Order stars and stability for delay differential equations

Summary.

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.