Numerical Regularity Of Discrete And Distributed Delay Differential Equations

For numerical methods, there has been an increasing interest in the investigation oftheir dynamical properties, such as convergence, stability, dissipativity, regularity, chaosand bifurcation. This paper focus on the regularity of general linear methods for solvingODEs with discrete and distributed time delays.In 1990, Hairer firstly proposed the concept of regularity. Since then, regularity hasbeen widely researched by many authors in the field of numerical methods. The existingresults are mainly concerned with hybrid methods, Runge-Kutta methods and general linearmethods for solving ODEs without time delays, and one-leg methods, Runge-Kutta methodsfor solving delay differential equations (DDEs). In this paper, we analyze the regularity ofgeneral linear methods for solving ODEs with discrete and distributed time delays.This paper is organized as follows. In chapter 1, we introduce the background, someexcellent results of regularity of numerical methods. The regularity of general linearmethods for solving ODEs with discrete delays is analyzed in chapter 2. We prove thatessentially one-stage (EOS) methods and the methods which can be folded to EOS areregular. Necessary and sufficient conditions of regularity of the non-EOS methods forsolving ODEs with discrete time delays are also given. In chapter 3, we discuss the generallinear methods for solving ODEs with distributed delays. Similar to the case of ODEswith discrete delays, we prove the regularity of the essentially one-stage (EOS) methodsfor solving ODEs with distributed delays. Also, necessary and sufficient conditions ofregularity of the non-EOS methods for solving ODEs with distributed time delays arepresented. Finally, in chapter 4 we give some numerical results to verify the main results ofthis paper.