Two Kinds Of Predictor Corrector Algorithms For Nonlinear Delay Differential Equations

In our daily life,differential equations and human society are closely related,these equations are used to establish many models,such as population development model,traffic flow model…However,due to the complexity of practical problems,the differential equations are complicated in structure,and it is very difficult to give analytical solutions.In view of this phenomenon,numerical methods are used,they are divided into two categories: explicit and implicit methods.Although the explicit method calculation process is simple,the error is relatively larger.On the other hand,the error of implicit method is smaller,but the calculation process is cumbersome.So,Experts and scholars combine the two methods,firstly,the explicit format is used to provide a preestimate.And then,the value will be plugged into the implicit format.The resulting value is called a correction value.This method is called predictor corrector algorithm,which combines the advantages of explicit method and implicit method,and makes up for their shortcomings.But in the past twenty years,predictor corrector algorithm was seldom researched.In this paper,one-leg predictor corrector algorithm and linear multi-step predictor corrector algorithm for the general scheme of nonlinear delay differential equations are introduced,including stability and convergence.And some general theoretical results are obtained,they are proved by numerical experiments finally.The main contributions are summarized as follows:In the first part,the background,significance and research status of this article are introduced.In the second part,the conclusion and the stability and convergence of this paper are given.In the third part,we construct one-leg predictor corrector algorithm for general schemes,then discuss the stability and convergence of the algorithm under certain conditions.Both the relationship between the stability of the predictor corrector algorithm and the stability of its sub-methods,and the quantitative relationship between the convergence order of the predictor corrector algorithm and the convergence order of its sub-methods are proved.In the fourth part,a general scheme of linear multi-step predictor corrector algorithm is constructed.According to the transformation relationship between the linear multi-step methods and one-leg methods,the stability and convergence of the linear multi-step predictor corrector algorithm are easily obtained.Lastly,we use numerical experiments to verify the results.