Stability of a class of Runge-Kutta methods for a family of pantograph equations of neutral type

This paper deals with the stability of Runge-Kutta methods for a class of neutral infinite delay-differential equations with different proportional delays. Under suitable conditions, the asymptotic stability of some Runge-Kutta methods with variable stepsize are considered by the stability function at infinity. It is proved that the even-stage Gauss-Legendre methods are not asymptotically stable, but the Radau IA methods, Radau IIA methods and Lobatto IIIC methods are all asymptotically stable. Furthermore, some numerical experiments are given to demonstrate the main conclusions.