Instability of Runge-Kutta methods when applied to linear systems of delay differential equations

Summary.

This paper investigates the stability of Runge-Kutta methods when they are applied to the complex linear system of delay differential equations $y^{\prime }\left( t\right) =Ly\left( t\right) +My\left( t-1\right)$, where $L,M\in \mathbb{C}^{d\times d}$. We prove that no Runge-Kutta method preserves asymptotic stability.