FINITE-DIFFERENCE METHODS PARTIAL DIFFERENTIAL EQUATIONS

Transcription

1 FINITE-DIFFERENCE METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS GEORGE E. FORSYTHE PROFESSOR OP MATHEMATICS STANFORD UNIVERSITY WOLFGANG R. WASOW PROFESSOR OF MATHEMATICS UNIVERSITY OF WISCONSIN Techni&che Univer&stat Darmstadt FACHBEREICH INFORMATIK B 1 BL 1 Inventar-Nr.: Sachgebtete: Standort fas 0 j T H 1 $ t LJJ K JOHN WILEY & SONS, INC. New York London Sydney

2 CONTENTS INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS AND COMPUTERS 1. Remarks on the Classification of Partial Differential Equations 1 2. Systems and Single Equations 6 3. Properties of Digital Computing Systems Desk computation Punched-card computers Automatic digital computers Demands of partial differential equations 11 1 HYPERBOLIC EQUATIONS IN TWO INDEPENDENT VARIABLES 4. A Finite-Difference Approximation to the Equation Uu u zx = Solution of the simplest initial-value problem for uu u X x = An approximating difference equation 4.3. Explicit solution of the difference equation for X < Solution of the difference equation by a finite Fourier series 4.5. The convergence to the solution of the differential problem 4.6. Stability 5. Further Aspects of the Concept of Stability 5.1. Definitions and simple examples 5.2. Application to the wave equation 6. Systems of Hyperbolic Differential Equations and Their Characteristics 6.1. The normal form 6.2. Examples 6.3. The canonical differential system for» = Remarks on the initial-value problem 7. Finite-Difference Methods for Systems of Quasilinear Hyperbolic Equations 7.1. Description of the procedure 7.2. A general scheme for proving the convergence of difference approximations 7.3. The convergence of the difference scheme for hyperbolic systems 7.4. Differences in a curvilinear net 7.5. The round-off errors 8. Integration Along Characteristics 8.1. The method of Massau 8.2. Quasilinear equations of order two 8.3. Another integration method for n depandent variables 9. Integration by Adams' Method vii

3 viii CONTENTS 10. Shock Waves The concept of shock waves Numerical solution of problems involving shock waves Calculation of shock fronts by means of simulated viscosity terms Integration of the true equations of viscous flow The difference method of Lax 84 2 PARABOLIC EQUATIONS 11. The Simplest Heat Flow Problem Preliminary remarks Solution of the initial-value problem The Simplest Finite-Difference Approximation The stability condition The convergence and the discretization error Linear Problems in a Finite Interval Differential problems A finite-difference approximation An implicit method The solution of the implicit difference equation The convergence of the implicit method More General Lineai>Parabolic Problems in Two Variables: Explicit Methods Formal explicit difference approximations Solution of nonhomogeneous linear difference problems by superposition Boundedness and stability properties of difference expressions of positive type The boundedness condition of John Further Explicit and Implicit Methods for Linear Problems A more general approach to implicit methods Explicit methods using more than two grid lines Problems of higher order Other Definitions of Convergence. The Theory of Lax and Richtmyer Remarks on functional analysis Convergence and stability in the sense of Lax and Richtmyer Nonlinear Problems Semilinear equations Examples of other parabolic problems ELLIPTIC EQUATIONS 18. Some Numerical Problems Involving Elliptic Partial Differential Equations General Laplacian boundary-value problem A water drainage problem An oil-flow problem 150

4 CONTENTS IX A stress problem A boundary-layer problem A membrane eigenvalue problem A simple reactor problem A biharmonic eigenvalue problem Plateau's problem Eigenvalue problem for the wave equation Selected Results from the Theory of Elliptic Partial Differential Equations Variational formulations Variational formulation of certain eigenvalue problems Self-adjointness Interface conditions Maximum principle Formulating Elliptic Difference Equation Problems Discretization and problems raised by it The method of lines Types of problems to be discretized Irregular nets Variational method of setting up difference equations Square nets: approximating the derivatives Square nets: approximating L(u) and AM Application of the variational method to a reactor diffusion equation Treatment of Dirichlet boundary conditions Normal derivative boundary conditions Singularities and free boundaries Classical Theory of Solving Elliptic Difference Equations The difference equations as a matrix equation Elimination methods Iterative methods Method of simultaneous displacements; gradient method Richardson's method Method of successive displacements Gauss-Southwell relaxation Explicit and Implicit Overrelaxation Methods The Young-Frankel theory of successive overrelaxation Overrelaxation without property (A) Implicit methods: overrelaxation by lines Implicit alternating-direction methods Summary of rates of convergence for a square Discretization and Round-Off Errors The method of Gerschgorin An integral equation with a Stieltjes kernel An appraisal of the solution of the integral equation Appraisal of the discretization error Summary of some further results concerning discretization errors for linear Dirichlet problems Green's function for discrete Dirichlet problems Discretization error for the Neumann and third boundary-value problems 318

5 x CONTENTS Round-off error in solving the Dirichlet difference problem Probabilistic estimate of round-off error The Membrane Eigenvalue Problem Introduction Upper bounds by difference methods A standard L-shaped membrane Lower bounds from difference equations: Weinberger's method Asymptotic lower bounds from difference equations Proof of Theorem Experiments with L-shaped membrane Numerical solution of the finite eigenvalue problem Solving Elliptic Partial Difference Equations on an Automatic Digital Computer Obtaining the equations in a digital computer Obtaining the difference equations when C is curved Plans for an integrated industrial program Use of graded nets Successive overrelaxation: estimating OJ Successive overrelaxation: time required Other methods for solving difference equations Solving eigenvalue problems on a computer Solving the Neumann problem on a computer INITIAL-VALUE PROBLEMS IN MORE THAN TWO INDEPENDENT VARIABLES 26. The Equation of Wave Propagation The differential equation The simplest difference approximation Characteristics in Several Dimensions A Meteorological Forecast Problem Forecasting directly from the primitive equations Modified approaches to forecasting One-dimensional model Two-dimensional model "Upwind" difference equations Three space dimensions A General Discussion of the Fourier Method for Difference and Differential Equations The problem Explicit solution by infinite Fourier series ^ Convergence of U(x, t) to u(x, t) Stability How to test for stability and convergence The Method of Peaceman and Rachford General formulation Application to the equation of heat flow in two dimensions 412 Bibliography and Author Index 415 Subject Index. 433