Stability Analysis of Runge-Kutta Methods for Nonlinear Neutral Volterra Delay-Integro-Differential Equations

This paper is concerned with the numerical stability of implicit Runge-Kutta methods for nonlinear neutral Volterra delay-integro-differential equations with constant delay. Using a Halanay inequality generalized by Liz and Trofimchuk, we give two sufficient conditions for the stability of the true solution to this class of equations. Runge-Kutta methods with compound quadrature rule are considered. Nonlinear stability conditions for the proposed methods are derived. As an illustration of the application of these investigations, the asymptotic stability of the presented methods for Volterra delay-integro-differential equations is proved under some weaker conditions than those in the literature. An extension of the stability results to such equations with weakly singular kernel is also discussed.