Asymptotic stability barriers for natural Runge--Kutta processes for delay equations

This paper investigates the asymptotic stability properties of a class of numerical methods for delay differential equations (DDEs), the so-called natural Runge–Kutta methods. We first examine the behavior of these methods when applied to the neutral model equation y'(t) = a y(t) + b y(t − 1) + c y'(t − 1) with a, b, c ∈ R (we also consider the case when a, b, c ∈ C) and provide a suitable geometric characterization of their asymptotic stability regions. Then, by means of the obtained results, in conjunction with those given in [N. Guglielmi and E. Hairer, IMA J. Numer. Anal., 21 (2001), pp. 439–450], we are able to give a final answer concerning the possible preservation of asymptotic stability of the considered class of methods when applied to systems of linear DDEs of the form y'(t) = L y(t) + M y(t − 1) with L, M ∈ R^d×d, d > 1. We are interested here in methods that produce stable numerical solutions for all values of the parameters (a, b, and c in the first equation and L and M in the second one) for which the exact solution tends to zero. To this aim we direct our attention to a novel stability setting, recently introduced and investigated for the scalar nonneutral case (see [N. Guglielmi, Numer. Math., 77 (1997), pp. 467–485, N. Guglielmi, IMA J. Numer. Anal., 18 (1998), pp. 399–418, N. Guglielmi and E. Hairer, Numer. Math., 83 (1999), pp. 371–383, N. Guglielmi and E. Hairer, IMA J. Numer. Anal., 21 (2001), pp. 439–450, S. Maset, Numer. Math., 87 (2000), pp. 355–371]).