{"title":"方形相场晶体模型无条件能量稳定二阶 BDF 方案的时间误差分析","authors":"Guomei Zhao , Shuaifei Hu","doi":"10.1016/j.apnum.2024.05.009","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we first propose and study the second-order time-discrete numerical scheme for the sixth-order nonlinear parabolic problem of the square phase-field crystal model. Then, we demonstrate the two-step backward differentiation formula (BDF-2) scheme with mass conservation and energy dissipation, where the higher order nonlinear term is treated implicitly. Moreover, a rigorous error analysis is presented and we prove the optimal second-order convergence rate <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>- norm, where <em>τ</em> is the time step. Finally, some numerical results are provided to confirm our theoretical analysis.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"202 ","pages":"Pages 222-245"},"PeriodicalIF":2.2000,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Temporal error analysis of an unconditionally energy stable second-order BDF scheme for the square phase-field crystal model\",\"authors\":\"Guomei Zhao , Shuaifei Hu\",\"doi\":\"10.1016/j.apnum.2024.05.009\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we first propose and study the second-order time-discrete numerical scheme for the sixth-order nonlinear parabolic problem of the square phase-field crystal model. Then, we demonstrate the two-step backward differentiation formula (BDF-2) scheme with mass conservation and energy dissipation, where the higher order nonlinear term is treated implicitly. Moreover, a rigorous error analysis is presented and we prove the optimal second-order convergence rate <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>- norm, where <em>τ</em> is the time step. Finally, some numerical results are provided to confirm our theoretical analysis.</p></div>\",\"PeriodicalId\":8199,\"journal\":{\"name\":\"Applied Numerical Mathematics\",\"volume\":\"202 \",\"pages\":\"Pages 222-245\"},\"PeriodicalIF\":2.2000,\"publicationDate\":\"2024-05-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Applied Numerical Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0168927424001144\",\"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 Numerical Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0168927424001144","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Temporal error analysis of an unconditionally energy stable second-order BDF scheme for the square phase-field crystal model
In this paper, we first propose and study the second-order time-discrete numerical scheme for the sixth-order nonlinear parabolic problem of the square phase-field crystal model. Then, we demonstrate the two-step backward differentiation formula (BDF-2) scheme with mass conservation and energy dissipation, where the higher order nonlinear term is treated implicitly. Moreover, a rigorous error analysis is presented and we prove the optimal second-order convergence rate in - norm, where τ is the time step. Finally, some numerical results are provided to confirm our theoretical analysis.
期刊介绍:
The purpose of the journal is to provide a forum for the publication of high quality research and tutorial papers in computational mathematics. In addition to the traditional issues and problems in numerical analysis, the journal also publishes papers describing relevant applications in such fields as physics, fluid dynamics, engineering and other branches of applied science with a computational mathematics component. The journal strives to be flexible in the type of papers it publishes and their format. Equally desirable are:
(i) Full papers, which should be complete and relatively self-contained original contributions with an introduction that can be understood by the broad computational mathematics community. Both rigorous and heuristic styles are acceptable. Of particular interest are papers about new areas of research, in which other than strictly mathematical arguments may be important in establishing a basis for further developments.
(ii) Tutorial review papers, covering some of the important issues in Numerical Mathematics, Scientific Computing and their Applications. The journal will occasionally publish contributions which are larger than the usual format for regular papers.
(iii) Short notes, which present specific new results and techniques in a brief communication.