Stability of θ-methods for advanced differential equations with piecewise continuous arguments

This paper deals with the stability analysis of numerical methods for the solution of advanced differential equations with piecewise continuous arguments. We focus on the behaviour of the one-leg θ-method and the linear θ-method in the solution of the equation x′(t) = ax(t + a0x([t]) + a1x([t+1]), with real a, a0, a1 and [·] designates the greatest-integer function. The stability regions of two θ-methods are determined. The conditions under which the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given.