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1、Numerical Method of Differential EquationInstructor: Dr. Xinming ZhangDepartment of Mathematics & Natural Sciences第1页第1页 About This Course - OutlineInitial-Value Problems for Ordinary Differential Equations Chapter 1 Introduction to the Partial Differential Equations and Finite Difference MethodChap

2、ter 3Chapter 2Boundary-Value Problems for Ordinary Differential Equations Finite Difference Method for Hyperbolic Partial Differential EquationsChapter 5Chapter 6Finite Difference Method for Elliptic Partial Differential EquationsFinite Difference Method for Parabolic Partial Differential EquationsC

3、hapter 4第2页第2页 Chapter 3 Introduction to PDE and FDM The chapter serves as an introduction to partial differential equation and to the subject of finite difference methods for solving partial differential equations.3.1 Introduction to Partial Differential Equation3.2 Introduction to Finite Differenc

4、e Method第3页第3页 3.1 Introduction to Partial Differential Equation3.1.1 Some classical PDEs Partial differential equation (PDEs) from the basis of very many mathematical models of physical, chemical and biological phenomena, and more recently their use has spread into economics, financial forecasting,

5、 image processing and other fields. To investigate the predictions of PDE models of such phenomena it is often necessary to approximate their solution numerically, commonly in combination with the analysis of simple special cases; while in some of the recent instances the numerical models play an al

6、most independent role. Let us consider an example. 第4页第4页 3.1 Introduction to Partial Differential Equation第5页第5页 3.1 Introduction to Partial Differential Equation If the boundary of the body is relatively simple, the solution to this equation can be found using Fourier series. In most situations wh

7、ere k, c and p are not constants or when the boundary is irregular, the solution to the partial differential equation must be obtained by approximation techniques. An introduction to techniques of this type is presented in this chapter. Of the many different approaches to solving partial differentia

8、l equations numerically (finite difference, finite elements, spectral methods, collocation methods, etc.), we shall study difference methods. Next, we will give some classical PDEs.第6页第6页 3.1 Introduction to Partial Differential Equation第7页第7页 3.1 Introduction to Partial Differential Equation第8页第8页

9、3.1 Introduction to Partial Differential Equation第9页第9页 3.1 Introduction to Partial Differential Equation第10页第10页 3.1 Introduction to Partial Differential Equation第11页第11页 3.1 Introduction to Partial Differential Equation3.1.2 Fixed Solution Problem In general, it is difficult to express the partial

10、 differential equation with general solution. PDEs are solved in certain conditions, we refer to the conditions as fixed solution conditions, including initial condition, boundary condition. The equations and the fixed solution conditions make a fixed solution problem.第12页第12页 3.1 Introduction to Pa

11、rtial Differential Equation第13页第13页 3.1 Introduction to Partial Differential Equation第14页第14页3.1 Introduction to Partial Differential Equation3.2 Introduction to Finite Difference MethodChapter 3 Introduction to PDE and FDM第15页第15页 3.2 Introduction to Finite Difference Method3.2.1 Get Started We con

12、sider the following initial boundary value problem第16页第16页 3.2 Introduction to Finite Difference Method第17页第17页 3.2 Introduction to Finite Difference Method第18页第18页 3.2 Introduction to Finite Difference Method第19页第19页 3.2 Introduction to Finite Difference Method第20页第20页 3.2 Introduction to Finite Di

13、fference Method第21页第21页 3.2 Introduction to Finite Difference Method第22页第22页 3.2 Introduction to Finite Difference Method第23页第23页 3.2 Introduction to Finite Difference Method We now have a numerical scheme to approximate the solution of initial-boundary-value problem. We call this scheme an explicit

14、 scheme because we are able to solve for the variable at the (n+1)st time level explicitly.第24页第24页 3.2 Introduction to Finite Difference Method第25页第25页 3.2 Introduction to Finite Difference Method第26页第26页 3.2 Introduction to Finite Difference MethodNext, we introduce some difference notations第27页第2

15、7页 3.2 Introduction to Finite Difference Method第28页第28页 3.2 Introduction to Finite Difference Method3.2.2 Convergence, Consistency and Stability3.2.2.1 Truncation error第29页第29页 3.2 Introduction to Finite Difference Method第30页第30页 3.2 Introduction to Finite Difference Method For implicit difference s

16、cheme (3.5), we have(3.2-1)(3.3)第31页第31页 3.2 Introduction to Finite Difference Method3.2.2.2 Convergence第32页第32页 3.2 Introduction to Finite Difference Method第33页第33页 3.2 Introduction to Finite Difference Method第34页第34页 3.2 Introduction to Finite Difference Method(3.13)(3.11-1)第35页第35页 3.2 Introducti

17、on to Finite Difference Method第36页第36页 3.2 Introduction to Finite Difference Method第37页第37页 3.2 Introduction to Finite Difference Method第38页第38页 3.2 Introduction to Finite Difference Method第39页第39页 3.2 Introduction to Finite Difference MethodInitial-Boundary-Value Problems第40页第40页 3.2 Introduction t

18、o Finite Difference Method第41页第41页 3.2 Introduction to Finite Difference Method3.2.2.3 Consistency第42页第42页 3.2 Introduction to Finite Difference Method第43页第43页 3.2 Introduction to Finite Difference Method第44页第44页 3.2 Introduction to Finite Difference Method第45页第45页 3.2 Introduction to Finite Differe

19、nce Method第46页第46页 3.2 Introduction to Finite Difference Method第47页第47页 3.2 Introduction to Finite Difference Method第48页第48页 3.2 Introduction to Finite Difference Method第49页第49页 3.2 Introduction to Finite Difference MethodInitial-Boundary-Value Problems第50页第50页 3.2 Introduction to Finite Difference

20、Method3.2.2.4 Stability第51页第51页 3.2 Introduction to Finite Difference Method第52页第52页 3.2 Introduction to Finite Difference Method第53页第53页 3.2 Introduction to Finite Difference Method第54页第54页 3.2 Introduction to Finite Difference Method3.2.2.5 The Lax Theorem第55页第55页 3.2 Introduction to Finite Differ

21、ence Method第56页第56页 3.2 Introduction to Finite Difference Method3.2.3 Proving stability of difference scheme In the previous section, we showed important stability is for proving convergence of difference scheme. This section is devoted to proving stability of difference scheme. This is done largely

22、 by introducing tools that can be used to prove stability of difference schemes, such as discrete Fourier transform. 第57页第57页 3.2 Introduction to Finite Difference Method3.2.3.1Initial Value Problem第58页第58页 3.2 Introduction to Finite Difference Method第59页第59页 3.2 Introduction to Finite Difference Me

23、thod第60页第60页 3.2 Introduction to Finite Difference Method第61页第61页 3.2 Introduction to Finite Difference Method第62页第62页 3.2 Introduction to Finite Difference Method第63页第63页 3.2 Introduction to Finite Difference Method第64页第64页 3.2 Introduction to Finite Difference Method第65页第65页 3.2 Introduction to Fi

24、nite Difference Method第66页第66页 3.2 Introduction to Finite Difference Method第67页第67页 3.2 Introduction to Finite Difference Method第68页第68页 3.2 Introduction to Finite Difference Method第69页第69页 3.2 Introduction to Finite Difference Method第70页第70页 3.2 Introduction to Finite Difference Method第71页第71页 3.2

25、Introduction to Finite Difference Method第72页第72页 3.2 Introduction to Finite Difference Method第73页第73页 3.2 Introduction to Finite Difference Method第74页第74页 3.2 Introduction to Finite Difference Method3.2.3.2 Initial Boundary Value Problems We now discuss stability for initial boundary value problems.

26、 We shall discuss only problems that are bounded in each spatial variable. We recall that a difference scheme for an initial boundary value problem consists of a difference equation approximating the partial differential equation and difference equations approximating each boundary condition. If the

27、 difference scheme is unstable without considering the boundary conditions (i.e. considering the difference scheme as an initial value scheme), then the scheme will also be unstable for the initial boundary value problem when the boundary condition equations are included. Hence, we obtain the following result.第75页第75页 3.2 Introduction to Finite Difference MethodProposition 3.9 Consider a difference scheme for an initial boundary value problem. The von Neumann condition for the difference sche