NP-stability of Runge-Kutta methods based on classical quadrature
This paper is concerned with the stability properties of implicit Runge-Kutta methods for neutral delay differential equations of type: (*) \(u'(t)= au(t)+ bu(t-\tau)+ cu' (t- \tau)\) where \(\tau>0\) is the constant delay and \(a,b\) and \(c\) complex parameters. In many stability studies a Runge-Kutta method is called NP-stable if for all sets of complex values \((a,b,c)\) such that the zero solution of (*) is asymptotically stable the numerical solution of the method equipped with a natural continuous extension shares the same property for all \(h= \tau/m\), when \(m\) is a positive integer. With this definition \textit{A. Bellen}, \textit{Z. Jackiewicz} and \textit{M. Zennaro} [Numer. Math. 52, No. 6, 605-619 (1988; Zbl 0644.65049)] have proved that all A-stable collocation methods are NP-stable. Here by using a different approach the author proves a general theorem which implies the NP-stability of a Runge-Kutta method under weaker assumptions. Then this theorem is applied to prove the NP-stability of Radau IA and Lobatto IIIC families of methods equipped with suitable continuous extensions.
scientific article; zbMATH DE number 1117813
scientific article