Heath - Scientific Computing (523150), страница 87
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If the coefficients arevariable, then the type of the equation can vary from one region to another, and if there325326CHAPTER 11. PARTIAL DIFFERENTIAL EQUATIONSis more than one equation in a system, each equation can be of a different type. And ofcourse, the problem may be nonlinear or of higher order or dimension. Nevertheless, theseterms are often used to describe PDEs even when the meaning is not so precise. Roughlyspeaking,• Hyperbolic PDEs describe time-dependent physical processes, such as wave motion, thatare not evolving toward a steady state.• Parabolic PDEs describe time-dependent physical processes, such as the diffusion of heat,that are evolving toward a steady state.• Elliptic PDEs describe processes that have already reached a steady state, or equilibrium,and hence are time-independent.11.2Time-Dependent ProblemsTime-dependent PDEs usually involve both initial values and boundary values.
For example, the region in which the solution is desired, as well as the initial and boundary conditionsthat must be specified, are shown for a problem with one space dimension in Fig. 11.1. Twoof the most commonly occurring examples of time-dependent PDEs are the heat equation,which is parabolic, and the wave equation, which is hyperbolic.t....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................boundaryvaluesboundaryvaluesproblem domainainitial valuesbxFigure 11.1: An initial-boundary value problem for a time-dependent PDE in one space dimension.In one space dimension, the heat equation has the formut = cuxx ,0 ≤ x ≤ L,t ≥ 0,with given initial and boundary conditionsu(0, x) = f (x),u(t, 0) = α,u(t, L) = β,and c a positive constant.
This equation models, for example, the diffusion of heat in a barof length L whose ends are maintained at temperatures given by the boundary conditionsand whose initial temperature distribution is given by the function f (x). The constant c,which governs the rate of diffusion, depends on physical properties of the material, such as11.2. TIME-DEPENDENT PROBLEMS327its thermal conductivity, specific heat, and density. The solution u to this equation givesthe subsequent temperature distribution as a function of both space and time.In one space dimension, the wave equation has the formutt = cuxx ,0 ≤ x ≤ L,t ≥ 0,with given initial conditionsu(0, x) = f (x),ut (0, x) = g(x),and boundary conditionsu(t, 0) = α,u(t, L) = β,and c a positive constant.
This equation models, for example, the vibrations of a violinstring of length L whose initial profile and velocity are given by the functions f (x) and g(x),respectively, and whose ends are anchored as given by the boundary conditions. Becauseit is second-order in time, this equation requires initial conditions for both the solutionfunction and its first derivative with respect to time. It turns out that the solution consists√of waves propagating to the left or right with speed c. More generally, this equationdescribes many types of wave motion, such as the propagation of sound waves in the air orwater waves in the ocean.For both the heat equation and wave equation, we have given only the simplest typeof boundary conditions.
More complicated boundary conditions may involve derivatives ofthe solution as well as its values, or combinations of these, or may require that the solutionbe periodic, for example.Problems with more space dimensions incur greater computational requirements, both instorage and execution time, but do not introduce significant additional conceptual difficulty,so we will focus on time-dependent problems having a single space dimension. We will alsofocus on relatively simple model problems, such as the heat and wave equations, ratherthan attempt a broader treatment of partial differential equations in general.
Nevertheless,these model problems illustrate most of the important issues in the numerical solution ofPDEs.11.2.1Semidiscrete Methods Using Finite DifferencesOne way to solve a time-dependent PDE is to discretize in space but leave the time variablecontinuous. This approach results in a system of ODEs, which can then be solved by themethods discussed in Chapter 9.
For example, consider the heat equationut = cuxx ,0 ≤ x ≤ 1,t ≥ 0,u(t, 0) = 0,u(t, 1) = 0.with initial and boundary conditionsu(0, x) = f (x),If we replace the derivative uxx with the finite difference approximationuxx ≈u(t, x + ∆x) − 2u(t, x) + u(t, x − ∆x),(∆x)2328CHAPTER 11. PARTIAL DIFFERENTIAL EQUATIONSwhere ∆x = 1/(n + 1), then we get a system of n ODEsyi0 (t) =c[yi+1 (t) − 2yi (t) + yi−1 (t)],(∆x)2i = 1, . . . , n,where yi (t) ≈ u(t, i∆x). From the boundary conditions, y0 (t) and yn+1 (t) are identicallyzero, and from the initial conditions, yi (0) = f (xi ), i = 1, . .
. , n. We can therefore use anODE method to solve the initial value problem for this system. This approach is called themethod of lines. If we think of the solution u(t, x) as a surface over the space-time plane,this method computes cross sections of that surface along a series of lines, each of which isparallel to the time axis and corresponds to one of the discrete spatial mesh points.The foregoing semidiscrete system can be written in matrix form ascy0 =(∆x)2−2101−210...1...···−2...00··· 0.. ....
... 0 y = Ay.... 1 1 −2The Jacobian matrix A of this system has eigenvalues between −4c/(∆x)2 and 0, whichmakes the ODE very stiff as the spatial mesh size ∆x becomes small. This stiffness, which istypical of ODEs derived from PDEs in this manner, must be taken into account in choosingan ODE method for solving the semidiscrete system (recall Section 9.5).11.2.2Semidiscrete Methods Using Finite ElementsSpatial discretization to convert a PDE into a system of ODEs can also be done by a finiteelement approach. As we did for two-point boundary problems for ODEs, we approximatethe solution by a linear combination of basis functions, except that now the coefficients aretime dependent. Thus, we seek an approximate solution of the formu(t, x) ≈nXαj (t)φj (x),j=1where the φj (x) are the basis function over the spatial domain and the αj (t) are the timedependent coefficients.
If we use collocation (we could also use Ritz or Galerkin methods),then we substitute this approximation into the PDE and require that the equation besatisfied exactly at a discrete set of points xi . For the heat equation, for example, thisyields a system of ODEsnXj=1αj0 (t)φj (xi )=cnXαj (t)φ00j (xi ),i = 1, .
. . , n,j=1whose solution is the set of coefficient functions αj (t) that determine the approximatesolution to the PDE.11.2. TIME-DEPENDENT PROBLEMS329The implicit form of the foregoing system of ODEs is not the explicit form required bystandard ODE methods, so we define the n × n matrices A and B byaij = φj (xi ),bij = φ00j (xi ).Assuming the matrix A is nonsingular, we then obtain the system of ODEsα0 (t) = cA−1 Bα(t),which is in a form suitable for solution with standard ODE software (as usual, the matrixA need not be inverted explicitly, but merely used to solve linear systems). We still needan initial condition for the ODE, however, which we can obtain by requiring that thesolution satisfy the given initial condition for the PDE at the points xi . Again, the matricesinvolved in this method will be sparse if the basis functions are “local,” such as B-splines.Alternatively, we could use eigenfunctions of the differential operator (e.g., trigonometricfunctions for uxx ) as basis functions, which would give a spectral method, or other basisfunctions, such as Legendre or Chebyshev polynomials, which would give a pseudospectralmethod.Unlike the finite difference method, the finite element method does not produce approximate values of the solution u directly, but rather it generates a representation of theapproximate solution as a linear combination of basis functions.
The basis functions depend only on the spatial variable, but the coefficients of the linear combination (given bythe solution to the system of ODEs) are time dependent. Thus, for any given time t, thecorresponding linear combination of basis functions generates a cross section of the solutionsurface parallel to the spatial axis.As with finite difference methods, systems of ODEs arising from semidiscretization of aPDE by finite elements tend to be stiff, which should be taken into account in choosing anODE method for solving them.11.2.3Fully Discrete MethodsFully discrete methods for PDEs discretize in both time and space dimensions. In a fullydiscrete finite difference method, we replace the continuous domain of the equation by adiscrete mesh of points, we replace the derivatives in the PDE by finite difference approximations, and we seek a numerical solution that is a table of approximate values at theselected points in space and time. In two dimensions (one space and one time), the resulting approximate solution values represent points on the solution surface over the problemdomain in the space-time plane.
The accuracy of the approximate solution depends on thestepsizes in both space and time.Replacement of all partial derivatives by finite differences results in a system of algebraicequations for the unknown solution at the discrete set of sample points. This system maybe linear or nonlinear, depending on the underlying PDE. With an initial-value problem,the solution is obtained by beginning with the initial values along some boundary of theproblem domain and marching forward in time step by step, generating successive rows inthe solution table. Such a time-stepping procedure may be explicit or implicit, dependingon whether the formula for the solution values at the next time step involves only pastinformation.330CHAPTER 11.
PARTIAL DIFFERENTIAL EQUATIONSWe would expect to obtain arbitrarily good accuracy by taking sufficiently small stepsizesin time and space. The two stepsizes cannot always be chosen independently of each other,however. For the approximate solution to converge to the true solution of the PDE as thestepsizes in time and space go to zero, two conditions must be met:• Consistency, which means that the local truncation error goes to zero as the stepsizes goto zero (i.e., the discrete problem approximates the right continuous problem).• Stability, which, as we have seen in other contexts, essentially means that the approximatesolution remains bounded.
More specifically, the global error is bounded by a constanttimes the local error.The Lax Equivalence Theorem says that consistency and stability are together necessary andsufficient for convergence. Neither condition alone is sufficient to guarantee convergence.Example 11.1 Heat Equation. As an example of full discretization, consider the heatequationut = cuxx , 0 ≤ x ≤ 1, t ≥ 0,with initial and boundary conditionsu(0, x) = f (x),u(t, 0) = α,u(t, 1) = β.We let ukj denote the approximate solution at xj = j∆x and tk = k∆t.
If we replace ut bya forward difference in time and uxx by a centered difference in space, with ∆x = 1/(n + 1),we get the schemeuk+1− ukjj∆toruk+1= ukj + cj=cukj+1 − 2ukj + ukj−1,(∆x)2j = 1, . . . , n,∆t(uk − 2ukj + ukj−1 ),(∆x)2 j+1j = 1, . .
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