{"title":"修正相场晶体模型的保正能量稳定方案的误差分析","authors":"","doi":"10.1016/j.apnum.2024.09.010","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper, we introduce second-order numerical schemes for the modified phase field crystal (MPFC) model that are decoupled, linear, positivity-preserving, and unconditionally energy-stable. These schemes adopt a positivity-preserving auxiliary variable method to explicitly handle the nonlinear potential function, resulting in decoupled linear systems with constant coefficients at each time step. We rigorously demonstrate that the auxiliary variables remain positive throughout all time steps and prove the unconditionally energy stability of these schemes. The stability pertains to a discrete modified energy, rather than the original free energy or the pseudo energy of the MPFC system. Moreover, a detailed error analysis is provided. A series of numerical experiments are conducted to validate the accuracy and efficiency of our proposed schemes.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Error analysis of positivity-preserving energy stable schemes for the modified phase field crystal model\",\"authors\":\"\",\"doi\":\"10.1016/j.apnum.2024.09.010\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>In this paper, we introduce second-order numerical schemes for the modified phase field crystal (MPFC) model that are decoupled, linear, positivity-preserving, and unconditionally energy-stable. These schemes adopt a positivity-preserving auxiliary variable method to explicitly handle the nonlinear potential function, resulting in decoupled linear systems with constant coefficients at each time step. We rigorously demonstrate that the auxiliary variables remain positive throughout all time steps and prove the unconditionally energy stability of these schemes. The stability pertains to a discrete modified energy, rather than the original free energy or the pseudo energy of the MPFC system. Moreover, a detailed error analysis is provided. A series of numerical experiments are conducted to validate the accuracy and efficiency of our proposed schemes.</div></div>\",\"PeriodicalId\":8199,\"journal\":{\"name\":\"Applied Numerical Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.2000,\"publicationDate\":\"2024-09-18\",\"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/S0168927424002472\",\"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/S0168927424002472","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Error analysis of positivity-preserving energy stable schemes for the modified phase field crystal model
In this paper, we introduce second-order numerical schemes for the modified phase field crystal (MPFC) model that are decoupled, linear, positivity-preserving, and unconditionally energy-stable. These schemes adopt a positivity-preserving auxiliary variable method to explicitly handle the nonlinear potential function, resulting in decoupled linear systems with constant coefficients at each time step. We rigorously demonstrate that the auxiliary variables remain positive throughout all time steps and prove the unconditionally energy stability of these schemes. The stability pertains to a discrete modified energy, rather than the original free energy or the pseudo energy of the MPFC system. Moreover, a detailed error analysis is provided. A series of numerical experiments are conducted to validate the accuracy and efficiency of our proposed schemes.
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.