Numerical Methods For Ordinary Differential Equations And Delay Differential Equations Are Then,

Recently, the dynamic features of the numerical methods has aroused people's extensive concern since the numerical methods appeared and developed. The dynamic features of the numerical methods include stability,monotonicity,disspativity,regularity and so on. In this paper the regularities of the numerical methods are studied. That is to say whether the system's balance is consistent with the equilibrium of the numerical methods. It also means whether false balance can be produced when a system is solved with a numerical method with a fixed step. The method is called regular when thus spurious balance can't be produced. The main work in this paper are done as follows.Firstly, we outline the regularity properties of Runge-Kutta methods for ordinary differential equations and delay differential equations and give several theorems and their corollaries.Secondly, the regularity properties of multi-derivative methods for delay differential equations are discussed. The concepts of regularity and strong regularity are given. Several theorems and corollaries are obtained which are used to judge regularity or strong regularity of the numerical methods and some examples are provided accordingly.Thirdly, the regularity properties of Runge-Kutta-Nystr(o|¨)m methods for two stage delay differential equations are observed. It gives the concepts of regularity and strong regularity for this method. Several theorems and corollaries are acquired to judge regularity or strong regularity of the numerical methods and one corresponding example is given.Finally, the summary of this thesis is given and several ideas for the follow-up work of this paper are put forward.