\u00e9e, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (1997)"},{"key":"9799_CR29","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1093\/imanum\/11.4.509","volume":"11","author":"Y Tourigny","year":"1991","unstructured":"Tourigny, Y.: Optimal $$H^1$$ H 1 estimates for two time-discrete Galerkin approximations of a nonlinear Schr\u00f6dinger equation. IMA J. Numer. Anal. 11, 509\u2013523 (1991)","journal-title":"IMA J. Numer. Anal."},{"key":"9799_CR30","doi-asserted-by":"crossref","first-page":"1604","DOI":"10.1016\/j.amc.2009.07.040","volume":"217","author":"T Wang","year":"2010","unstructured":"Wang, T., Guo, B., Zhang, L.: New conservative difference schemes for a coupled nonlinear Schr\u00f6dinger system. Appl. Math. Comput. 217, 1604\u20131619 (2010)","journal-title":"Appl. Math. Comput."},{"key":"9799_CR31","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1137\/S0036142901399781","volume":"41","author":"H Wu","year":"2003","unstructured":"Wu, H., Ma, H., Li, H.: Optimal error estimates of the Chebyshev-Legendre spectral method for solving the generalized Burgers equation. SIAM J. Numer. Anal. 41, 659\u2013672 (2003)","journal-title":"SIAM J. Numer. Anal."},{"key":"9799_CR32","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1137\/0710022","volume":"10","author":"M Zl\u00e1mal","year":"1973","unstructured":"Zl\u00e1mal, M.: Curved elements in the finite element method. I*. SIAM J. Numer. Anal. 10, 229\u2013240 (1973)","journal-title":"SIAM J. Numer. Anal."},{"key":"9799_CR33","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1051\/m2an:2001121","volume":"35","author":"GE Zouraris","year":"2001","unstructured":"Zouraris, G.E.: On the convergence of a linear two-step finite element method for the nonlinear Schr\u00f6dinger equation. M2AN Math. Model. Numer. Anal. 35, 389\u2013405 (2001)","journal-title":"M2AN Math. Model. Numer. Anal."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-013-9799-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-013-9799-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-013-9799-4","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,8,7]],"date-time":"2020-08-07T09:50:29Z","timestamp":1596793829000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-013-9799-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,11,8]]},"references-count":33,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2014,8]]}},"alternative-id":["9799"],"URL":"https:\/\/doi.org\/10.1007\/s10915-013-9799-4","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,11,8]]}}}