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Numerical Method of Differential EquationInstructor:Dr.Xinming ZhangDepartment of Mathematics&Natural Sciences第第1页页 About This Course-OutlineInitial-Value Problems for Ordinary Differential Equations Chapter 1Chapter 1 Introduction to the Partial Differential Equations and Finite Difference MethodChapter 3Chapter 3Chapter 2Chapter 2Boundary-Value Problems for Ordinary Differential Equations Finite Difference Method for Hyperbolic Partial Differential EquationsChapter 5Chapter 5Chapter 6Chapter 6Finite Difference Method for Elliptic Partial Differential EquationsFinite Difference Method for Parabolic Partial Differential EquationsChapter 4Chapter 4第第2页页 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 Difference Method第第3页页 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,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 almost independent role.Let us consider an example.第第4页页 3.1 Introduction to Partial Differential Equation第第5页页 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 where 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 differential equations numerically(finite difference,finite elements,spectral methods,collocation methods,etc.),we shall study difference methods.Next,we will give some classical PDEs.第第6页页 3.1 Introduction to Partial Differential Equation第第7页页 3.1 Introduction to Partial Differential Equation第第8页页 3.1 Introduction to Partial Differential Equation第第9页页 3.1 Introduction to Partial Differential Equation第第10页页 3.1 Introduction to Partial Differential Equation第第11页页 3.1 Introduction to Partial Differential Equation3.1.2 Fixed Solution Problem In general,it is difficult to express the partial 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.第第12页页 3.1 Introduction to Partial Differential Equation第第13页页 3.1 Introduction to Partial Differential Equation第第14页页3.1 Introduction to Partial Differential Equation3.2 Introduction to Finite Difference MethodChapter 3 Introduction to PDE and FDM第第15页页 3.2 Introduction to Finite Difference Method3.2.1 Get Started We consider the following initial boundary value problem第第16页页 3.2 Introduction to Finite Difference Method第第17页页 3.2 Introduction to Finite Difference Method第第18页页 3.2 Introduction to Finite Difference Method第第19页页 3.2 Introduction to Finite Difference Method第第20页页 3.2 Introduction to Finite Difference Method第第21页页 3.2 Introduction to Finite Difference Method第第22页页 3.2 Introduction to Finite Difference Method第第23页页 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 scheme because we are able to solve for the variable at the(n+1)st time level explicitly.第第24页页 3.2 Introduction to Finite Difference Method第第25页页 3.2 Introduction to Finite Difference Method第第26页页 3.2 Introduction to Finite Difference MethodNext,we introduce some difference notations第第27页页 3.2 Introduction to Finite Difference Method第第28页页 3.2 Introduction to Finite Difference Method3.2.2 Convergence,Consistency and Stability3.2.2.1 Truncation error第第29页页 3.2 Introduction to Finite Difference Method第第30页页 3.2 Introduction to Finite Difference Method For implicit difference scheme(3.5),we have(3.2-1)(3.3)第第31页页 3.2 Introduction to Finite Difference Method3.2.2.2 Convergence第第32页页 3.2 Introduction to Finite Difference Method第第33页页 3.2 Introduction to Finite Difference Method第第34页页 3.2 Introduction to Finite Difference Method(3.13)(3.11-1)第第35页页 3.2 Introduction to Finite Difference Method第第36页页 3.2 Introduction to Finite Difference Method第第37页页 3.2 Introduction to Finite Difference Method第第38页页 3.2 Introduction to Finite Difference Method第第39页页 3.2 Introduction to Finite Difference MethodInitial-Boundary-Value Problems第第40页页 3.2 Introduction to Finite Difference Method第第41页页 3.2 Introduction to Finite Difference Method3.2.2.3 Consistency第第42页页 3.2 Introduction to Finite Difference Method第第43页页 3.2 Introduction to Finite Difference Method第第44页页 3.2 Introduction to Finite Difference Method第第45页页 3.2 Introduction to Finite Difference Method第第46页页 3.2 Introduction to Finite Difference Method第第47页页 3.2 Introduction to Finite Difference Method第第48页页 3.2 Introduction to Finite Difference Method第第49页页 3.2 Introduction to Finite Difference MethodInitial-Boundary-Value Problems第第50页页 3.2 Introduction to Finite Difference Method3.2.2.4 Stability第第51页页 3.2 Introduction to Finite Difference Method第第52页页 3.2 Introduction to Finite Difference Method第第53页页 3.2 Introduction to Finite Difference Method第第54页页 3.2 Introduction to Finite Difference Method3.2.2.5 The Lax Theorem第第55页页 3.2 Introduction to Finite Difference Method第第56页页 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 by introducing tools that can be used to prove stability of difference schemes,such as discrete Fourier transform.第第57页页 3.2 Introduction to Finite Difference Method3.2.3.1Initial Value Problem第第58页页 3.2 Introduction to Finite Difference Method第第59页页 3.2 Introduction to Finite Difference Method第第60页页 3.2 Introduction to Finite Difference Method第第61页页 3.2 Introduction to Finite Difference Method第第62页页 3.2 Introduction to Finite Difference Method第第63页页 3.2 Introduction to Finite Difference Method第第64页页 3.2 Introduction to Finite Difference Method第第65页页 3.2 Introduction to Finite Difference Method第第66页页 3.2 Introduction to Finite Difference Method第第67页页 3.2 Introduction to Finite Difference Method第第68页页 3.2 Introduction to Finite Difference Method第第69页页 3.2 Introduction to Finite Difference Method第第70页页 3.2 Introduction to Finite Difference Method第第71页页 3.2 Introduction to Finite Difference Method第第72页页 3.2 Introduction to Finite Difference Method第第73页页 3.2 Introduction to Finite Difference Method第第74页页 3.2 Introduction to Finite Difference Method3.2.3.2 Initial Boundary Value Problems We now discuss stability for initial boundary value problems.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 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.第第75页页 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 scheme considered as a difference scheme for an initial value problem is a necessary condition for stability.第第76页页 3.2 Introduction to Finite Difference Method第第77页页 3.2 Introduction to Finite Difference Method For example,we could write the scheme(3.32)as the following matrix equation:第第78页页 3.2 Introduction to Finite Difference Method第第79页页 3.2 Introduction to Finite Difference Method第第80页页 3.2 Introduction to Finite Difference Method第第81页页 3.2 Introduction to Finite Difference Method第第82页页 3.2 Introduction to Finite Difference Method第第83页页 3.2 Introduction to Finite Difference Method第第84页页 3.2 Introduction to Finite Difference Method第第85页页 3.2 Introduction to Finite Difference Method第第86页页 3.2 Introduction to Finite Difference Method第第87页页 3.2 Introduction to Finite Difference Method第第88页页Instructor:Dr.Xinming ZhangDepartment of Mathematics&Natural Sciences第第89页页