Solution Convergence Of The Pre-conditions Of A Class Of Differential Equations

In this thesis we mainly research on the numerical solution of differential equations. To get the numerical solution of differential functions, we usually use difference method to get the linear system, then solve it. In according to the needs of practical problems, this linear system is usually a large sparse system.Therefore, this article mainly discuss how to increase the convergence speed of large sparse linear systems. For such systems, it's difficult to use the direct method to get exact solutions. Therefore we usually use the iterative method to solve, then it's worth to discuss the convergence and the convergence speed of the iterative method. Non-convergence or slow convergence of the iterative method is of no practical value.In recent years, the preconditioned method is widely studied, it can speed up the convergence rate of the iterative process greatly. This article main discusses this method, how to choose the good preconditoner, makes the pre-condition method to speed up the convergence rate. Because of the convergence speed is related with the spectrum radius of the iteration matrix, therefore this article is mainly comparing the spectrum radius.Main text includes five chapters. The first chapter is the introduction part, first export the linear system from the differential equation, then introduce iterative method of linear systems, and give the forms of several common iterative methods, at last introduce the precondition method; The second chapter is preliminary, it mainly lists the definition and the lemma which will be used in this article; The third chapter is relevant knowledge, it mainly introduce several preconditoners, and describe briefly the development of precondition method. From Chapter four , it is main body. Below we give the detailed explanation.The fourth chapter talks about when A is the non-singular irreducible M- matrix, how to choose S in the preconditioner P = I + Scan get a very good comparison theorem between the pre-condition method and the original iterative method, and give the example to proof the theorem. In addition by comparing the convergence speed, I explained how to choose the better preconditioner. The fifth chapter discusses the preconditioner P₁ specifically, first I explain the pre-condition method with P₁ is convergence and it can speed up the original iterative method the convergence rate; Then we discuss how to choice the best factor of this kind of precondition method; Then through the comparison we show P₁ is better than others; Finally we give the numerical example to confirmation theorem the accuracy and give the research prospect briefly.