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 θ -methods for the solution of alternately advanced and retarded differential equations with piecewise continuous arguments. Using two θ -methods, namely the one-leg θ -method and the linear θ -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 θ -methods are also obtained. It is proved that oscillations of the analytic solution are preserved by the θ -methods. Furthermore, the relationships between stability and oscillations are revealed. Some numerical experiments are presented to illustrate our results.