The Analysis Of Stability Regions For Several Boundary Value Methods

As one of the branch of mathematics, differential equations have a wide range of applications in science and technology, economics, humanities and other fields. However, even for the simplest differential equation, its solution is quite complicated. Since in some actual problems, we do not need the real solution but the numerical solution of differential equations, it is important to use numerical methods to solve practical problems.There are many methods to deal with the numerical solutions for differential equations, such as Euler methods, Runge-Kutta methods and linear multistep methods. As a kind of simple and convenient numerical methods, linear multistep methods have been studied deeply and widely by scholars. With the further exploration for linear multistep methods, the defects which seem difficult to be overcome are waiting to be solved. At this time, BVMs(Boundary Value Methods) came into being. BVMs, the promotion of linear multistep methods, can overcome the defects and are known with great stability properties.In this paper, three-step BVMs and four-step BVMs are mainly studied.First, we introduce the background of differential equations and BVMs. At the same time, the status of research of BVMs is expounded.Then, we give difference schemes of three-step BVMs and four-step BVMs on the basis of order conditions. Meanwhile, we give the stability definitions corresponding to three-step BVMs and four-step BVMs by applying the techniques used in two-step BVMs.After that, we analyze the stability and convergence properties of the schemes of three-step BVMs and four-step BVMs primarily. In the process of stability analysis, we introduce the notion of type of a polynomial and give boundary value stability conditions for three-step BVMs and four-step BVMs by applying some results of Schur polynomial. In the process of convergence analysis, we give the proof of convergence of three-step BVMs and four-step BVMs and provide the order of convergence according to the related properties and conclusions of Toeplitz matrices.At last, numerical examples are given to verify the conclusions referring to three-step BVMs and four-step BVMs.