The Numerical Stability Analysis Of Two-Step Runge-Kutta Methods For Delay Differential Equations

Delay differential equations(DDE_s) provide a powerful model of many phenomena in ap-plied sciences, such as control theory, environment science, biology, economics and so on. As weknow, most of them cannot be solved analytically , so the numerical treatments of DDE_s becomesvery necessary. Furthermore, numerical stability is an important part in numerical analysis.The numerical solution of DDE_s has been the subject of intense research activity in the pastfew years,many numerical methods have been proposed for DDE_s,for example,θ-methods,multistep linear methods, Runge-Kutta methods and so on. In this paper, we are concerned withtwo-step Runge-Kutta methods(TSRK).In Chapter 2, the sufficient conditions of L-stability of two-step Runge-Kutta methods forordinary differential equation(ODE) are discussed. In Chapter 3, we discusses the GPLm-stabilityof TSRK with many delays. We show that two-step Runge-Kutta is GPLm-stable for DDE ifand only if the corresponding methods for ODE is L-stable; In Chapter 4, we discuss the GR(l)-stability, GAR(l)-stability and weak GAR(l)-stability on the basis of (k,l)-algebraically stability oftwo-step Runge-Kutta methods. In Chapter 5, we give some numerical examples of DDE_s. Theseexamples can confirm the theoretical results.