Stability of numerical methods for delay differential equations

AbstractConsider the following delay differential equation (DDE) y′=ƒ(t,y(t),y(t−τ(t))), t⩾t0,with the initial condition y(t)=Φ for t⩽t0,where ƒ and Φ are such that (0.1), (0.2) has a unique solution y(t). The author gives sufficient conditions for the asymptotic stability of the equation (0.1) for which he introduces new definitions of numerical stability. The approach is reminiscent of that from the nonlinear, stiff ordinary differential equation (ODE) field. He investigates stability properties of the class of one-point collocation rules. In particular, the backward Euler method turns out to be stable with respect to all the given definitions