{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T15:56:08Z","timestamp":1777564568433,"version":"3.51.4"},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2023,10,23]],"date-time":"2023-10-23T00:00:00Z","timestamp":1698019200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,10,23]],"date-time":"2023-10-23T00:00:00Z","timestamp":1698019200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12131005"],"award-info":[{"award-number":["12131005"]}],"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":["12071020"],"award-info":[{"award-number":["12071020"]}],"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":["U1930402"],"award-info":[{"award-number":["U1930402"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2023,12]]},"DOI":"10.1007\/s10444-023-10077-5","type":"journal-article","created":{"date-parts":[[2023,10,23]],"date-time":"2023-10-23T10:01:42Z","timestamp":1698055302000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Mass- and energy-conserving Gauss collocation methods for the nonlinear Schr\u00f6dinger equation with a wave operator"],"prefix":"10.1007","volume":"49","author":[{"given":"Shu","family":"Ma","sequence":"first","affiliation":[]},{"given":"Jilu","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Mingyan","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Zhimin","family":"Zhang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,10,23]]},"reference":[{"key":"10077_CR1","unstructured":"Berge, L., Colin, T.: Un probl\u00e8me de perturbation singuli\u00e8re pour une \u00e9quation d\u2019eveloppe en physique des plasmas. Comptes rendus de l\u2019Academie des sciences. S\u00e9rie 1, Math\u00e9matique. 320(1), 31\u201334 (1999)"},{"issue":"13","key":"10077_CR2","doi-asserted-by":"publisher","first-page":"1120","DOI":"10.1016\/j.physd.2010.03.002","volume":"239","author":"W Bao","year":"2010","unstructured":"Bao, W., Dong, X., Xin, J.: Comparisons between sine-Gordon and perturbed nonlinear Schr\u00f6dinger equations for modeling light bullets beyond critical collapse. Physica D 239(13), 1120\u20131134 (2010)","journal-title":"Physica D"},{"issue":"6","key":"10077_CR3","doi-asserted-by":"publisher","first-page":"637","DOI":"10.1016\/0362-546X(84)90008-7","volume":"8","author":"M Tsutsumi","year":"1984","unstructured":"Tsutsumi, M.: Nonrelativistic approximation of nonlinear Klein-Gordon equations in two space dimensions. Nonlinear Analysis: Theory, Methods & Applications. 8(6), 637\u2013643 (1984)","journal-title":"Nonlinear Analysis: Theory, Methods & Applications."},{"issue":"2\u20133","key":"10077_CR4","first-page":"165","volume":"71","author":"Z Fei","year":"1995","unstructured":"Fei, Z., P\u00e9rez-Garc\u00eda, V.M., V\u00e1zquez, L.: Numerical simulation of nonlinear Schr\u00f6dinger systems: a new conservative scheme. Appl. Math. Comput. 71(2\u20133), 165\u2013177 (1995)","journal-title":"Appl. Math. Comput."},{"issue":"2","key":"10077_CR5","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1016\/0021-9991(81)90052-8","volume":"44","author":"M Delfour","year":"1981","unstructured":"Delfour, M., Fortin, M., Payr, G.: Finite-difference solutions of a non-linear Schr\u00f6dinger equation. J. Comput. Phys. 44(2), 277\u2013288 (1981)","journal-title":"J. Comput. Phys."},{"issue":"1","key":"10077_CR6","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1093\/imanum\/13.1.115","volume":"13","author":"GD Akrivis","year":"1993","unstructured":"Akrivis, G.D.: Finite difference discretization of the cubic Schr\u00f6dinger equation. IMA J. Numer. Anal. 13(1), 115\u2013124 (1993)","journal-title":"IMA J. Numer. Anal."},{"issue":"1","key":"10077_CR7","doi-asserted-by":"publisher","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(1), 31\u201353 (1991)","journal-title":"Numer. Math."},{"key":"10077_CR8","doi-asserted-by":"crossref","unstructured":"Antoine, X., Bao, W., Besse, C.: Computational methods for the dynamics of the nonlinear Schr\u00f6dinger\/Gross\u2013Pitaevskii equations. Computer Physics Communications. 184(12), 2621\u20132633 (2013)","DOI":"10.1016\/j.cpc.2013.07.012"},{"key":"10077_CR9","doi-asserted-by":"publisher","first-page":"423","DOI":"10.1016\/j.jcp.2012.10.054","volume":"235","author":"W Bao","year":"2013","unstructured":"Bao, W., Tang, Q., Xu, Z.: Numerical methods and comparison for computing dark and bright solitons in the nonlinear Schr\u00f6dinger equation. J. Comput. Phys. 235, 423\u2013445 (2013)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"10077_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":"5","key":"10077_CR11","doi-asserted-by":"publisher","first-page":"1365","DOI":"10.4208\/cicp.2019.js60.05","volume":"26","author":"X Feng","year":"2019","unstructured":"Feng, X., Liu, H., Ma, S.: Mass- and energy-conserved numerical schemes for nonlinear Schr\u00f6dinger equations. Communications in Computational Physics. 26(5), 1365\u20131396 (2019)","journal-title":"Communications in Computational Physics."},{"issue":"4","key":"10077_CR12","doi-asserted-by":"publisher","first-page":"593","DOI":"10.1016\/j.apnum.2010.12.004","volume":"61","author":"Z Gao","year":"2011","unstructured":"Gao, Z., Xie, S.: Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schr\u00f6dinger equations. Appl. Numer. Math. 61(4), 593\u2013614 (2011)","journal-title":"Appl. Numer. Math."},{"key":"10077_CR13","doi-asserted-by":"crossref","unstructured":"Sanz-Serna, J.: Methods for the numerical solution of the nonlinear Schr\u00f6dinger equation. mathematics of computation. 43(167), 21\u201327 (1984)","DOI":"10.1090\/S0025-5718-1984-0744922-X"},{"issue":"6","key":"10077_CR14","doi-asserted-by":"publisher","first-page":"814","DOI":"10.1016\/j.apnum.2005.06.006","volume":"56","author":"J Hong","year":"2006","unstructured":"Hong, J., Liu, Y., Munthe-Kaas, H., Zanna, A.: Globally conservative properties and error estimation of a multi-symplectic scheme for Schr\u00f6dinger equations with variable coefficients. Appl. Numer. Math. 56(6), 814\u2013843 (2006)","journal-title":"Appl. Numer. Math."},{"key":"10077_CR15","doi-asserted-by":"crossref","unstructured":"Liu, H., Huang, Y., Lu, W., Yi, N.: On accuracy of the mass-preserving dg method to multi-dimensional Schr\u00f6dinger equations. IMA Journal of Numerical Analysis. 39(2), 760\u2013791 (2019)","DOI":"10.1093\/imanum\/dry012"},{"key":"10077_CR16","doi-asserted-by":"crossref","unstructured":"Wang, J.: A new error analysis of Crank\u2013Nicolson Galerkin FEMs for a generalized nonlinear Schr\u00f6dinger equation. Journal of Scientific Computing. 60(2), 390\u2013407 (2014)","DOI":"10.1007\/s10915-013-9799-4"},{"issue":"1","key":"10077_CR17","doi-asserted-by":"publisher","first-page":"72","DOI":"10.1016\/j.jcp.2004.11.001","volume":"205","author":"Y Xu","year":"2005","unstructured":"Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for nonlinear Schr\u00f6dinger equations. J. Comput. Phys. 205(1), 72\u201397 (2005)","journal-title":"J. Comput. Phys."},{"issue":"2","key":"10077_CR18","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":"2","key":"10077_CR19","doi-asserted-by":"publisher","first-page":"622","DOI":"10.1007\/s10915-014-9977-z","volume":"65","author":"L Guo","year":"2015","unstructured":"Guo, L., Xu, Y.: Energy conserving local discontinuous Galerkin methods for the nonlinear Schr\u00f6dinger equation with wave operator. J. Sci. Comput. 65(2), 622\u2013647 (2015)","journal-title":"J. Sci. Comput."},{"issue":"3","key":"10077_CR20","doi-asserted-by":"publisher","first-page":"862","DOI":"10.1002\/num.22033","volume":"32","author":"H Hu","year":"2016","unstructured":"Hu, H., Chen, Y.: A conservative difference scheme for two-dimensional nonlinear Schr\u00f6dinger equation with wave operator. Numer. Methods Partial Differential Equations. 32(3), 862\u2013876 (2016)","journal-title":"Numer. Methods Partial Differential Equations."},{"key":"10077_CR21","doi-asserted-by":"crossref","unstructured":"Wang, T.-c., Zhang, L.-m.: Analysis of some new conservative schemes for nonlinear Schr\u00f6dinger equation with wave operator. Applied Mathematics and Computation. 182(2), 1780\u20131794 (2006)","DOI":"10.1016\/j.amc.2006.06.015"},{"issue":"2\u20133","key":"10077_CR22","first-page":"603","volume":"145","author":"L Zhang","year":"2003","unstructured":"Zhang, L., Chang, Q.: A conservative numerical scheme for a class of nonlinear Schr\u00f6dinger equation with wave operator. Appl. Math. Comput. 145(2\u20133), 603\u2013612 (2003)","journal-title":"Appl. Math. Comput."},{"issue":"6","key":"10077_CR23","first-page":"3187","volume":"219","author":"X Li","year":"2012","unstructured":"Li, X., Zhang, L., Wang, S.: A compact finite difference scheme for the nonlinear Schr\u00f6dinger equation with wave operator. Appl. Math. Comput. 219(6), 3187\u20133197 (2012)","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"10077_CR24","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1007\/s12190-016-1000-4","volume":"54","author":"X Li","year":"2017","unstructured":"Li, X., Zhang, L., Zhang, T.: A new numerical scheme for the nonlinear Schr\u00f6dinger equation with wave operator. J. Appl. Math. Comput. 54(1), 109\u2013125 (2017)","journal-title":"J. Appl. Math. Comput."},{"key":"10077_CR25","doi-asserted-by":"crossref","unstructured":"Cheng, X., Wu, F.: Several conservative compact schemes for a class of nonlinear Schr\u00f6dinger equations with wave operator. Boundary Value Problems. 2018(1), 1\u201317 (2018)","DOI":"10.1186\/s13661-018-0956-4"},{"key":"10077_CR26","doi-asserted-by":"crossref","unstructured":"Cai, W., He, D., Pan, K.: A linearized energy\u2013conservative finite element method for the nonlinear Schr\u00f6dinger equation with wave operator. Applied Numerical Mathematics. 140, 183\u2013198 (2019)","DOI":"10.1016\/j.apnum.2019.02.005"},{"key":"10077_CR27","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1016\/j.cam.2021.113762","volume":"400","author":"X Cheng","year":"2022","unstructured":"Cheng, X., Qin, H., Zhang, J.: Convergence of an energy-conserving scheme for nonlinear space fractional Schr\u00f6dinger equations with wave operator. J. Comput. Appl. Math. 400, 113\u2013762 (2022)","journal-title":"J. Comput. Appl. Math."},{"key":"10077_CR28","doi-asserted-by":"crossref","unstructured":"Cheng, X., Yan, X., Qin, H., Wang, H.: Optimal $$l^\\infty $$ error estimates of the conservative scheme for two-dimensional Schr\u00f6dinger equations with wave operator. Computers & Mathematics with Applications 100, 74\u201382 (2021)","DOI":"10.1016\/j.camwa.2021.08.026"},{"key":"10077_CR29","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 flows. J. Comput. Phys. 353, 407\u2013416 (2018)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"10077_CR30","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":"10077_CR31","doi-asserted-by":"publisher","first-page":"294","DOI":"10.1016\/j.jcp.2016.09.029","volume":"327","author":"X Yang","year":"2016","unstructured":"Yang, X.: Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J. Comput. Phys. 327, 294\u2013316 (2016)","journal-title":"J. Comput. Phys."},{"key":"10077_CR32","doi-asserted-by":"publisher","first-page":"691","DOI":"10.1016\/j.cma.2016.10.041","volume":"315","author":"X Yang","year":"2017","unstructured":"Yang, X., Ju, L.: Efficient linear schemes with unconditional energy stability for the phase field elastic bending energy model. Comput. Methods Appl. Mech. Eng. 315, 691\u2013712 (2017)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"10077_CR33","doi-asserted-by":"crossref","unstructured":"Yang, X., Ju, L.: Linear and unconditionally energy stable schemes for the binary fluid\u2013surfactant phase field model. Computer Methods in Applied Mechanics and Engineering. 318, 1005\u20131029 (2017)","DOI":"10.1016\/j.cma.2017.02.011"},{"key":"10077_CR34","doi-asserted-by":"crossref","unstructured":"Yang, X., Zhao, J., Wang, Q., Shen, J.: Numerical approximations for a three component Cahn\u2013Hilliard phase-field model based on the invariant energy quadratization method. Mathematical Models and Methods in Applied Sciences. 27(11), 1993\u20132030 (2017)","DOI":"10.1142\/S0218202517500373"},{"key":"10077_CR35","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1016\/j.cnsns.2017.12.018","volume":"60","author":"L Brugnano","year":"2018","unstructured":"Brugnano, L., Zhang, C., Li, D.: A class of energy-conserving Hamiltonian boundary value methods for nonlinear Schr\u00f6dinger equation with wave operator. Commun. Nonlinear Sci. Numer. Simul. 60, 33\u201349 (2018)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"issue":"1","key":"10077_CR36","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10915-021-01519-7","volume":"88","author":"X Li","year":"2021","unstructured":"Li, X., Gong, Y., Zhang, L.: Linear high-order energy-preserving schemes for the nonlinear Schr\u00f6dinger equation with wave operator using the scalar auxiliary variable approach. J. Sci. Comput. 88(1), 1\u201325 (2021)","journal-title":"J. Sci. Comput."},{"issue":"3","key":"10077_CR37","doi-asserted-by":"publisher","first-page":"1566","DOI":"10.1137\/20M1344998","volume":"59","author":"X Feng","year":"2021","unstructured":"Feng, X., Li, B., Ma, S.: High-order mass-and energy-conserving SAV-Gauss collocation finite element methods for the nonlinear Schro\u00f6dinger equation. SIAM J. Numer. Anal. 59(3), 1566\u20131591 (2021)","journal-title":"SIAM J. Numer. Anal."},{"key":"10077_CR38","doi-asserted-by":"crossref","unstructured":"Evans, L.C.: Partial differential equations. American Mathematical Soc. (2010)","DOI":"10.1090\/gsm\/019"},{"key":"10077_CR39","doi-asserted-by":"crossref","unstructured":"Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. Springer (2008)","DOI":"10.1007\/978-0-387-75934-0"},{"key":"10077_CR40","doi-asserted-by":"crossref","unstructured":"Shen, J., Tang, T., Wang, L.-L.: Spectral methods, volume 41 of Springer Series in Computational Mathematics. Springer (2011)","DOI":"10.1007\/978-3-540-71041-7"},{"key":"10077_CR41","doi-asserted-by":"crossref","unstructured":"Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods. Springer. Fundamentals in single domains (2006)","DOI":"10.1007\/978-3-540-30726-6"},{"key":"10077_CR42","doi-asserted-by":"crossref","unstructured":"Golub, G.H., Welsch, J.H.: Calculation of Gauss quadrature rules. Mathematics of computation 23(106), 221\u2013230 (1969)","DOI":"10.1090\/S0025-5718-69-99647-1"},{"key":"10077_CR43","doi-asserted-by":"crossref","unstructured":"Kopriva, D.A.: Implementing spectral methods for partial differential equations: algorithms for scientists and engineers. Springer (2009)","DOI":"10.1007\/978-90-481-2261-5"}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-023-10077-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-023-10077-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-023-10077-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,22]],"date-time":"2023-12-22T12:25:29Z","timestamp":1703247929000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-023-10077-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,23]]},"references-count":43,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2023,12]]}},"alternative-id":["10077"],"URL":"https:\/\/doi.org\/10.1007\/s10444-023-10077-5","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,23]]},"assertion":[{"value":"3 December 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 September 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 October 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of interest"}}],"article-number":"77"}}