Stability of Runge-Kutta methods for linear delay differential equations

Summary.

This paper investigates the stability of Runge-Kutta methods when they are applied to the complex linear scalar delay differential equation $y^{\prime }\left( t\right) =ay\left(t\right) +by\left( t-1\right)$. This kind of stability is called $\tau -$stability. We give a characterization of $\tau -$ stable Runge-Kutta methods and then we prove that implicit Euler method is $\tau -$stable.