On the dynamics of a discretized neutral equation

In this paper, we investigate the stability properties of numerical methods for solving the differential equation of the neutral type y'(t) = Ay(t) +By(qi) + Cy'ipt), y(0) = 1, where p, q e (0, 1), A, B,Ce<€ and A =£0. Sufficient conditions for the asymptotic stability of a discretized model of this equation are given when p = q = L~l, where L 5= 2 is an integer. These coincide with conditions for asymptotic stability of the original equation, as reported in (Iserles, 1991), except that the step-length need be restricted and, at the limit, asymptotic stability is retained only if \C\ < 1. Moreover, for a particular choice of A these conditions are also shown to be necessary.