Numerical stability and oscillation of the Runge-Kutta methods for the differential equations with piecewise continuous arguments alternately of retarded and advanced type

摘要

This paper is concerned with the numerical properties of Runge-Kutta methods for the alternately of retarded and advanced equation x ˙ ( t ) = a x ( t ) + a 0 x ( 2 [ t + 1 2 ] ) . The stability region of Runge-Kutta methods is determined. The conditions that the analytic stability region is contained in the numerical stability region are obtained. A?necessary and sufficient condition for the oscillation of the numerical solution is given. And it is proved that the Runge-Kutta methods preserve the oscillations of the analytic solutions. Some numerical experiments are illustrated.