Semi-Implicit R-K Methods And Exponential Rosenbrock Methods Of Delay Differential Equations

Delay differential equations (DDEs) have been widely used in many fields such as economy, biology, physics, automation etc. But because of complexity of DDEs, it is usually difficult to obtain the analytic expression of solutions of the systems. So many people dedicate themselves to the research of the numerical methods of DDEs.This thesis studies two kinds of numerical methods for DDEs. Firstly, we have investigated the numerical stability of explicit and semi-implicit Runge-Kutta methods for delay differential equations with piecewise continuous argument (EPCA); Secondly, stability of the exponential Rosenbrock methods for DDEs is considered.The paper begins with original and applications of delay differential equations. Some applied examples for them are shown. It continues to present information with respect to analytic and numerical stability for delay differential equations for recent 50 years, and the paper's main work.On the stability of the numerical methods for EPCA, many conclusions have been obtained, but until now few conclusions have been obtained for the stability of explicit and semi-implicit Runge-Kutta methods. So in chapter 2, we discuss the properties of Order star of explicit and semi-implicit Runge-Kutta methods and investigate the numerical stability of these numerical methods for EPCA. We obtain the sufficient conditions of asymptotic stability, and we validate these conclusions by numerical experiments.Exponential Runge-Kutta methods and exponential Rosenbrock methods are invented recently. In chapter 3, we construct a kind of new exponential Rosenbrock methods for DDEs, and we proof that the exponential Rosenbrock method is GP-stable if and only if the corresponding method for ordinary differential equations is A-stable.