{"title":"非线性耦合Schrödinger系统新型结构保持非协调BDF2-FEM的无条件超收敛分析","authors":"Zhenqi Qi , Dongyang Shi , Xiaozhong Yang","doi":"10.1016/j.cnsns.2025.109132","DOIUrl":null,"url":null,"abstract":"<div><div>This article mainly addresses a new mass and energy preserving two-step backward differential formula (BDF2) fully discretization scheme for the coupled nonlinear Schrödinger (CNLS) equations with nonconforming <span><math><mrow><mi>E</mi><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>r</mi><mi>o</mi><mi>t</mi></mrow></msubsup></mrow></math></span> element. By means of the fashionable spatiotemporal splitting approach and the mathematical induction, the boundedness in <span><math><msup><mrow><mi>W</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>∞</mi></mrow></msup></math></span>-norm of the numerical solution is derived strictly. Then, based on the special virtues of <span><math><mrow><mi>E</mi><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>r</mi><mi>o</mi><mi>t</mi></mrow></msubsup></mrow></math></span> element, through subtracting the error equations of two adjacent time layers and the discrete derivative transfer trick, the unconditional superclose result in the broken <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm, with no limitation between grid size and time step, is deduced rigorously. Whereafter, the corresponding unconditional superconvergence error estimate is presented using the interpolation post-processing skill. Finally, five numerical examples are offered to confirm the theoretical analysis. It is worth noting that the outcomes gained herein also apply to anisotropic mesh, which are numerically testified, moreover, they improve the previous studies which focused primarily on optimal error estimate with conforming elements on regular mesh for dealing with CNLS system.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"152 ","pages":"Article 109132"},"PeriodicalIF":3.8000,"publicationDate":"2025-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Unconditional superconvergence analysis of a novel structure preserving nonconforming BDF2-FEM for coupled nonlinear Schrödinger system\",\"authors\":\"Zhenqi Qi , Dongyang Shi , Xiaozhong Yang\",\"doi\":\"10.1016/j.cnsns.2025.109132\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>This article mainly addresses a new mass and energy preserving two-step backward differential formula (BDF2) fully discretization scheme for the coupled nonlinear Schrödinger (CNLS) equations with nonconforming <span><math><mrow><mi>E</mi><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>r</mi><mi>o</mi><mi>t</mi></mrow></msubsup></mrow></math></span> element. By means of the fashionable spatiotemporal splitting approach and the mathematical induction, the boundedness in <span><math><msup><mrow><mi>W</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>∞</mi></mrow></msup></math></span>-norm of the numerical solution is derived strictly. Then, based on the special virtues of <span><math><mrow><mi>E</mi><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>r</mi><mi>o</mi><mi>t</mi></mrow></msubsup></mrow></math></span> element, through subtracting the error equations of two adjacent time layers and the discrete derivative transfer trick, the unconditional superclose result in the broken <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm, with no limitation between grid size and time step, is deduced rigorously. Whereafter, the corresponding unconditional superconvergence error estimate is presented using the interpolation post-processing skill. Finally, five numerical examples are offered to confirm the theoretical analysis. It is worth noting that the outcomes gained herein also apply to anisotropic mesh, which are numerically testified, moreover, they improve the previous studies which focused primarily on optimal error estimate with conforming elements on regular mesh for dealing with CNLS system.</div></div>\",\"PeriodicalId\":50658,\"journal\":{\"name\":\"Communications in Nonlinear Science and Numerical Simulation\",\"volume\":\"152 \",\"pages\":\"Article 109132\"},\"PeriodicalIF\":3.8000,\"publicationDate\":\"2025-07-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications in Nonlinear Science and Numerical Simulation\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S100757042500543X\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Nonlinear Science and Numerical Simulation","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S100757042500543X","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Unconditional superconvergence analysis of a novel structure preserving nonconforming BDF2-FEM for coupled nonlinear Schrödinger system
This article mainly addresses a new mass and energy preserving two-step backward differential formula (BDF2) fully discretization scheme for the coupled nonlinear Schrödinger (CNLS) equations with nonconforming element. By means of the fashionable spatiotemporal splitting approach and the mathematical induction, the boundedness in -norm of the numerical solution is derived strictly. Then, based on the special virtues of element, through subtracting the error equations of two adjacent time layers and the discrete derivative transfer trick, the unconditional superclose result in the broken -norm, with no limitation between grid size and time step, is deduced rigorously. Whereafter, the corresponding unconditional superconvergence error estimate is presented using the interpolation post-processing skill. Finally, five numerical examples are offered to confirm the theoretical analysis. It is worth noting that the outcomes gained herein also apply to anisotropic mesh, which are numerically testified, moreover, they improve the previous studies which focused primarily on optimal error estimate with conforming elements on regular mesh for dealing with CNLS system.
期刊介绍:
The journal publishes original research findings on experimental observation, mathematical modeling, theoretical analysis and numerical simulation, for more accurate description, better prediction or novel application, of nonlinear phenomena in science and engineering. It offers a venue for researchers to make rapid exchange of ideas and techniques in nonlinear science and complexity.
The submission of manuscripts with cross-disciplinary approaches in nonlinear science and complexity is particularly encouraged.
Topics of interest:
Nonlinear differential or delay equations, Lie group analysis and asymptotic methods, Discontinuous systems, Fractals, Fractional calculus and dynamics, Nonlinear effects in quantum mechanics, Nonlinear stochastic processes, Experimental nonlinear science, Time-series and signal analysis, Computational methods and simulations in nonlinear science and engineering, Control of dynamical systems, Synchronization, Lyapunov analysis, High-dimensional chaos and turbulence, Chaos in Hamiltonian systems, Integrable systems and solitons, Collective behavior in many-body systems, Biological physics and networks, Nonlinear mechanical systems, Complex systems and complexity.
No length limitation for contributions is set, but only concisely written manuscripts are published. Brief papers are published on the basis of Rapid Communications. Discussions of previously published papers are welcome.