The Numerical Asymptotically Stability of a Linear Differential Equation with Piecewise Constant Arguments of Mixed Type

In this paper, we mainly consider the stability of numerical solution for the differential equation with piecewise constant arguments of mixed type. By the technique of solving differential equations, the concrete form of analytic solution is derived. Furthermore, the conditions under which the analytic solution is asymptotically stable are obtained. Then the Runge-Kutta methods are applied to the equation, using the theory of characteristic, the conditions under which the numerical solution is asymptotically stable are presented. Moreover, the necessary and sufficient conditions under which the numerical stability regions contain the analytical stability regions are determined. In particular, for -methods, we give the corresponding results of stability, which are the generalization of conclusions in the existed paper. Finally, some numerical experiments are being included to support the theoretical results.