---

Order stars and stability for delay differential equations

---

Nicola Guglielmi and Ernst Hairer

---

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 $a$ and $b$ are complex numbers.

---

Key Words. Dealay differential equations, Runge-Kutta methods, order stars, asymptotic stability, root locus technique.

---