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

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