The stability of linear multistep methods for linear systems of neutral differential equations

This paper deals with the numerical solution of initial value problems for systems of neutral differential equations y ' (t) = f(t,y(t), y(t-tau), y ' (t-tau)) t >0, y(t) = phi (t) t <0, where tau > 0, f and phi denote given vector-valued functions. The numerical stability of a linear multistep method is investigated by analysing the solution of the test equations y ' (t) = Ay(t) + By(t - tau) + Cy ' (t - tau), where A, B and C denote constant complex N x N-matrices, and tau > 0. We investigate the properties of adaptation of the linear multistep method and the characterization of the stability region. It is proved that the linear multistep method is NGP-stable if and only if it is A-stable for ordinary differential equations.