Order stars and stability for delay differential equations

Abstract

We consider Runge–Kutta methods applied to delay differential equations \(y'(t)=ay(t)+by(t-1)\) with real a and b. If the numerical solution tends to zero whenever the exact solution does, the method is called \(\tau (0)\)-stable. Using the theory of order stars we characterize high-order symmetric methods with this property. In particular, we prove that all Gauss methods are \(\tau (0)\)-stable. Furthermore, we present sufficient conditions and we give evidence that also the Radau methods are \(\tau (0)\)-stable. We conclude this article with some comments on the case where a andb are complex numbers.