Convergence and Stability Analysis of Exponential General Linear Methods for Delay Differential Equations

In this paper, we study the convergence and stability properties of explicit exponential general linear methods for delay differential equations. We prove that, under some assumptions, for delay differential equations in Banach spaces, these numerical methods converge essentially with the order min{P, Q + 1}, where P and Q denote the order and stage order of the methods for ordinary differential equations, respectively. By using an interpolation procedure for the delay term, we analyze the linear and non-linear stability of exponential general linear methods for two classes of delay differential equations. The sufficient conditions on the stability of exponential general linear methods for the test delay differential equations are provided. Several numerical experiments are given to demonstrate the conclusions.