Locally One-dimensional Methods For Solving A Class Of Nonlinear Parabolic Equations

In the research field of computational mathematics, the theory and method of numerical solution of PDE is a very challenging task. With the increasing development of computer technology, people use numerical computation to solve mathematical model in the actual physical process, and it has increasingly become the forefront of the research and development at home and abroad. Parabolic equations are an important class of partial differential equations, and their theoretical study and solution are well studied in recent years, which make their research and application more widely in the major subject areas. At present, the finite difference method (FDM) is usually used to solve these equations. Because of its simple and intuitive operation, the FDM plays a very important role in many methods. Among them, the explicit scheme is easy to implement, but it often has low precision and poor stability. The stability of the implicit scheme is better than that of explicit scheme, but in the actual numerical simulation process, it is often required to calculate high dimensional or nonlinear equations, which leads to a large amount of computation when dealing with high dimensional problems.In this paper, locally one-dimensional methods are considered for solving the following nonlinear parabolic equationsIn these equations, the initial boundary value problem is mainly discussed. For the above problems, in most cases, people work on the simple case a?a(u). That is, the a has nothing to do with u. But now we are going to discuss the case a= a(u). This paper applies the technique of locally one-dimensional (abbreviated as LOD) method proposed by Dyakonov and Yanenko. This method is primarily used to solve high dimensional equations. However, due to the inherent characteristics of nonlinear problems, the nonlinear term is difficult to handle and it is difficult to determine the value of the buffer layer, so the process is hard to solve. Even though some simplified methods are given, the calculation accuracy is also affected. Focusing on a series of questions for the above problems, on the basis of the existing LOD method, this paper suggests to use two methods to deal with nonlinear terms a(u). Compared with the methods given in the previous literature, the difference scheme constructed in this paper has the advantages of less computation, easy to grasp and easy to implement and so on.The main works of the thesis are arranged as follows:The first chapter introduces the research background and significance of the nonlinear parabolic equation, as well as the existing theory and method for solving this kind of nonlinear parabolic equations. In Chapter 2, some related contents of several different schemes are introduced. Chapter 3 describes the locally one-dimension format and gives its discrete process. The truncation error and stability analysis of linear parabolic equations are also carried out. And its effectiveness is showed in the numerical experiment results. In Chapter 4, two processing methods for nonlinear parabolic equations are introduced. This chapter makes a further improvement for the result of the last chapter, and two sets of numerical experiments are also given in this part. Finally, a summary chapter is given.