非均匀网格上一类时间分式方程离散化的一些格伦沃尔不等式

IF 2.8 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis Pub Date : 2024-09-18 DOI:10.1137/24m1631614

Yuanyuan Feng, Lei Li, Jian-Guo Liu, Tao Tang

{"title":"非均匀网格上一类时间分式方程离散化的一些格伦沃尔不等式","authors":"Yuanyuan Feng, Lei Li, Jian-Guo Liu, Tao Tang","doi":"10.1137/24m1631614","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2196-2221, October 2024. <br/> Abstract. We consider the completely positive discretizations of fractional ordinary differential equations (FODEs) on nonuniform meshes. Making use of the resolvents for nonuniform meshes, we first establish comparison principles for the discretizations. Then we prove some discrete Grönwall inequalities using the comparison principles and careful analysis of the solutions to the time continuous FODEs. Our results do not have restriction on the step size ratio. The Grönwall inequalities for dissipative equations can be used to obtain the uniform-in-time error control and decay estimates of the numerical solutions. The Grönwall inequalities are then applied to subdiffusion problems and the time fractional Allen–Cahn equations for illustration.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"63 1","pages":""},"PeriodicalIF":2.8000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Some Grönwall Inequalities for a Class of Discretizations of Time Fractional Equations on Nonuniform Meshes\",\"authors\":\"Yuanyuan Feng, Lei Li, Jian-Guo Liu, Tao Tang\",\"doi\":\"10.1137/24m1631614\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2196-2221, October 2024. <br/> Abstract. We consider the completely positive discretizations of fractional ordinary differential equations (FODEs) on nonuniform meshes. Making use of the resolvents for nonuniform meshes, we first establish comparison principles for the discretizations. Then we prove some discrete Grönwall inequalities using the comparison principles and careful analysis of the solutions to the time continuous FODEs. Our results do not have restriction on the step size ratio. The Grönwall inequalities for dissipative equations can be used to obtain the uniform-in-time error control and decay estimates of the numerical solutions. The Grönwall inequalities are then applied to subdiffusion problems and the time fractional Allen–Cahn equations for illustration.\",\"PeriodicalId\":49527,\"journal\":{\"name\":\"SIAM Journal on Numerical Analysis\",\"volume\":\"63 1\",\"pages\":\"\"},\"PeriodicalIF\":2.8000,\"publicationDate\":\"2024-09-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"SIAM Journal on Numerical Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1137/24m1631614\",\"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":"SIAM Journal on Numerical Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/24m1631614","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

摘要

SIAM 数值分析期刊》第 62 卷第 5 期第 2196-2221 页，2024 年 10 月。摘要。我们考虑在非均匀网格上对分数常微分方程（FODE）进行完全正离散化。利用非均匀网格的解析式，我们首先建立了离散化的比较原则。然后，我们利用比较原理和对时间连续 FODE 解的仔细分析，证明了一些离散格伦沃不等式。我们的结果对步长比没有限制。耗散方程的格伦沃尔不等式可用于获得数值解的时间均匀误差控制和衰减估计。然后将格伦沃尔不等式应用于亚扩散问题和时间分数 Allen-Cahn 方程以作说明。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Some Grönwall Inequalities for a Class of Discretizations of Time Fractional Equations on Nonuniform Meshes

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2196-2221, October 2024.
Abstract. We consider the completely positive discretizations of fractional ordinary differential equations (FODEs) on nonuniform meshes. Making use of the resolvents for nonuniform meshes, we first establish comparison principles for the discretizations. Then we prove some discrete Grönwall inequalities using the comparison principles and careful analysis of the solutions to the time continuous FODEs. Our results do not have restriction on the step size ratio. The Grönwall inequalities for dissipative equations can be used to obtain the uniform-in-time error control and decay estimates of the numerical solutions. The Grönwall inequalities are then applied to subdiffusion problems and the time fractional Allen–Cahn equations for illustration.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

SIAM Journal on Numerical Analysis 数学-应用数学

CiteScore

4.80

自引率

6.90%

发文量

110

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Numerical Analysis (SINUM) contains research articles on the development and analysis of numerical methods. Topics include the rigorous study of convergence of algorithms, their accuracy, their stability, and their computational complexity. Also included are results in mathematical analysis that contribute to algorithm analysis, and computational results that demonstrate algorithm behavior and applicability.