A Decomposition Algorithm For KdV-Burgers Equation

In1895, the famous Dutch mathematician D. Korteweg and his studen-t G. de Vries gave us the KdV equation when they researched the shallow water. On the assumption of long wave approximation and small amplitude, they obtained the one-way movement for the shallow water equations. KdV equation is a relatively simple and very classical nonlinear model of these in mathematics and physics, such as the magnetic plasma wave and ion sound waves. And a lot of approximate hyperbolic equations can be transformed into KdV equation, in reality KdV equation is so important model that many people did a lot of research on it.Many fields of applied mathematics often involve Burgers equation, which has been discussed as a mathematical model of a flow phenomenon of fluid, such as the gas dynamic model and the traffic flow model. Burgers equation has some characteristics of Navier-Stokes equation. Therefore, researches on numerical method for Burgers equation are of important academic meaning besides obtaining approximate solution of Burgers’equation. So there are many finite difference methods and finite element methods for solving Burgers equation.According to domain decomposition algorithm, the calculated area is di-vided into several small areas which are easy to calculate. Finally, we hope the shape of these small areas as much regular as possible. The major dif-ficulties of domain decomposition algorithm are how to define the values of the boundary points and how to select the reasonable approximate solutions in these small areas. At the same time, as the rapid development of the com-puter, science and engineering calculation has made great progress. With the efficient parallel processing computer and the opportunity of the application and practice increased, it has greatly pushed forward the further research of domain decomposition algorithm and parallel computing numerical method.This paper has three chapters. The first chapter is "Introduction" sec-tion, we mainly make some preliminary knowledge of domain decomposition algorithm and the finite difference algorithm. In Chapter Two, we firstly give the finite difference domain decomposition algorithm for linear convection dif-fusion equation, and then solve Burgers equation with the new finite difference domain decomposition algorithm. We construct the new finite difference do-main decomposition algorithm with using the second category of Saul’yev asymmetric format, and then give the solution with the group explicit method at boundary points. And then we compare with the algorithms which were given by Dawson and the other persons, the numerical experiments show that we obtain a good stability and very high precision new algorithm. In Chapter Three, we combine KdV equation with Burgers equation, KdV-Burgers equa-tion arc given a class of alternating segment explicit and implicit difference algorithm. And we show that the algorithm is linear absolute stability and has the very good accuracy.