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 is an element of R (we also consider the case when a, b, c is an element of 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 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 is an element of R-dxd, 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 rst 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]).