Spectral Analysis of Preconditioned Matrices Arising from Stage-Parallel Implicit Runge–Kutta Methods of Arbitrarily High Order

IF 1.5 2区数学 Q2 MATHEMATICS, APPLIED

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-05-13 DOI:10.1137/23m1552498

Ivo Dravins, Stefano Serra-Capizzano, Maya Neytcheva

{"title":"Spectral Analysis of Preconditioned Matrices Arising from Stage-Parallel Implicit Runge–Kutta Methods of Arbitrarily High Order","authors":"Ivo Dravins, Stefano Serra-Capizzano, Maya Neytcheva","doi":"10.1137/23m1552498","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 2, Page 1007-1034, June 2024. <br/> Abstract. The use of high order fully implicit Runge–Kutta methods is of significant importance in the context of the numerical solution of transient partial differential equations, in particular when solving large scale problems due to fine space resolution with many millions of spatial degrees of freedom and long time intervals. In this study we consider strongly [math]-stable implicit Runge–Kutta methods of arbitrary order of accuracy, based on Radau quadratures, for which efficient preconditioners have been introduced. A refined spectral analysis of the corresponding matrices and matrix sequences is presented, both in terms of localization and asymptotic global distribution of the eigenvalues. Specific expressions of the eigenvectors are also obtained. The given study fully agrees with the numerically observed spectral behavior and substantially improves the theoretical studies done in this direction so far. Concluding remarks and open problems end the current work, with specific attention to the potential generalizations of the hereby suggested general approach.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5000,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Matrix Analysis and Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/23m1552498","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

Abstract

SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 2, Page 1007-1034, June 2024.
Abstract. The use of high order fully implicit Runge–Kutta methods is of significant importance in the context of the numerical solution of transient partial differential equations, in particular when solving large scale problems due to fine space resolution with many millions of spatial degrees of freedom and long time intervals. In this study we consider strongly [math]-stable implicit Runge–Kutta methods of arbitrary order of accuracy, based on Radau quadratures, for which efficient preconditioners have been introduced. A refined spectral analysis of the corresponding matrices and matrix sequences is presented, both in terms of localization and asymptotic global distribution of the eigenvalues. Specific expressions of the eigenvectors are also obtained. The given study fully agrees with the numerically observed spectral behavior and substantially improves the theoretical studies done in this direction so far. Concluding remarks and open problems end the current work, with specific attention to the potential generalizations of the hereby suggested general approach.

查看原文本刊更多论文

任意高阶阶段并行隐式 Runge-Kutta 方法产生的预条件矩阵的频谱分析

SIAM 矩阵分析与应用期刊》，第 45 卷第 2 期，第 1007-1034 页，2024 年 6 月。摘要。高阶全隐式 Runge-Kutta 方法的使用在瞬态偏微分方程数值求解中具有重要意义，特别是在求解具有数百万空间自由度和长时间跨度的精细空间分辨率的大规模问题时。在本研究中，我们考虑了基于 Radau quadratures 的任意精度阶的强[数学]稳定隐式 Runge-Kutta 方法，并引入了高效预处理。从特征值的局部分布和渐近全局分布两个方面，对相应矩阵和矩阵序列进行了精细的谱分析。同时还获得了特征向量的具体表达式。所给出的研究完全符合数值观测到的频谱行为，并大大改进了迄今为止在此方向上所做的理论研究。结束语和有待解决的问题结束了当前的工作，并特别关注了本文提出的一般方法的潜在概括性。

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分钟内获得全文求助全文

来源期刊

SIAM Journal on Matrix Analysis and Applications 数学-应用数学

CiteScore

2.90

自引率

6.70%

发文量

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Matrix Analysis and Applications contains research articles in matrix analysis and its applications and papers of interest to the numerical linear algebra community. Applications include such areas as signal processing, systems and control theory, statistics, Markov chains, and mathematical biology. Also contains papers that are of a theoretical nature but have a possible impact on applications.