Stability and oscillations of numerical solutions for differential equations with piecewise continuous arguments of alternately advanced and retarded type

This paper is concerned with the numerical properties of θθ-methods for the solution of alternately advanced and retarded differential equations with piecewise continuous arguments. Using two θθ-methods, namely the one-leg θθ-method and the linear θθ-method, the necessary and sufficient conditions under which the analytic stability region is contained in the numerical stability region are obtained, and the conditions of oscillations for the θθ-methods are also obtained. It is proved that oscillations of the analytic solution are preserved by the θθ-methods. Furthermore, the relationships between stability and oscillations are revealed. Some numerical experiments are presented to illustrate our results.