Nonlinear Stability Of One-leg Methods For Delay Differential Equations

Delay differential equations(DDEs) arise widely in physics, biology, engineering, medical science, economics and so on. It is meaningful to investigate the theory of numerical methods for the solution of DDEs.In 1989, Torelli discussed the nonlinear stability of numerical methods for DDEs for the first time. He introduced the concepts of RN-and GRN-stability and proved that implicit Euler method is GRN-stable. Unfortunately, the following studies showed that a very restrictive order barrier exists because the requirements of RN-stability are too strong. In 1999, Huang chengming relaxed the requirements and introduced the concepts of R- and GR-stability and proved that any A-stable one-leg methods is R-stable and with linear interpolation is GR-stable. Under above basis, the purpose of the present thesis is further to discuss the nonlinear stability of G(c,p,q)-algebraically stable one-leg methods for DDEs of the class of Kα,β,γ. In chapter 1, we provide a generalintroduction to the numerical analysis of DDEs. In chapter 2, we prove that G(c,p,q)-algebraically stable one-leg methods with linear interpolation are GR(p/2,q/2)-stable and weak GAR(p/2,q/2)-stable for c<1 and GAR(p/2,q/2)-stable for c