The stability of natural Runge-Kutta methods for nonlinear delay differential equations

Abstract

A natural Runge-Kutta method is a special type of Runge-Kutta method for delay differential equations (DDEs); it is known that any collocation method is equivalent to one of such methods. In this paper, stability properties of natural Runge-Kutta methods are studied using nonlinear DDEs which have a quadratic Liapunov functional. A discrete analogue of the functional is defined for each method, and the stability of the method is examined on the basis of this analogue. In particular, it is shown that an algebracially stable method, if it satisfies an additional condition, preserves the asymptotic properties of the original equations for every stepsize.