Stability Of Runge-Kutta Methods For Nonlinear Neutral Functional Differential Equations

Neutral functional differential equations (NFDEs) arise in many scientific fields such as biology, economics, nonlinear dynamics, and so on. It is important to study their numerical simulation, since it is difficult to obtain their analytic solution. In this thesis, we try to construct a class of preserving-stability diagonal implicit Runge-Kutta methods and investigate the stability of Runge-Kutta methods for nonlinear NFDEs based on existed results. This thesis includes the following several parts:The first Chapter is Introduction in which we introduce the origin of the diagonal implicit Runge-Kutta methods and the newest research results on the stability when they are applied to NFDEs. In this Chapter, we also emphasize the research significance of this work.In Chapter 2, we try to construct a class of preserving-stability diagonal implicit Runge-Kutta methods. Compared with fully implicit Runge-Kutta methods, the com-putational work of diagonal implicit Runge-Kutta methods drastically decreases. So some researcher have investigated the stability of diagonal implicit Runge-Kutta meth-ods for ordinary differential equations (ODEs) and functional differential equations (FDEs). By studying Bl-consistency, Bl-stability, and order conditions of diagonal implicit Runge-Kutta methods, we give the equivalent conditions of Bl-consistency and Bi-stability, and construct some 2-stage 2-order B0-stable and B1-stable diag-onal implicit Runge-Kutta methods. For 2-stage 2-order B2-stable diagonal implicit Runge-Kutta methods, theoretical analysis reveals that no such method exists.Chapter 3 concerns the stability of Runge-Kutta methods for nonlinear neutral functional differential equations (NFDEs). Based on the B-theory of Runge-Kutta methods for Volterra functional differential equations (VFDEs), we introduce the con-cepts of EB-stability of Runge-Kutta methods for NFDEs and obtain EB-stability results on variable stepsize Runge-Kutta methods for this class of equations. These results are new even for neutral delay differential equations and neutral delay integro-differential equations.In the final chapter, we give some numerical examples which further confirm our theoretical results.