Volume 18, Issue 1
The GPL-Stability of Runge-Kutta Methods Fordelay Differential Systems
DOI:

J. Comp. Math., 18 (2000), pp. 75-82

Published online: 2000-02

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• Abstract

This paper deals with the GPL-stability of the Implicit Runge-Kutta methods for the numerical solutions of the systems of delay differential equations. We focus on the stability behaviour of the Implicit Runge-Kutta(IRK) methods in the solutions of the following test systems with a delay term $$y'(t) = Ly(t) + My(t-\tau), t\ge 0,$$ $$y(t)=\Phi(t), t\le 0,$$ where $L, M$ are $N \times N$ complex matrices, $\tau \gt 0$, $\Phi(t)$ is a given vector function. We shall show that the IRK methods is GPL-stable if and only if it is L-stable, when we use the IRK methods to the test systems above.

• Keywords

Delay differential equation Implicit Runge-Kutta methods GPL- stability

@Article{JCM-18-75, author = {}, title = {The GPL-Stability of Runge-Kutta Methods Fordelay Differential Systems}, journal = {Journal of Computational Mathematics}, year = {2000}, volume = {18}, number = {1}, pages = {75--82}, abstract = { This paper deals with the GPL-stability of the Implicit Runge-Kutta methods for the numerical solutions of the systems of delay differential equations. We focus on the stability behaviour of the Implicit Runge-Kutta(IRK) methods in the solutions of the following test systems with a delay term $$y'(t) = Ly(t) + My(t-\tau), t\ge 0,$$ $$y(t)=\Phi(t), t\le 0,$$ where $L, M$ are $N \times N$ complex matrices, $\tau \gt 0$, $\Phi(t)$ is a given vector function. We shall show that the IRK methods is GPL-stable if and only if it is L-stable, when we use the IRK methods to the test systems above. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9024.html} }
TY - JOUR T1 - The GPL-Stability of Runge-Kutta Methods Fordelay Differential Systems JO - Journal of Computational Mathematics VL - 1 SP - 75 EP - 82 PY - 2000 DA - 2000/02 SN - 18 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9024.html KW - Delay differential equation KW - Implicit Runge-Kutta methods KW - GPL- stability AB - This paper deals with the GPL-stability of the Implicit Runge-Kutta methods for the numerical solutions of the systems of delay differential equations. We focus on the stability behaviour of the Implicit Runge-Kutta(IRK) methods in the solutions of the following test systems with a delay term $$y'(t) = Ly(t) + My(t-\tau), t\ge 0,$$ $$y(t)=\Phi(t), t\le 0,$$ where $L, M$ are $N \times N$ complex matrices, $\tau \gt 0$, $\Phi(t)$ is a given vector function. We shall show that the IRK methods is GPL-stable if and only if it is L-stable, when we use the IRK methods to the test systems above.