{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T15:56:14Z","timestamp":1777564574478,"version":"3.51.4"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2019,9,28]],"date-time":"2019-09-28T00:00:00Z","timestamp":1569628800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,9,28]],"date-time":"2019-09-28T00:00:00Z","timestamp":1569628800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11701196"],"award-info":[{"award-number":["11701196"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100007847","name":"Natural Science Foundation of Jilin Province","doi-asserted-by":"publisher","award":["11701197"],"award-info":[{"award-number":["11701197"]}],"id":[{"id":"10.13039\/100007847","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11701081"],"award-info":[{"award-number":["11701081"]}],"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":["11501447"],"award-info":[{"award-number":["11501447"]}],"id":[{"id":"10.13039\/501100001809","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":[[2019,11]]},"DOI":"10.1007\/s10915-019-01050-w","type":"journal-article","created":{"date-parts":[[2019,9,28]],"date-time":"2019-09-28T07:02:36Z","timestamp":1569654156000},"page":"965-989","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":36,"title":["Error Analysis and Numerical Simulations of Strang Splitting Method for Space Fractional Nonlinear Schr\u00f6dinger Equation"],"prefix":"10.1007","volume":"81","author":[{"given":"Shuying","family":"Zhai","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dongling","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhifeng","family":"Weng","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xuan","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,9,28]]},"reference":[{"issue":"3","key":"1050_CR1","doi-asserted-by":"publisher","first-page":"485","DOI":"10.1103\/PhysRev.115.485","volume":"115","author":"Y Aharonov","year":"1959","unstructured":"Aharonov, Y., Bohm, D.: Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115(3), 485\u2013491 (1959)","journal-title":"Phys. Rev."},{"key":"1050_CR2","doi-asserted-by":"publisher","first-page":"298","DOI":"10.1016\/S0375-9601(00)00201-2","volume":"268","author":"N Laskin","year":"2000","unstructured":"Laskin, N.: Fractional quantum mechanics and L\u00e9vy path integrals. Phys. Lett. A 268, 298\u2013305 (2000)","journal-title":"Phys. Lett. A"},{"key":"1050_CR3","doi-asserted-by":"publisher","first-page":"056108","DOI":"10.1103\/PhysRevE.66.056108","volume":"66","author":"N Laskin","year":"2002","unstructured":"Laskin, N.: Fractional Schr\u00f6dinger equation. Phys. Rev. E 66, 056108 (2002)","journal-title":"Phys. Rev. E"},{"key":"1050_CR4","doi-asserted-by":"publisher","first-page":"082104","DOI":"10.1063\/1.2235026","volume":"47","author":"X Guo","year":"2006","unstructured":"Guo, X., Xu, M.: Some physical applications of fractional Schr\u00f6dinger equation. J. Math. Phys. 47, 082104 (2006)","journal-title":"J. Math. Phys."},{"key":"1050_CR5","doi-asserted-by":"publisher","first-page":"1117","DOI":"10.1364\/OL.40.001117","volume":"40","author":"S Longhi","year":"2015","unstructured":"Longhi, S.: Fractional Schr\u00f6dinger equation in optics. Opt. Lett. 40, 1117 (2015)","journal-title":"Opt. Lett."},{"key":"1050_CR6","doi-asserted-by":"publisher","first-page":"195310","DOI":"10.1103\/PhysRevB.92.195310","volume":"92","author":"F Pinsker","year":"2015","unstructured":"Pinsker, F., Bao, W., Zhang, Y., Ohadi, H., Dreismann, A., Baumberg, J.J.: Fractional quantum mechanics in polariton condensates with velocity dependent mass. Phys. Rev. B 92, 195310 (2015)","journal-title":"Phys. Rev. B"},{"key":"1050_CR7","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1007\/s002450010014","volume":"42","author":"Y Hu","year":"2000","unstructured":"Hu, Y., Kallianpur, G.: Schr\u00f6dinger equations with fractional Laplacians. Appl. Math. Optim. 42, 281\u2013290 (2000)","journal-title":"Appl. Math. Optim."},{"issue":"15","key":"1050_CR8","doi-asserted-by":"publisher","first-page":"350","DOI":"10.1016\/j.jcp.2016.01.003","volume":"309","author":"W Chen","year":"2016","unstructured":"Chen, W., Pang, G.F.: A new definition of fractional Laplacian with application to modeling three-dimensional nonlocal heat conduction. J. Comput. Phys. 309(15), 350\u2013367 (2016)","journal-title":"J. Comput. Phys."},{"key":"1050_CR9","doi-asserted-by":"publisher","first-page":"1159","DOI":"10.1137\/100800634","volume":"33","author":"QQ Yang","year":"2011","unstructured":"Yang, Q.Q., Turner, I., Liu, F.W., Ili\u2019c, M.: Novel numerical methods for solving the time-space fractional diffusion equation in 2D. SIAM J. Sci. Comput. 33, 1159\u20131180 (2011)","journal-title":"SIAM J. Sci. Comput."},{"issue":"2","key":"1050_CR10","doi-asserted-by":"publisher","first-page":"1315","DOI":"10.1016\/j.apm.2015.07.021","volume":"40","author":"SY Zhai","year":"2016","unstructured":"Zhai, S.Y., Weng, Z.F., Feng, X.L.: Fast explicit operator splitting method and time-step adaptivity for fractional non-local Allen-Cahn model. Appl. Math. Model. 40(2), 1315\u20131324 (2016)","journal-title":"Appl. Math. Model."},{"key":"1050_CR11","doi-asserted-by":"publisher","first-page":"A1320","DOI":"10.1137\/16M1078197","volume":"39","author":"FY Song","year":"2017","unstructured":"Song, F.Y., Xu, C.J., Karniadakis, G.E.: Computing fractional Laplacians on complex-geometry domains: algorithms and simulations. SIAM J. Sci. Comput. 39, A1320\u2013A1344 (2017)","journal-title":"SIAM J. Sci. Comput."},{"key":"1050_CR12","doi-asserted-by":"publisher","first-page":"670","DOI":"10.1016\/j.jcp.2013.02.037","volume":"242","author":"DL Wang","year":"2013","unstructured":"Wang, D.L., Xiao, A.G., Yang, W.: Crank\u2013Nicolson difference scheme for the coupled nonlinear Schr\u00f6dinger equations with the Riesz space fractional derivative. J. Comput. Phys. 242, 670\u2013681 (2013)","journal-title":"J. Comput. Phys."},{"key":"1050_CR13","doi-asserted-by":"publisher","first-page":"644","DOI":"10.1016\/j.jcp.2014.04.047","volume":"272","author":"DL Wang","year":"2014","unstructured":"Wang, D.L., Xiao, A.G., Yang, W.: A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schr\u00f6dinger equations. J. Comput. Phys. 272, 644\u2013655 (2014)","journal-title":"J. Comput. Phys."},{"key":"1050_CR14","doi-asserted-by":"publisher","first-page":"238","DOI":"10.1016\/j.jcp.2014.03.037","volume":"293","author":"PD Wang","year":"2015","unstructured":"Wang, P.D., Huang, C.M.: An energy conservative difference scheme for the nonlinear fractional Schr\u00f6dinger equations. J. Comput. Phys. 293, 238\u2013251 (2015)","journal-title":"J. Comput. Phys."},{"key":"1050_CR15","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1016\/j.cam.2016.04.017","volume":"306","author":"PD Wang","year":"2016","unstructured":"Wang, P.D., Huang, C.M., Zhao, L.B.: Point-wise error estimate of a conservative difference scheme for the fractional Schr\u00f6dinger equation. J. Comput. Appl. Math. 306, 231\u2013247 (2016)","journal-title":"J. Comput. Appl. Math."},{"key":"1050_CR16","doi-asserted-by":"publisher","first-page":"64","DOI":"10.1016\/j.cnsns.2016.04.026","volume":"41","author":"MH Ran","year":"2016","unstructured":"Ran, M.H., Zhang, C.J.: A conservative difference scheme for solving the strongly coupled nonlinear fractional Schr\u00f6dinger equations. Commun. Nonlinear Sci. Numer. Simul. 41, 64\u201383 (2016)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"1050_CR17","doi-asserted-by":"publisher","first-page":"A2865","DOI":"10.1137\/140961560","volume":"36","author":"X Zhao","year":"2014","unstructured":"Zhao, X., Sun, Z.Z., Hao, Z.P.: A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schr\u00f6dinger equation. SIAM J. Sci. Comput. 36, A2865\u2013A2886 (2014)","journal-title":"SIAM J. Sci. Comput."},{"key":"1050_CR18","doi-asserted-by":"publisher","first-page":"428","DOI":"10.1016\/j.cnsns.2017.06.018","volume":"54","author":"T Aboelenen","year":"2018","unstructured":"Aboelenen, T.: A high-order nodal discontinuous Galerkin method for nonlinear fractional Schr\u00f6dinger type equations. Commun. Nonlinear Sci. Numer. Simul. 54, 428\u2013452 (2018)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"1050_CR19","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1016\/j.apnum.2016.09.009","volume":"111","author":"AH Bhrawy","year":"2017","unstructured":"Bhrawy, A.H., Zaky, M.A.: An improved collocation method for multi-dimensional space time variable-order fractional Schr\u00f6dinger equations. Appl. Numer. Math. 111, 197\u2013218 (2017)","journal-title":"Appl. Numer. Math."},{"key":"1050_CR20","doi-asserted-by":"publisher","first-page":"506","DOI":"10.1137\/0705041","volume":"5","author":"G Strang","year":"1968","unstructured":"Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 506\u2013517 (1968)","journal-title":"SIAM J. Numer. Anal."},{"key":"1050_CR21","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1016\/j.jcp.2015.09.005","volume":"303","author":"YZ Cheng","year":"2015","unstructured":"Cheng, Y.Z., Kurganov, A., Qu, Z.L., Tang, T.: Fast and stable explicit operator splitting methods for phase-field models. J. Comput. Phys. 303, 45\u201365 (2015)","journal-title":"J. Comput. Phys."},{"key":"1050_CR22","doi-asserted-by":"publisher","first-page":"4462","DOI":"10.1002\/mma.4317","volume":"40","author":"ZF Weng","year":"2017","unstructured":"Weng, Z.F., Zhuang, Q.Q.: Numerical approximation of the conservative Allen\u2013Cahn equation by operator splitting method. Math. Method Appl. Sci. 40, 4462\u20134480 (2017)","journal-title":"Math. Method Appl. Sci."},{"key":"1050_CR23","doi-asserted-by":"publisher","first-page":"2621","DOI":"10.1016\/j.cpc.2013.07.012","volume":"184","author":"XA Antoine","year":"2013","unstructured":"Antoine, X.A., Bao, W.Z., Besse, C.: Computational methods for the dynamics of the nonlinear Schrodinger\/Gross\u2013Pitaevskii equations. Comput. Phys. Commun. 184, 2621\u20132633 (2013)","journal-title":"Comput. Phys. Commun."},{"key":"1050_CR24","doi-asserted-by":"publisher","first-page":"5449","DOI":"10.1016\/j.jcp.2011.03.051","volume":"230","author":"WZ Bao","year":"2011","unstructured":"Bao, W.Z., Dong, X.C.: Numerical methods for computing ground state and dynamics of nonlinear relativistic Hartree equation for boson stars. J. Comput. Phys. 230, 5449\u20135469 (2011)","journal-title":"J. Comput. Phys."},{"key":"1050_CR25","doi-asserted-by":"publisher","first-page":"1","DOI":"10.3934\/krm.2013.6.1","volume":"6","author":"W Bao","year":"2013","unstructured":"Bao, W., Cai, Y.: Mathematical theory and numerical methods for Bose\u2013Einstein condensation. Kinet. Relat. Models 6, 1\u2013135 (2013)","journal-title":"Kinet. Relat. Models"},{"key":"1050_CR26","volume-title":"Spectral Methods Algorithms: Analyses and Applications","author":"J Shen","year":"2010","unstructured":"Shen, J., Tang, T., Wang, L.L.: Spectral Methods Algorithms: Analyses and Applications, 1st edn. Springer, Berlin (2010)","edition":"1"},{"key":"1050_CR27","doi-asserted-by":"publisher","first-page":"1423","DOI":"10.1080\/00207160.2018.1437911","volume":"95","author":"X Antoine","year":"2018","unstructured":"Antoine, X., Tang, Q.L., Zhang, J.W.: On the numerical solution and dynamical laws of nonlinear fractional Schr\u00f6dinger\/Gross\u2013Pitaevskii equations. Int. J. Comput. Math. 95, 1423\u20131443 (2018)","journal-title":"Int. J. Comput. Math."},{"key":"1050_CR28","doi-asserted-by":"publisher","first-page":"2141","DOI":"10.1090\/S0025-5718-08-02101-7","volume":"77","author":"C Lubich","year":"2008","unstructured":"Lubich, C.: On splitting methods for Schr\u00f6dinger\u2013Poisson and cubic nonlinear Schr\u00f6dinger equations. Math. Comput. 77, 2141\u20132153 (2008)","journal-title":"Math. Comput."},{"key":"1050_CR29","doi-asserted-by":"publisher","first-page":"3231","DOI":"10.1137\/120866373","volume":"50","author":"M Thalhammer","year":"2012","unstructured":"Thalhammer, M.: Convergence analysis of high-order time-splitting pseudospectral methods for nonlinear Schr\u00f6dinger equations. SIAM J. Numer. Anal. 50, 3231\u20133258 (2012)","journal-title":"SIAM J. Numer. Anal."},{"key":"1050_CR30","doi-asserted-by":"publisher","first-page":"140","DOI":"10.1137\/130918599","volume":"52","author":"L Einkemmer","year":"2014","unstructured":"Einkemmer, L., Ostermann, A.: Convergence analysis of Strang splitting for Vlasov-type equations. SIAM J. Numer. Anal. 52, 140\u2013155 (2014)","journal-title":"SIAM J. Numer. Anal."},{"key":"1050_CR31","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. Springer, Berlin (2011)"},{"key":"1050_CR32","doi-asserted-by":"publisher","first-page":"1689","DOI":"10.1137\/16M1075302","volume":"55","author":"M Ainsworth","year":"2017","unstructured":"Ainsworth, M., Mao, Z.P.: Analysis and approximation of a fractional Cahn\u2013Hilliard equation. SIAM J Numer Anal. 55, 1689\u20131718 (2017)","journal-title":"SIAM J Numer Anal."},{"key":"1050_CR33","doi-asserted-by":"publisher","first-page":"1510","DOI":"10.1016\/j.camwa.2011.05.039","volume":"62","author":"JQ Hu","year":"2011","unstructured":"Hu, J.Q., Xin, J., Lu, H.: The global solution for a class of systems of fractional nonlinear Schr\u00f6dinger equations with periodic boundary condition. Comput. Math. Appl. 62, 1510\u20131521 (2011)","journal-title":"Comput. Math. Appl."},{"key":"1050_CR34","doi-asserted-by":"publisher","first-page":"1297","DOI":"10.1137\/18M120703X","volume":"51","author":"Y Hong","year":"2019","unstructured":"Hong, Y., Yang, C.: Strong convergence for discrete nonlinear Schr\u00f6dinger equations in the Continuum Limit. SIAM J. Math. Anal. 51, 1297\u20131320 (2019)","journal-title":"SIAM J. Math. Anal."},{"key":"1050_CR35","doi-asserted-by":"publisher","first-page":"1237","DOI":"10.1017\/S0308210511000746","volume":"142","author":"P Felmer","year":"2012","unstructured":"Felmer, P., Quaas, A., Tan, J.: Positive solutions of the nonlinear Schr\u00f6dinger equation with the fractional Laplacian. Proc. R. Soc. Edinb. Sect. A 142, 1237\u20131262 (2012)","journal-title":"Proc. R. Soc. Edinb. Sect. A"},{"key":"1050_CR36","doi-asserted-by":"publisher","first-page":"735","DOI":"10.1023\/A:1022396519656","volume":"40","author":"T Jahnke","year":"2000","unstructured":"Jahnke, T., Lubich, C.: Error bounds for exponential operator splitting. BIT Numer. Math. 40, 735\u2013744 (2000)","journal-title":"BIT Numer. Math."},{"key":"1050_CR37","unstructured":"Willoughby, M.R.: High-order time-adaptive numerical methods for the Allen\u2013Cahn and Cahn\u2013Hilliard equations. Master\u2019s thesis, Mathematics, University of British Columbia, Vancouver (2011)"},{"key":"1050_CR38","doi-asserted-by":"publisher","first-page":"180403","DOI":"10.1103\/PhysRevLett.115.180403","volume":"115","author":"YQ Zhang","year":"2015","unstructured":"Zhang, Y.Q., Liu, X., Beli\u0107, M.R., Zhong, W.P., Zhang, Y.P., Xiao, M.: Propagation dynamics of a light beam in a fractional Schr\u00f6dinger equation. Phys. Rev. Lett. 115, 180403 (2015)","journal-title":"Phys. Rev. Lett."},{"key":"1050_CR39","doi-asserted-by":"publisher","first-page":"861","DOI":"10.1016\/j.cpc.2009.12.022","volume":"181","author":"ZF Tian","year":"2010","unstructured":"Tian, Z.F., Yu, P.X.: High-order compact ADI (HOC-ADI) method for solving unsteady 2D Schr\u00f6dinger equation. Comput. Phys. Commun. 181, 861\u2013868 (2010)","journal-title":"Comput. Phys. Commun."},{"key":"1050_CR40","doi-asserted-by":"publisher","first-page":"382","DOI":"10.1016\/j.jcp.2013.03.007","volume":"243","author":"TC Wang","year":"2013","unstructured":"Wang, T.C., Guo, B.L., Xu, Q.B.: Fourth-order compact and energy conservative difference schemes for the nonlinear Schr\u00f6dinger equation in two dimensions. J. Comput. Phys. 243, 382\u2013399 (2013)","journal-title":"J. Comput. Phys."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-019-01050-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-019-01050-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-019-01050-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,9,26]],"date-time":"2020-09-26T23:37:27Z","timestamp":1601163447000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-019-01050-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,9,28]]},"references-count":40,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,11]]}},"alternative-id":["1050"],"URL":"https:\/\/doi.org\/10.1007\/s10915-019-01050-w","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,9,28]]},"assertion":[{"value":"22 October 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 June 2019","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 September 2019","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 September 2019","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}