Regularity properties of Runge-Kutta methods for delay differential equations
type: Item
instance of: scholarly article
title: Regularity properties of Runge-Kutta methods for delay differential equations
zbMATH Open document ID: 0879.65052
DOI: 10.1016/S0168-9274(97)00025-1
published in: Applied Numerical Mathematics
publication date: 1998-01-22T00:00:00Z
review text: The authors extend well-known results on Runge-Kutta methods for ordinary differential equations to delay-differential equations. Consider the initial value problem with constant delay \(\tau>0\) \[ y'(t)= g(y(t),y(t-\tau)) \quad\text{for }t\geq t_0,\qquad y(t)= \varphi(t)\quad\text{for }t\in[t_0-\tau,t_0] \tag{1} \] with continuous functions \(g:\mathbb{R}^q\times \mathbb{R}^q\to\mathbb{R}^q\) and a given initial function \(\varphi\in C([t_0-\tau,t_0],\mathbb{R}^q)\). To solve (1) consider Runge-Kutta methods of the form \[ \begin{aligned} Y_i^{n+1}&= y_h(t_n)+h \sum_{j=1}^s a_{ij} g(Y_j^{n+1}, y_h(t_{n-m}+c_jh)),\\ y_h(t_n+\theta h)&= y_h(t_n)+h \sum_{j=1}^s b_j(\theta)\cdot g(Y_j^{n+1}, y_h(t_{n-m}+ c_jh)), \end{aligned}\tag{2} \] \(i=1,2,\dots, s\), where \(s\) is the number of stages, \(n\in\mathbb{N}_0\), \(\theta\in[0,1]\), \(h\) is a constant stepsize such that \(mh=\tau\) for some integer \(m>0\), \(y_h\) is the numerical approximation to the solution \(y\) of (1) at the gridpoints and \(b_j(\theta)\) are polynomials. Now regularity means that if the numerical approximations \(y_h(t_n)\) at the gridpoints \(t_n\) converge to \(\overline{y}\) as \(n\to\infty\), then \(\overline{y}\in \{y\in\mathbb{R}^q\), \(g(y,y)=0\}\), i.e. Runge-Kutta methods (2) cannot have spurious asymptotics. Strong regularity means that not only \(y_h(t_n)\), but also the stage values \(Y_i^{n+1}\) obtained by interpolation, tend to the equilibrium point of (1) as \(n\to\infty\). The authors derive conditions which characterize regular and strongly regular Runge-Kutta methods for systems (1). These conditions are then used to derive examples of strongly regular methods with \(s=p=2\) and \(s=p=3\).
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 34K05
Mathematics Subject Classification ID: 65L05
zbMATH DE Number: 1036658
zbMATH Keywords: Runge-Kutta methods
zbMATH Keywords: delay-differential equations
zbMATH Keywords: regularity
author: Zdzislaw Jackiewicz
author: R. Vermiglio
author: M. Zennaro
reviewed by: Hans Ade
MaRDI profile type: Publication
cites work: Q4725642
cites work: Equilibria of Runge-Kutta methods
cites work: Q4839883
cites work: Stability and Dynamics of Numerical Methods for Nonlinear Ordinary Differential Equations
cites work: Regularity properties of Runge-Kutta methods for ordinary differential equations
cites work: Q4000373
cites work: A stability property of \(A\)-stable natural Runge-Kutta methods for systems of delay differential equations
cites work: Delay differential equations: with applications in population dynamics
cites work: Natural Continuous Extensions of Runge-Kutta Methods
cites work: P-stability properties of Runge-Kutta methods for delay differential equations
cites work: Natural Runge-Kutta and projection methods
Recommended article: Regularity properties of Runge-Kutta methods for ordinary differential equations
Recommended article: Q4860604
Recommended article: Convergence of Runge-Kutta methods for delay differential equations
Recommended article: On the asymptotic stability properties of Runge-Kutta methods for delay differential equations
Recommended article: The stability of a class of Runge-Kutta methods for delay differential equations
Recommended article: On stability of Runge-Kutta methods for delay integral equations
Recommended article: Stability of Runge-Kutta methods for delay integro-differential equations
Recommended article: P-stability properties of Runge-Kutta methods for delay differential equations
Recommended article: Nonlinear stability and D-convergence of Runge-Kutta methods for delay differential equations
Recommended article: Stability of Runge-Kutta methods for linear delay differential equations
description: scientific article; zbMATH DE number 1036658