The Research Of Reduced High Order Compact Difference Algorithms Based On POD Method For Numerical Solution Of Some Partial Differential Equations

In this thesis,several high order compact finite difference algorithms for parabolic equations and Fisher-Kolmogorov equations are mainly studied.In the computation of large engineering problems,the high order compact finite difference scheme will generate tens of million of unknown quantities,which takes up a lot of time.In order to overcome the deficiency,we utilize the Proper Orthogonal Decomposition(POD)to develop the reduced high order compact finite difference scheme and optimize the process of computation.This reduced high order compact finite difference scheme based on POD method not only possesses theses advantages such as high accuracy using considerably few grid points and high spectral like-resolution and universal boundary treatment,but also reduces computational time dramatically and alleviate the calculation load and CPU burden.The numerical examples verify the correctness of the theoretical analysis and the reduced finite difference scheme can greatly improve the computational efficiency under guarantee of the sufficiently precision.Therefore,the numerical results show the feasibility and effectiveness of the reduced finite difference scheme based on POD method.The full text is divided into five chapters.The main contents are as follows:The Chapter 1 of this paper is the introduction part,which mainly gives a simple overview of partial differential equations,and briefly introduces some forms of finite difference scheme and the background knowledge and application of POD method.In the Chapter 2,the reduced fourth-order compact difference scheme for parabolic equations is investigated.Firstly,based on Taylor's formula,we give the detailed derivation steps of the fourth-order compact difference scheme for parabolic equations.Then,by introducing POD method,we get a reduced compact finite difference scheme and error estimation formulas of two schemes.Finally,we use several numerical examples to illustrate that the proposed method is efficient for seeking the numerical solution of the parabolic equation.The Chapter 3,we firstly introduces the detailed construction of the sixth order compact difference scheme for one-dimensional parabolic equation.Then,we employ the POD method to the solution of six-order compact difference scheme for parabolic equation and establish the reduced six-order compact difference scheme which can ensure the accuracy and save a lot of the computational time of six-order compact difference scheme.The method is extended to multi-dimension problems successfully by splitting method.The Chapter 4 expands and improves on the basis of Chapter three.This method is used to solve the extended Fisher-Kolmogorov equation,which is more complex in form than parabolic equations and has mixed derivatives,making it more difficult to solve.In this chapter,the whole process of the algorithm are given.Finally,a numerical example is given to illustrate the high accuracy,high efficiency and feasibility of the method.The Chapter 5 briefly summarizes the main work and innovations of this paper and gives the need for in-depth study of the content to continue to discuss in the future.