{"title":"导数非线性薛定谔方程保守差分方案的误差估计","authors":"","doi":"10.1016/j.aml.2024.109283","DOIUrl":null,"url":null,"abstract":"<div><p>In this letter, we propose and rigorously analyze a fully implicit difference scheme for the derivative nonlinear Schrödinger equation. We show that the numerical scheme at least preserves two discrete conserved quantities. Next, to facilitate error estimate, the numerical scheme is converted into an equivalent system, which can be regarded as one-stage Gaussian–Legendre Runge–Kutta method in time. Furthermore, with the help of the cut-off function technique, we prove the convergence of the equivalent system for the first time with the convergence order <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> under discrete <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span>-norm without any restriction on step ratio. Finally, the numerical results confirm theoretical findings and capacity in long-time simulations.</p></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":null,"pages":null},"PeriodicalIF":2.9000,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Error estimate of the conservative difference scheme for the derivative nonlinear Schrödinger equation\",\"authors\":\"\",\"doi\":\"10.1016/j.aml.2024.109283\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this letter, we propose and rigorously analyze a fully implicit difference scheme for the derivative nonlinear Schrödinger equation. We show that the numerical scheme at least preserves two discrete conserved quantities. Next, to facilitate error estimate, the numerical scheme is converted into an equivalent system, which can be regarded as one-stage Gaussian–Legendre Runge–Kutta method in time. Furthermore, with the help of the cut-off function technique, we prove the convergence of the equivalent system for the first time with the convergence order <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> under discrete <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span>-norm without any restriction on step ratio. Finally, the numerical results confirm theoretical findings and capacity in long-time simulations.</p></div>\",\"PeriodicalId\":55497,\"journal\":{\"name\":\"Applied Mathematics Letters\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.9000,\"publicationDate\":\"2024-08-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Applied Mathematics Letters\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0893965924003033\",\"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":"Applied Mathematics Letters","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0893965924003033","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Error estimate of the conservative difference scheme for the derivative nonlinear Schrödinger equation
In this letter, we propose and rigorously analyze a fully implicit difference scheme for the derivative nonlinear Schrödinger equation. We show that the numerical scheme at least preserves two discrete conserved quantities. Next, to facilitate error estimate, the numerical scheme is converted into an equivalent system, which can be regarded as one-stage Gaussian–Legendre Runge–Kutta method in time. Furthermore, with the help of the cut-off function technique, we prove the convergence of the equivalent system for the first time with the convergence order under discrete -norm without any restriction on step ratio. Finally, the numerical results confirm theoretical findings and capacity in long-time simulations.
期刊介绍:
The purpose of Applied Mathematics Letters is to provide a means of rapid publication for important but brief applied mathematical papers. The brief descriptions of any work involving a novel application or utilization of mathematics, or a development in the methodology of applied mathematics is a potential contribution for this journal. This journal''s focus is on applied mathematics topics based on differential equations and linear algebra. Priority will be given to submissions that are likely to appeal to a wide audience.