{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,13]],"date-time":"2026-08-13T00:26:56Z","timestamp":1786580816269,"version":"build-2736575974"},"reference-count":35,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100002855","name":"Ministry of Science and Technology of the People's Republic of China","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100002855","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Communications in Nonlinear Science and Numerical Simulation"],"published-print":{"date-parts":[[2026,11]]},"DOI":"10.1016\/j.cnsns.2026.110547","type":"journal-article","created":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T15:42:43Z","timestamp":1783698163000},"page":"110547","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"P2","title":["Unconditionally optimal L2 error estimates of a linearized leap-frog FE scheme for the Kuramoto\u2013Tsuzuki equation"],"prefix":"10.1016","volume":"163","author":[{"given":"Zimin","family":"Xia","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wentao","family":"Cai","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wen-Sheng","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Absalom El-Shamir","family":"Ezugwu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"issue":"3","key":"10.1016\/j.cnsns.2026.110547_bib0001","doi-asserted-by":"crossref","first-page":"678","DOI":"10.1143\/PTP.54.687","article-title":"On the formation of dissipative structures in reaction-diffusion systems","volume":"54","author":"Kuramoto","year":"1975","journal-title":"Prog Theor Phys"},{"key":"10.1016\/j.cnsns.2026.110547_bib0002","doi-asserted-by":"crossref","first-page":"1292","DOI":"10.1007\/BF01098786","article-title":"Classification of solutions of a system of nonlinear diffusion equations in a neighborhood of a bifurcation point","volume":"41","author":"Akhromeeva","year":"1988","journal-title":"J Math Sci"},{"key":"10.1016\/j.cnsns.2026.110547_bib0003","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1016\/j.apnum.2022.11.004","article-title":"A new approach for numerical solution of Kuramoto\u2013Tsuzuki equation","volume":"184","author":"Labidi","year":"2023","journal-title":"Appl Numer Math"},{"key":"10.1016\/j.cnsns.2026.110547_bib0004","series-title":"Chemical oscillations, waves, and turbulence","author":"Kuramoto","year":"1984"},{"key":"10.1016\/j.cnsns.2026.110547_bib0005","first-page":"40","article-title":"On the convergence of difference schemes for the Kuramoto\u2013Tsuzuki equation and reaction-diffusion type systems","volume":"31","author":"Tsertsvadze","year":"1992","journal-title":"Comput Math Math Phys"},{"issue":"2","key":"10.1016\/j.cnsns.2026.110547_bib0006","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/S0377-0427(98)00135-6","article-title":"On tsertsvadze\u2019s difference scheme for the Kuramoto\u2013Tsuzuki equation","volume":"98","author":"Sun","year":"1998","journal-title":"J Comput Appl Math"},{"issue":"6","key":"10.1016\/j.cnsns.2026.110547_bib0007","doi-asserted-by":"crossref","first-page":"2620","DOI":"10.1016\/j.cnsns.2010.09.013","article-title":"Convergence of a nonlinear finite difference scheme for the Kuramoto\u2013Tsuzuki equation","volume":"16","author":"Wang","year":"2011","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"10.1016\/j.cnsns.2026.110547_bib0008","doi-asserted-by":"crossref","first-page":"961","DOI":"10.1002\/num.20070","article-title":"Convergence of galerkin approximations for the Kuramoto\u2013Tsuzuki equation","volume":"21","author":"Omrani","year":"2005","journal-title":"Numer Methods Partial Differ Equ"},{"issue":"3","key":"10.1016\/j.cnsns.2026.110547_bib0009","doi-asserted-by":"crossref","first-page":"838","DOI":"10.4208\/cicp.OA-2018-0208","article-title":"Optimal error estimates of a linearized Crank\u2013Nicolson Galerkin fem for the Kuramoto\u2013Tsuzuki equations","volume":"26","author":"Li","year":"2019","journal-title":"Commun Comput Phys"},{"issue":"2","key":"10.1016\/j.cnsns.2026.110547_bib0010","doi-asserted-by":"crossref","first-page":"211","DOI":"10.4208\/jcm.2107-m2020-0243","article-title":"Unconditionally optimal error analysis of the second-order BDF finite element method for the Kuramoto\u2013Tsuzuki equation","volume":"41","author":"Li","year":"2022","journal-title":"J Comput Math"},{"issue":"8","key":"10.1016\/j.cnsns.2026.110547_bib0011","doi-asserted-by":"crossref","first-page":"3161","DOI":"10.1016\/j.cnsns.2011.12.020","article-title":"A fourth-order split-step pseudospectral scheme for the Kuramoto\u2013Tsuzuki equation","volume":"17","author":"Dong","year":"2012","journal-title":"Commun Nonlinear Sci Numer Simul"},{"issue":"3","key":"10.1016\/j.cnsns.2026.110547_bib0012","first-page":"622","article-title":"Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations","volume":"10","author":"Li","year":"2013","journal-title":"Int J Numer Anal Model"},{"issue":"4","key":"10.1016\/j.cnsns.2026.110547_bib0013","doi-asserted-by":"crossref","first-page":"1959","DOI":"10.1137\/120871821","article-title":"Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media","volume":"51","author":"Li","year":"2013","journal-title":"SIAM J Numer Anal"},{"issue":"4","key":"10.1016\/j.cnsns.2026.110547_bib0014","doi-asserted-by":"crossref","first-page":"1141","DOI":"10.4208\/cicp.080313.051213s","article-title":"The stability and convergence of fully discrete Galerkin\u2013Galerkin FEMs for porous medium flows","volume":"15","author":"Li","year":"2014","journal-title":"Commun Comput Phys"},{"issue":"327","key":"10.1016\/j.cnsns.2026.110547_bib0015","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1090\/mcom\/3561","article-title":"New analysis of Galerkin-mixed fems for incompressible miscible flow in porous media","volume":"90","author":"Sun","year":"2021","journal-title":"Math Comput"},{"issue":"6","key":"10.1016\/j.cnsns.2026.110547_bib0016","doi-asserted-by":"crossref","first-page":"3000","DOI":"10.1137\/130939717","article-title":"A new error analysis of characteristics-mixed FEMs for miscible displacement in porous media","volume":"52","author":"Wang","year":"2014","journal-title":"SIAM J Numer Anal"},{"issue":"3","key":"10.1016\/j.cnsns.2026.110547_bib0017","doi-asserted-by":"crossref","first-page":"390","DOI":"10.1007\/s10915-013-9799-4","article-title":"A new error analysis of Crank\u2013Nicolson Galerkin fems for a generalized nonlinear Schr\u00f6dinger equation","volume":"60","author":"Wang","year":"2014","journal-title":"J Sci Comput"},{"key":"10.1016\/j.cnsns.2026.110547_bib0018","doi-asserted-by":"crossref","first-page":"1093","DOI":"10.1007\/s10915-017-0390-2","article-title":"Unconditional superconvergence analysis of a Crank\u2013Nicolson Galerkin fem for nonlinear Schr\u00f6dinger equation","volume":"72","author":"Shi","year":"2017","journal-title":"J Sci Comput"},{"issue":"3","key":"10.1016\/j.cnsns.2026.110547_bib0019","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.1137\/130918678","article-title":"Optimal error estimates of linearized Crank\u2013Nicolson Galerkin fems for the time-dependent Ginzburg\u2013Landau equations in superconductivity","volume":"52","author":"Gao","year":"2014","journal-title":"SIAM J Numer Anal"},{"issue":"5","key":"10.1016\/j.cnsns.2026.110547_bib0020","doi-asserted-by":"crossref","first-page":"2574","DOI":"10.1137\/130936476","article-title":"Optimal error estimates of a linearized backward Euler fem for the Landau\u2013Lifshitz equation","volume":"52","author":"Gao","year":"2014","journal-title":"SIAM J Numer Anal"},{"issue":"3","key":"10.1016\/j.cnsns.2026.110547_bib0021","doi-asserted-by":"crossref","first-page":"1291","DOI":"10.1137\/17M113544X","article-title":"Analysis of Galerkin FEMs for mixed formulation of time-dependent Ginzburg\u2013Landau equations under temporal gauge","volume":"56","author":"Wu","year":"2018","journal-title":"SIAM J Numer Anal"},{"issue":"5","key":"10.1016\/j.cnsns.2026.110547_bib0022","doi-asserted-by":"crossref","first-page":"2892","DOI":"10.1093\/imanum\/drad081","article-title":"Uniform L\u221e-bounds for energy-conserving higher-order time integrators for the gross\u2013Pitaevskii equation with rotation","volume":"44","author":"D\u00f6ding","year":"2024","journal-title":"IMA J Numer Anal"},{"issue":"2","key":"10.1016\/j.cnsns.2026.110547_bib0023","doi-asserted-by":"crossref","first-page":"933","DOI":"10.1137\/120892465","article-title":"Unconditionally optimal error estimates of a Crank\u2013Nicolson Galerkin method for the nonlinear thermistor equations","volume":"52","author":"Li","year":"2014","journal-title":"SIAM J Numer Anal"},{"key":"10.1016\/j.cnsns.2026.110547_bib0024","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1007\/s10915-013-9746-4","article-title":"Optimal error analysis of Galerkin FEMs for nonlinear joule heating equations","volume":"58","author":"Gao","year":"2014","journal-title":"J Sci Comput"},{"issue":"6","key":"10.1016\/j.cnsns.2026.110547_bib0025","doi-asserted-by":"crossref","first-page":"2623","DOI":"10.1137\/13093769X","article-title":"Linearized FE approximations to a nonlinear gradient flow","volume":"52","author":"Li","year":"2014","journal-title":"SIAM J Numer Anal"},{"key":"10.1016\/j.cnsns.2026.110547_bib0026","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/j.apnum.2019.02.005","article-title":"A linearized energy-conservative finite element method for the nonlinear Schr\u00f6dinger equation with wave operator","volume":"140","author":"Cai","year":"2019","journal-title":"Appl Numer Math"},{"issue":"3","key":"10.1016\/j.cnsns.2026.110547_bib0027","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1007\/s10915-021-01510-2","article-title":"Unconditional optimal error estimates of linearized, decoupled and conservative Galerkin fems for the Klein\u2013Gordon\u2013Schr\u00f6dinger equation","volume":"87","author":"Yang","year":"2021","journal-title":"J Sci Comput"},{"key":"10.1016\/j.cnsns.2026.110547_bib0028","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.camwa.2021.02.019","article-title":"A leap-frog finite element method for wave propagation of Maxwell\u2013Schr\u00f6dinger equations with nonlocal effect in metamaterials","volume":"90","author":"Yao","year":"2021","journal-title":"Comput Math Appl"},{"key":"10.1016\/j.cnsns.2026.110547_bib0029","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1016\/j.jcp.2016.08.016","article-title":"Time-filtered leapfrog integration of Maxwell equations using unstaggered temporal grids","volume":"325","author":"Mahalov","year":"2016","journal-title":"J Comput Phys"},{"key":"10.1016\/j.cnsns.2026.110547_bib0030","article-title":"On the numerical approximation of a phase-field volume reconstruction model: linear and energy-stable leap-frog finite difference scheme","volume":"152","author":"Fu","year":"2025","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"10.1016\/j.cnsns.2026.110547_bib0031","first-page":"372","article-title":"Numerical solution of the coupled viscous burgers equations by Chebyshev\u2013Legendre pseudo-spectral method","volume":"245","author":"Rashid","year":"2014","journal-title":"Appl Math Comput"},{"issue":"3","key":"10.1016\/j.cnsns.2026.110547_bib0032","doi-asserted-by":"crossref","first-page":"989","DOI":"10.1093\/imanum\/dru025","article-title":"Numerical analysis of very weakly well-posed hyperbolic Cauchy problems","volume":"35","author":"Colombini","year":"2015","journal-title":"IMA J Numer Anal"},{"key":"10.1016\/j.cnsns.2026.110547_bib0033","series-title":"Galerkin finite element methods for parabolic problems","author":"Thom\u00e9e","year":"2006"},{"key":"10.1016\/j.cnsns.2026.110547_bib0034","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1137\/0727022","article-title":"Finite-element approximation of the nonstationary Navier\u2013Stokes problem. Part IV: error analysis for the second-order time discretization","volume":"27","author":"Heywood","year":"1990","journal-title":"SIAM J Numer Anal"},{"key":"10.1016\/j.cnsns.2026.110547_bib0035","series-title":"The mathematical theory of finite element methods","author":"Brenner","year":"1994"}],"container-title":["Communications in Nonlinear Science and Numerical Simulation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S1007570426009019?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S1007570426009019?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,8,12]],"date-time":"2026-08-12T23:27:54Z","timestamp":1786577274000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S1007570426009019"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,11]]},"references-count":35,"alternative-id":["S1007570426009019"],"URL":"https:\/\/doi.org\/10.1016\/j.cnsns.2026.110547","relation":{},"ISSN":["1007-5704"],"issn-type":[{"value":"1007-5704","type":"print"}],"subject":[],"published":{"date-parts":[[2026,11]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Unconditionally optimal L2 error estimates of a linearized leap-frog FE scheme for the Kuramoto\u2013Tsuzuki equation","name":"articletitle","label":"Article Title"},{"value":"Communications in Nonlinear Science and Numerical Simulation","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.cnsns.2026.110547","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"110547"}}