Euler-Maclaurin Method for Linear Differential Equations with Piecewise Constant Arguments with One Delay: Stability and Oscillations

Abstract

This paper focuses on the stability and oscillations of Euler-Maclaurin method for linear differential equations with piecewise constant arguments $u^{\prime }\mathbf{\left(}t\mathbf{\right)}\hspace{0.17em}\mathbf{=}\hspace{0.17em}au\mathbf{\left(}t\mathbf{\right)}\hspace{0.17em}\mathbf{+}\hspace{0.17em}bu\mathbf{\left(}\mathbf{\right[}t\mathbf{\left]}\mathbf{\right)}$. The necessary and sufficient conditions under which the numerical stability region contains the analytical stability region are given. Furthermore, the conditions of oscillation for the Euler-Maclaurin method are obtained. We prove that the Euler-Maclaurin method preserves the oscillations of the analytic solution. Moreover, the relationships between stability and oscillations are discussed for analytic solution and numerical solution, respectively. Finally, some numerical experiments for verifying the theoretical analysis are also provided.