Stability of Runge–Kutta methods for the alternately advanced and retarded differential equations with piecewise continuous arguments

This paper deals with the numerical properties of Runge–Kutta methods for the solution of u′(t)=au(t)+a0u([t+12]). It is shown that the Runge–Kutta method can preserve the convergence order. The necessary and sufficient conditions under which the analytical stability region is contained in the numerical stability region are obtained. It is interesting that the θθ-methods with 0⩽θ<12 are asymptotically stable. Some numerical experiments are given.