| Delay differential equations(DDEs) have a wide range of applications in real life, however, only a few classes can we get the analytic solution. Since in the implementation of numerical schemes it will become complicated and some desirable accuracy and stability properties of the underlying ordinary differential equation method can be destroyed when the method is applied to a delay differential equations, so it is especially important to analyze the numerical solution. This paper mainly studies the convergence and stability of collocation exponential Runge-Kutta method(CERKM) for semi-linear DDEs.The paper is orginazed as follows:The first chapter we introduce the background of DDEs and exponential integrator, and give some example to explain the application of DDEs in the real life. Then brifely discribe the research of DDEs and exponential intergrator.In chapter two, we give the concepts of GRN -stability and GDN- stability, introduce the concept of strong exponential algebraical stability. Firstly we study the stability of analytic solution, then prove that the exponential Euler method is GRN -stable, and CERKM of the strong exponential algebraical stability isGDN- stable.In chapter three, we define the D- convergence, diagonally stability, and the order of Lagrangian interpolation. Then introduce the concept of exponential stage order, and derive the condition of it. At last we show that a strong exponential algebraical stable and diagonally stable CERKM with exponential stage order p , together with a Lagrangian interpolation of order q is at least D? convergent of order min{p,q+1}In chapter four, we give some schemes of CERKM of stage one and two, study their GDN- stability and D- convergence. Then apply them to the specific equations to prove the results.Finally, we conclude the whole paper, and shows the further research. |
