The paper discusses the analytical stability and numerical stability of differential equations with piecewise constant arguments with matrix coefficients. It is proved that the Runge-Kutta method can keep the convergence order. The recurrence relation of the Runge-Kutta method applied to the equations is obtained. Then, the stability conditions of the numerical solution under different matrix coefficients are given by Pade approximation and order star theory. Finally, the conclusions are verified by several numerical experiments.

The paper deals with three dynamic properties of the numerical solution for differential equations with piecewise constant arguments of advanced and retarded type: oscillation, stability and convergence. The Euler-Maclaurin methods are used to discretize the equations. According to the characteristic theory of the difference equation, the oscillation and stability conditions of the numerical solution are obtained. It is proved that the convergence order of numerical method is 2n + 2. Furthermore, the relationship between stability and oscillation is discussed for analytic solution and numerical solution, respectively. Finally, several numerical examples confirm the corresponding conclusions.