On the dynamics of a discretized neutral equation

1 July 1992

journal article
Published by Oxford University Press (OUP) in IMA Journal of Numerical Analysis

Vol. 12 (3) , 339-363
https://doi.org/10.1093/imanum/12.3.339

Abstract

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(qt) + Cy′(pt), y(0) = 1, where p, q ε (0, 1), A, B, C ε 𝒞 and A ≠ 0. Sufficient conditions for the asymptotic stability of a discretized model of this equation are given when p = q = L⁻¹, where L ≥ 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.

Keywords

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