On the stability of Runge–Kutta methods for arbitrarily large systems of ODEs

IF 3.1 1区数学 Q1 MATHEMATICS Communications on Pure and Applied Mathematics Pub Date : 2024-11-20 DOI:10.1002/cpa.22238

Eitan Tadmor

引用次数: 0

Abstract

We prove that Runge–Kutta (RK) methods for numerical integration of arbitrarily large systems of Ordinary Differential Equations are linearly stable. Standard stability arguments—based on spectral analysis, resolvent condition or strong stability, fail to secure the stability of RK methods for arbitrarily large systems. We explain the failure of different approaches, offer a new stability theory based on the numerical range of the underlying large matrices involved in such systems, and demonstrate its application with concrete examples of RK stability for hyperbolic methods of lines.

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论任意大的 ODE 系统的 Runge-Kutta 方法的稳定性

我们证明了用于任意大常微分方程系统数值积分的 Runge-Kutta (RK) 方法是线性稳定的。基于谱分析、分解条件或强稳定性的标准稳定性论证无法确保任意大系统的 RK 方法的稳定性。我们解释了不同方法的失效原因，提出了基于此类系统所涉及的底层大矩阵数值范围的新稳定性理论，并通过双曲线性方法的 RK 稳定性的具体实例演示了其应用。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Communications on Pure and Applied Mathematics 数学-数学

CiteScore

6.70

自引率

3.30%

发文量

审稿时长

>12 weeks

期刊介绍： Communications on Pure and Applied Mathematics (ISSN 0010-3640) is published monthly, one volume per year, by John Wiley & Sons, Inc. © 2019. The journal primarily publishes papers originating at or solicited by the Courant Institute of Mathematical Sciences. It features recent developments in applied mathematics, mathematical physics, and mathematical analysis. The topics include partial differential equations, computer science, and applied mathematics. CPAM is devoted to mathematical contributions to the sciences; both theoretical and applied papers, of original or expository type, are included.

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