Asymptotic stability analysis of Runge-Kutta methods for nonlinear systems of delay differential equations
The initial value problem for a system of delay differential equations and its approximate solution by means of RN-stable continuous Runge-Kutta methods are considered. The notion of RN-stability has been introduced recently by \textit{L. Torelli} [J. Comput. Appl. Math. 25, No. 1, 15-26 (1989; Zbl 0664.65073)] for numerical methods applied to dissipative nonlinear systems of delay differential equations. The authors show that RN-stable Runge-Kutta methods also preserve this asymptotic stability property. Essentially, RN-stability ensures that even the discrete solution possesses the contractivity property known from the analytic solution.
scientific article; zbMATH DE number 1096029