Using resolvent conditions to obtain new stability results for -methods for delay differential equations

Abstract

By using the Kreiss resolvent condition we establish upper bounds for the growth of errors in two simple, but typical, numerical processes for solving delay differential equations. We show that, when the θ‐methods are used, with a suitably restricted stepsize, for solving the linear test problems Z′(t) = λZ(t) + μZ(t − τ), errors grow at most linearly with the number of timesteps and with the dimension involved. Moreover, we investigate whether this kind of error growth is valid uniformly within the so‐called stability regions of these methods. Finally some numerical experiments are carried out with a representative member of the class of processes under consideration, and it is shown that our theoretical bounds for the error growth are of practical interest.