Computing stability regions—Runge-Kutta methods for delay differential equations

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Abstract
We discuss the practical determination of stability regions when various fixed-stepsize Runge-Kutta (RK) methods, combined with continuous extensions, are applied to the linear delay differential equation (DDE) y′(t)= λy(t)+µ(t–τ) (t≥) with fixed delay τ. It is significant that the delay is not limited to an integer multiple of the stepsize, and that we consider various continuous extensions. The stability loci obtained in practice indicate that the standard boundary-locus technique (BLT) can fail to map the RK DDE stability region correctly. The aim of this paper is to present an alternative stability boundary algorithm that overcomes the difficulties encountered using the BLT. This new algorithm can be used for both explicit and implicit RK methods.

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