《数字信号处理》教学参考资料（Numerical Recipes in C，The Art of Scientific Computing Second Edition）Chapter 19.0 Partial Differential Equations 19.0 Introduction

Chapter 19.Partial Differential Equations .com or call 19.0 Introduction 11-800-72 Cambridge The numerical treatment of partial differential equations is,by itself,a vast NUMERICAL RECIPES IN subject.Partial differential equations are at the heart of many,if not most, server computer analyses or simulations of continuous physical systems,such as fluids, electromagnetic fields,the human body,and so on.The intent of this chapter is to compu give the briefest possible useful introduction.Ideally,there would be an entire second Press C:THEA volume of Numerical Recipes dealing with partial differential equations alone.(The 号 references [1-4]provide,of course,available alternatives.) g In most mathematics books,partial differential equations(PDEs)are classified into the three categories,hyperbolic,parabolic,and elliptic,on the basis of their SCIENTIFIC characteristics,or curves of information propagation.The prototypical example of a hyperbolic equation is the one-dimensional wave equation 6 02u 品 19.0.1) COMPUTING 1189-19 0x2 where v =constant is the velocity of wave propagation.The prototypical parabolic rica Furthe equation is the diffusion equation Recipes SBN0-6211 Ou du D t (19.0.2) (outside where D is the diffusion coefficient. The prototypical elliptic equation is the Poisson equation Software. 02u,02u 02+0n2 =p(x,) (19.0.3) America) visit website machine- where the source term p is given.If the source term is equal to zero,the equation is Laplace's equation. From a computational point of view,the classification into these three canonical types is not very meaningful-or at least not as important as some other essential distinctions.Equations (19.0.1)and (19.0.2)both define initial value or Cauchy problems:If information on u(perhaps including time derivative information)is 827

Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). Chapter 19. Partial Differential Equations 19.0 Introduction The numerical treatment of partial differential equations is, by itself, a vast subject. Partial differential equations are at the heart of many, if not most, computer analyses or simulations of continuous physical systems, such as fluids, electromagnetic fields, the human body, and so on. The intent of this chapter is to give the briefest possible useful introduction. Ideally, there would be an entire second volume of Numerical Recipes dealing with partial differential equations alone. (The references [1-4] provide, of course, available alternatives.) In most mathematics books, partial differential equations (PDEs) are classified into the three categories, hyperbolic, parabolic, and elliptic, on the basis of their characteristics, or curves of information propagation. The prototypical example of a hyperbolic equation is the one-dimensional wave equation ∂2u ∂t2 = v2 ∂2u ∂x2 (19.0.1) where v = constant is the velocity of wave propagation. The prototypical parabolic equation is the diffusion equation ∂u ∂t = ∂ ∂x
D ∂u ∂x (19.0.2) where D is the diffusion coefficient. The prototypical elliptic equation is the Poisson equation ∂2u ∂x2 + ∂2u ∂y2 = ρ(x, y) (19.0.3) where the source term ρ is given. If the source term is equal to zero, the equation is Laplace’s equation. From a computational point of view, the classification into these three canonical types is not very meaningful — or at least not as important as some other essential distinctions. Equations (19.0.1) and (19.0.2) both define initial value or Cauchy problems: If information on u (perhaps including time derivative information) is 827

828 Chapter 19. Partial Differential Equations boundary conditions http://www.nr. ● 83 (including this one) granted for i 11-800-872 initial values (a) ● boundary 7423 (North America to any server computer,is strictly prohibited. users to make one paper 1988-1992 by Cambridge University Press.Programs Copyright(C) from NUMERICAL RECIPES IN C: THE values ● only),or ● copy for their ● ● ● ● email to directcustsen ● ● (b) v@cambr 1988-1992 by Numerical Recipes ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Figure 19.0.1.Initial value problem (a)and boundary value problem(b)are contrasted.In (a)initial values are given on one "time slice,"and it is desired to advance the solution in time,computing successive rows of open dots in the direction shown by the arrows.Boundary conditions at the left and right edges of each row (must also be supplied,but only one row at a time.Only one,or a few,previous rows (outside need be maintained in memory.In (b),boundary values are specified around the edge of a grid,and an iterative process is employed to find the values of all the internal points (open circles).All grid points must be maintained in memory. North Software. given at some initial time to for all z,then the equations describe how u(x,t) visit website propagates itself forward in time.In other words,equations(19.0.1)and(19.0.2) machine describe time evolution.The goal of a numerical code should be to track that time evolution with some desired accuracy. By contrast,equation(19.0.3)directs us to find a single"static"functionu(,y) which satisfies the equation within some(z,y)region of interest,and which-one must also specify-has some desired behavior on the boundary of the region of interest.These problems are called boundary value problems.In general it is not

828 Chapter 19. Partial Differential Equations Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). . . . . . . . . . . . . . . . . . . . . . boundary conditions initial values (a) boundary values (b) Figure 19.0.1. Initial value problem (a) and boundary value problem (b) are contrasted. In (a) initial values are given on one “time slice,” and it is desired to advance the solution in time, computing successive rows of open dots in the direction shown by the arrows. Boundary conditions at the left and right edges of each row (⊗) must also be supplied, but only one row at a time. Only one, or a few, previous rows need be maintained in memory. In (b), boundary values are specified around the edge of a grid, and an iterative process is employed to find the values of all the internal points (open circles). All grid points must be maintained in memory. given at some initial time t0 for all x, then the equations describe how u(x, t) propagates itself forward in time. In other words, equations (19.0.1) and (19.0.2) describe time evolution. The goal of a numerical code should be to track that time evolution with some desired accuracy. By contrast, equation (19.0.3) directs us to find a single “static” function u(x, y) which satisfies the equation within some (x, y) region of interest, and which — one must also specify — has some desired behavior on the boundary of the region of interest. These problems are called boundary value problems. In general it is not

19.0 Introduction 829 possible stably to just"integrate in from the boundary"in the same sense that an initial value problem can be"integrated forward in time."Therefore,the goal of a numerical code is somehow to converge on the correct solution everywhere at once. This,then,is the most important classification from a computational point of view:Is the problem at hand an initial value (time evolution)problem?or is it a boundary value (static solution)problem?Figure 19.0.1 emphasizes the distinction.Notice that while the italicized terminology is standard,the terminology in parentheses is a much better description of the dichotomy from a computational perspective.The subclassification of initial value problems into parabolic and hyperbolic is much less important because(i)many actual problems are of a mixed type,and(ii)as we will see,most hyperbolic problems get parabolic pieces mixed into them by the time one is discussing practical computational schemes. Initial Value Problems An initial value problem is defined by answers to the following questions: What are the dependent variables to be propagated forward in time? What is the evolution equation for each variable?Usually the evolution equations will all be coupled,with more than one dependent variable appearing on the right-hand side of each equation. What is the highest time derivative that occurs in each variable's evolution equation?If possible,this time derivative should be put alone on the equation's left-hand side.Not only the value of a variable,but also the Programs value of all its time derivatives-up to the highest one-must be specified to define the evolution. .What special equations(boundary conditions)govern the evolution in time 、兰三猴 of points on the boundary of the spatial region of interest?Examples. to dir Dirichlet conditions specify the values of the boundary points as a function of time;Neumann conditions specify the values of the normal gradients on the boundary;outgoing-wave boundary conditions are just what they say. Sections 19.1-19.3 of this chapter deal with initial value problems of several different forms.We make no pretence of completeness,but rather hope to convey a 10621 certain amount of generalizable information through a few carefully chosen model examples.These examples will illustrate an important point:One's principal Fuurggoglrion Numerical Recipes 43108 computational concern must be the stability of the algorithm.Many reasonable- looking algorithms for initial value problems just don't work-they are numerically (outside unstable Software. Boundary Value Problems ying of The questions that define a boundary value problem are: What are the variables? What equations are satisfied in the interior of the region of interest? What equations are satisfied by points on the boundary of the region of interest?(Here Dirichlet and Neumann conditions are possible choices for elliptic second-order equations,but more complicated boundary conditions can also be encountered.)

19.0 Introduction 829 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). possible stably to just “integrate in from the boundary” in the same sense that an initial value problem can be “integrated forward in time.” Therefore, the goal of a numerical code is somehow to converge on the correct solution everywhere at once. This, then, is the most important classification from a computational point of view: Is the problem at hand an initial value (time evolution) problem? or is it a boundary value (static solution) problem? Figure 19.0.1 emphasizes the distinction. Notice that while the italicized terminology is standard, the terminology in parentheses is a much better description of the dichotomy from a computational perspective. The subclassification of initial value problems into parabolic and hyperbolic is much less important because (i) many actual problems are of a mixed type, and (ii) as we will see, most hyperbolic problems get parabolic pieces mixed into them by the time one is discussing practical computational schemes. Initial Value Problems An initial value problem is defined by answers to the following questions: • What are the dependent variables to be propagated forward in time? • What is the evolution equation for each variable? Usually the evolution equations will all be coupled, with more than one dependent variable appearing on the right-hand side of each equation. • What is the highest time derivative that occurs in each variable’s evolution equation? If possible, this time derivative should be put alone on the equation’s left-hand side. Not only the value of a variable, but also the value of all its time derivatives — up to the highest one — must be specified to define the evolution. • What special equations (boundary conditions) govern the evolution in time of points on the boundary of the spatial region of interest? Examples: Dirichlet conditionsspecify the values of the boundary points as a function of time; Neumann conditionsspecify the values of the normal gradients on the boundary; outgoing-wave boundary conditions are just what they say. Sections 19.1–19.3 of this chapter deal with initial value problems of several different forms. We make no pretence of completeness, but rather hope to convey a certain amount of generalizable information through a few carefully chosen model examples. These examples will illustrate an important point: One’s principal computational concern must be the stability of the algorithm. Many reasonable￾looking algorithms for initial value problems just don’t work — they are numerically unstable. Boundary Value Problems The questions that define a boundary value problem are: • What are the variables? • What equations are satisfied in the interior of the region of interest? • What equations are satisfied by points on the boundary of the region of interest? (Here Dirichlet and Neumann conditions are possible choices for elliptic second-order equations, but more complicated boundary conditions can also be encountered.)

830 Chapter 19.Partial Differential Equations In contrast to initial value problems,stability is relatively easy to achieve for boundary value problems.Thus,the efficiency of the algorithms,both in computational load and storage requirements,becomes the principal concern. Because all the conditions on a boundary value problem must be satisfied "simultaneously,"these problems usually boil down,at least conceptually,to the solution of large numbers of simultaneous algebraic equations.When such equations are nonlinear,they are usually solved by linearization and iteration;so without much loss of generality we can view the problem as being the solution of special,large linear sets of equations. As an example,one which we will refer to in 8819.4-19.6 as our "model problem,"let us consider the solution of equation (19.0.3)by the finite-difference method.We represent the function u(,y)by its values at the discrete set of points x5=0+j△，j=0,1,,J (19.0.4) =0+1△，1=0,1，，L ⊙ RECIPES where A is the grid spacing.From now on,we will write uj.for u(j,y),and Pj.!for p(j,).For(19.0.3)we substitute a finite-difference representation (see 9 Figure 19.0.2). 42+11-241+4-业+41-2%+=p5 42 42 (19.0.5) so 9 or equivalently u+1,1+4)-1,1+4,1+1+4.l-1-4u1,1=△2p5.2 (19.0.6) 6 To write this system of linear equations in matrix form we need to make a vector out of u.Let us number the two dimensions of grid points in a single one-dimensional sequence by defining i三L+1)+I for j=0,1,,J,1=0,1,,L (19.0.7) Numerical In other words,i increases most rapidly along the columns representing y values. -431 Equation (19.0.6)now becomes Recipes i+L+1+u1-(L+1)+ui+1+-1-4=△2P5 (outside (19.0.8) North This equation holds only at the interior pointsj=1,2,...J-1;1=1,2,..., L-1. The points where 1=0 [i.e,i=0,,L j=J [i.e.,i=J(L+1),,J(L+1)+ (19.0.9) 1=0 [i.e,i=0,L+1,J(L+1] I=L [i.e,i=L,L+1+L,,J(L+1)+

830 Chapter 19. Partial Differential Equations Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). In contrast to initial value problems, stability is relatively easy to achieve for boundary value problems. Thus, the efficiency of the algorithms, both in computational load and storage requirements, becomes the principal concern. Because all the conditions on a boundary value problem must be satisfied “simultaneously,” these problems usually boil down, at least conceptually, to the solution of large numbers of simultaneous algebraic equations. When such equations are nonlinear, they are usually solved by linearization and iteration; so without much loss of generality we can view the problem as being the solution of special, large linear sets of equations. As an example, one which we will refer to in §§19.4–19.6 as our “model problem,” let us consider the solution of equation (19.0.3) by the finite-difference method. We represent the function u(x, y) by its values at the discrete set of points xj = x0 + j∆, j = 0, 1, ..., J yl = y0 + l∆, l = 0, 1, ..., L (19.0.4) where ∆ is the grid spacing. From now on, we will write uj,l for u(xj , yl), and ρj,l for ρ(xj , yl). For (19.0.3) we substitute a finite-difference representation (see Figure 19.0.2), uj+1,l − 2uj,l + uj−1,l ∆2 + uj,l+1 − 2uj,l + uj,l−1 ∆2 = ρj,l (19.0.5) or equivalently uj+1,l + uj−1,l + uj,l+1 + uj,l−1 − 4uj,l = ∆2ρj,l (19.0.6) To write this system of linear equations in matrix form we need to make a vector out of u. Let us number the two dimensions of grid points in a single one-dimensional sequence by defining i ≡ j(L + 1) + l for j = 0, 1, ..., J, l = 0, 1, ..., L (19.0.7) In other words, i increases most rapidly along the columns representing y values. Equation (19.0.6) now becomes ui+L+1 + ui−(L+1) + ui+1 + ui−1 − 4ui = ∆2ρi (19.0.8) This equation holds only at the interior points j = 1, 2, ..., J − 1; l = 1, 2, ..., L − 1. The points where j = 0 j = J l = 0 l = L [i.e., i = 0, ..., L] [i.e., i = J(L + 1), ..., J(L + 1) + L] [i.e., i = 0, L + 1, ..., J(L + 1)] [i.e., i = L, L +1+ L, ..., J(L + 1) + L] (19.0.9)

19.0 Introduction 831 http://www.nr.com or call 1-800-872- Permission is read able files A (including this one) granted fori 1988-1992 by Cambridge -7423 (North America to any server computer,is tusers to make one paper from NUMERICAL RECIPES IN C: e University Press. THE only),or st st Programs Xo Copyright (C) Figure 19.0.2.Finite-difference representation of a second-order elliptic equation on a two-dimensional to dir grid.The second derivatives at the point A are evaluated using the points to which A is shown connected. The second derivatives at point B are evaluated using the connected points and also using"right-hand ART OF SCIENTIFIC COMPUTING(ISBN side"boundary information,shown schematically as. rectcustser are boundary points where either u or its derivative has been specified.If we pull v@cam all this "known"information over to the right-hand side of equation (19.0.8),then 1988-1992 by Numerical Recipes 10-521 the equation takes the form 43108 A·u=b (19.0.10) where A has the form shown in Figure 19.0.3.The matrix A is called"tridiagonal (outside with fringes."A general linear second-order elliptic equation North Software. 02u du 02u a(z,y) 2+b( +c(,) 7+d红，而 (19.0.11) 02u visit website machine +e(x,y) xOy +f(x,y)u=g(x,y) will lead to a matrix of similar structure except that the nonzero entries will not be constants. As a rough classification,there are three different approaches to the solution of equation (19.0.10),not all applicable in all cases:relaxation methods,"rapid" methods (e.g.,Fourier methods),and direct matrix methods

19.0 Introduction 831 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). yL ∆ y1 y0 x0 xJ x1 ... ∆ A B Figure 19.0.2. Finite-difference representation of a second-order elliptic equation on a two-dimensional grid. The second derivatives at the point A are evaluated using the points to which A is shown connected. The second derivatives at point B are evaluated using the connected points and also using “right-hand side” boundary information, shown schematically as ⊗. are boundary points where either u or its derivative has been specified. If we pull all this “known” information over to the right-hand side of equation (19.0.8), then the equation takes the form A · u = b (19.0.10) where A has the form shown in Figure 19.0.3. The matrix A is called “tridiagonal with fringes.” A general linear second-order elliptic equation a(x, y) ∂2u ∂x2 + b(x, y) ∂u ∂x + c(x, y) ∂2u ∂y2 + d(x, y) ∂u ∂y + e(x, y) ∂2u ∂x∂y + f(x, y)u = g(x, y) (19.0.11) will lead to a matrix of similar structure except that the nonzero entries will not be constants. As a rough classification, there are three different approaches to the solution of equation (19.0.10), not all applicable in all cases: relaxation methods, “rapid” methods (e.g., Fourier methods), and direct matrix methods