Numerical solution of retarded functional differential equations as abstract Cauchy problems

An approach for the numerical solution of linear delay differential equations, different from the classical step-by-step integration, was presented in (Numer. Math. 84 (2000) 351). The problem is restated as an abstract Cauchy problem (or as the advection equation with a particular nonstandard boundary condition) and then, by using a scheme of order one, it is discretized as a system of ordinary differential equations by the method of lines. In this paper we introduce a class of related schemes of arbitrarily high order and we then extend the approach to general retarded functional differential equations. An analysis of convergence, and of asymptotic stability when the numerical schemes are applied to the complex scalar equation y′(t)=ay(t)+by(t−1), is provided.