General linear methods for the numerical solution of functional-differential equations (Q5951203)

This paper is concerned with the numerical solution of initial value problems for delay differential equations with the form: \( x' = f(t, x_t(\cdot)), t\in [\alpha, \beta]\), \(x(t)= \varphi(t)\), \(t \in [\alpha - \tau, \alpha]\) where \( \tau>0\) is a given constant and \( x_t( \cdot)= \{ x(t+s)\); \(\alpha - \tau \leq t+s \leq t \}\) is the history of the phase vector at time \(t\). By introducing suitable assumptions on \(f\), the author develops the well known relations between the concepts of consistency, stability and convergence for linear step by step methods on uniform grids. The theory is applied to explicit Runge-Kutta methods and to a general class of linear methods \( u_{n+1} = S_n u_n + h \Phi (t_n, u_n,h)\) which are assumed to be strongly stable giving necessary and sufficient conditions for convergence of order \(p\) in terms of consistency conditions.