Stability of Discontinuous Galerkin Methods for Volterra Integral Equations

IF 1.8 3区数学 Q1 MATHEMATICS, APPLIED Mathematical Methods in the Applied Sciences Pub Date : 2025-01-06 DOI:10.1002/mma.10649

Jiao Wen, Min Li, Hongbo Guan

{"title":"Stability of Discontinuous Galerkin Methods for Volterra Integral Equations","authors":"Jiao Wen, Min Li, Hongbo Guan","doi":"10.1002/mma.10649","DOIUrl":null,"url":null,"abstract":"<div>\n \n <p>We conduct the stability analysis of discontinuous Galerkin methods applied to Volterra integral equations in this paper. Stability conditions with respect to both the basic and convolution test equations are derived. Our findings indicate that the methods with orders up to 6 exhibit \n<span></span><math>\n <semantics>\n <mrow>\n <mi>A</mi>\n </mrow>\n <annotation>$$ A $$</annotation>\n </semantics></math>-stability when applied with the basic test equation, while demonstrating unbounded stability regions when applied to the convolution test equation. Additionally, the results of \n<span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mrow>\n <mi>V</mi>\n </mrow>\n <mrow>\n <mn>0</mn>\n </mrow>\n </msub>\n </mrow>\n <annotation>$$ {V}_0 $$</annotation>\n </semantics></math>-stability for the semidiscretized variants (quadrature discontinuous Galerkin methods) and fully discretized versions (fully discretized discontinuous Galerkin methods) with orders 1 and 2 are presented when solving the convolution test equation. To corroborate these theoretical results, we provide some numerical experiments for validation.</p>\n </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 5","pages":"5972-5986"},"PeriodicalIF":1.8000,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Methods in the Applied Sciences","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mma.10649","RegionNum":3,"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

Abstract

We conduct the stability analysis of discontinuous Galerkin methods applied to Volterra integral equations in this paper. Stability conditions with respect to both the basic and convolution test equations are derived. Our findings indicate that the methods with orders up to 6 exhibit $A$ -stability when applied with the basic test equation, while demonstrating unbounded stability regions when applied to the convolution test equation. Additionally, the results of $V_{0}$ -stability for the semidiscretized variants (quadrature discontinuous Galerkin methods) and fully discretized versions (fully discretized discontinuous Galerkin methods) with orders 1 and 2 are presented when solving the convolution test equation. To corroborate these theoretical results, we provide some numerical experiments for validation.

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

Volterra 积分方程的非连续 Galerkin 方法的稳定性

本文对应用于Volterra积分方程的不连续伽辽金方法进行了稳定性分析。导出了基本方程和卷积方程的稳定性条件。我们的研究结果表明，当应用于基本测试方程时，阶数高达6的方法表现出A $$ A $$ -稳定性，而当应用于卷积测试方程时，则表现出无界的稳定性区域。此外，给出了1阶和2阶半离散型（正交不连续Galerkin方法）和完全离散型（完全离散不连续Galerkin方法）的v0 $$ {V}_0 $$ -稳定性的结果卷积检验方程。为了证实这些理论结果，我们提供了一些数值实验来验证。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助

来源期刊

Mathematical Methods in the Applied Sciences 数学-应用数学

CiteScore

4.90

自引率

6.90%

发文量

798

审稿时长

6 months

期刊介绍： Mathematical Methods in the Applied Sciences publishes papers dealing with new mathematical methods for the consideration of linear and non-linear, direct and inverse problems for physical relevant processes over time- and space- varying media under certain initial, boundary, transition conditions etc. Papers dealing with biomathematical content, population dynamics and network problems are most welcome. Mathematical Methods in the Applied Sciences is an interdisciplinary journal: therefore, all manuscripts must be written to be accessible to a broad scientific but mathematically advanced audience. All papers must contain carefully written introduction and conclusion sections, which should include a clear exposition of the underlying scientific problem, a summary of the mathematical results and the tools used in deriving the results. Furthermore, the scientific importance of the manuscript and its conclusions should be made clear. Papers dealing with numerical processes or which contain only the application of well established methods will not be accepted. Because of the broad scope of the journal, authors should minimize the use of technical jargon from their subfield in order to increase the accessibility of their paper and appeal to a wider readership. If technical terms are necessary, authors should define them clearly so that the main ideas are understandable also to readers not working in the same subfield.

期刊最新文献

Issue Information Issue Information Issue Information How Far Are Two Symmetric Matrices From Commuting? With an Application to Object Characterisation and Identification in Metal Detection Strictly Positive Solutions of Neumann Boundary Value Problems and Applications to Duffing Type Models