Explicit Linear Multistep Methods And Exponential Integral Methods For Burgers Equations

Differential equations are important cognition tools to represent the low of the nature, since they were established. They are widely used in the fields of Ecology, Environmental science, Economics, Electrical power Engineering, Automation and so on. So, it turns very essential to make research on the numerical solution of differential equations and the stability of the numerical methods. In this paper, we focus on the linear multistep methods with an increased region of absolute stability, the stability character of the neutral delay differential equations with linear multistep methods, and a kind of explicit stable method to solve the Burgers equations.The linear multistep methods have the simple format and small error constant when they are used to solve the problem of ordinary differential equations, so they are in common used. But because of the restriction of the stability, they must use very small step length to integrate when they are used to solve the problem with long time step that make the efficiency very low. So in the second section we ameliorate the stability of the linear multistep methods, and find the methods which have the optimum stable character.The neutral delay differential equation is an important sort of delay differential equations. They are widely used in many fields, so the research is practicality. In the third section of this paper, we discuss the stability character of the neutral delay differential equations with A-stable linear multistep method.Professor Feng brought the ideas of structure-preserving to the classical numerical analysis at 1984, it has gotten a lot of attention and interest from all around the world, and many structure-preserving algorithms come into being with this idea. Lie-group methods which are used to construct the numerical solution of differential equations evolving on the manifold are kinds of structure-preserving algorithms which are developed recently. In the forth section of this paper, we construct the exponential integral methods based on Magnus expansion, and discrete the burgers equations at spatial direction, than transform it to a system of ordinary differential equations which can be solved by the exponential integral methods. Numerical results show that the Lie-group methods have the same accuracy as the corresponding Runge-Kutta methods and have more preponderance at stability and step strict.