Numerical solution of constant coefficient linear delay differential equations as abstract Cauchy problems

Summary. In this paper we present an approach for the numerical solution of delay differential equations

\begin{equation} \left\{ \begin{array}{l} y^{\prime }\left( t\right) =Ly\left( t\right) +My\left( t-\tau \right) \;\;t\geq 0 y\left( t\right) =\varphi \left( t\right) \;\;-\tau \leq t\leq 0, \end{array} \right. \end{equation}

where $\tau >0$, $L,M\in \mathbb{C}^{m\times m}$ and $\varphi \in C\left( \left[ -\tau ,0\right] ,\mathbb{C}^m\right)$, different from the classical step-by-step method. We restate (1) as an abstract Cauchy problem and then we discretize it in a system of ordinary differential equations. The scheme of discretization is proved to be convergent. Moreover the asymptotic stability is investigated for two significant classes of asymptotically stable problems (1).