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A Publication of Engineering Education System TM, Wichita, Kansas, 67208-1078, USA

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STEVE T. CHIANG

KLAUS A. HOFFMANN

COMPUTATIONAL FLUID DYNAMICSVOLUME I

Fourth Edition

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ISBN 0-9623731-0-9First Print: August 2000

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Copyright ©2000, 1998, 1993, 1989 by Engineering Education System. All rightsreserved. No part of this publication may be reproduced or distributed in anyform or by any means, mechanical or electronic, including photocopying, recording,storage or retrieval system, without prior written permission from the publisher.

2.1 Introductory Remarks 292.2 Taylor Series Expansion 292.3 Finite Differenceby Polynomials 352.4 Finite DifferenceEquations 37

Chapter Two:Finite DifferenceFormulations 29

1.1 Introductory Remarks 31.2 Linear and Nonlinear Partial Differential Equations 31.3 Second-Order Partial Differential Equations 41.4 Elliptic Equations 6

1.5 Parabolic Equations 61.6 Hyperbolic Equations 81.7 Model Equations 101.8 System of First-Order Partial Differential Equations 111.9 System of Second-Order Partial Differential Equations 161.10 Initial and Boundary Conditions 201.11 Remarks and Definitions 22

1.12 Summary Objectives 241.13 Problems 25

Chapter One:Classification of Partial Differential Equations 3

PrefaceIntrod uction 1

CONTENTS

3.1 Introductory Remarks 603.2 Finite Difference Formulations 603.3 Explicit Methods 64

3.3.1 The Forward Time/Central Space Method 643.3.2 The Richardson Method 643.3.3 The DuFort-Frankel Method 64

3.4 Implicit Methods 653.4.1 The Laasonen Method 663.4.2 The Crank-Nicolson Method 663.4.3 The Beta Formulation 67

3.5 Applications 673.6 Analysis 723.7 Parabolic Equations in Two-Space Dimensions 763.8 Approximate Factorization 853.9 Fractional Step Methods 873.10 Extension to Three-Space Dimensions 873.11 Consistency Analysis of Finite DifferenceEquations 883.12 Linearization 903.13 Irregular Boundaries 923.14 Summary Objectives 943.15 Problems 96

60Chapter Three:Parabolic Partial Differential Equations

525355

405151

2.5 Applications2.6 Finite DifferenceApproximation of Mixed Partial Derivatives

2.6.1 Taylor Series Expansion2.6.2 The Use of Partial Derivatives with Respect to

One Independent Variable2.7 Summary Objectives2.8 Problems

Contents11

Chapter Five:Elliptic Equations 152

5.1 Introductory Remarks 1525.2 Finite Difference Formulations 1525.3 Solution Algorithms 156

5.3.1 The Jacobi Iteration Method 1575.3.2 The Point Gauss-Seidel Iteration Method 1605.3.3 The Line Gauss-Seidel Iteration Method 1625.3.4 Point SuccessiveOver-Relaxation Method (PSOR) 1645.3.5 Line SuccessiveOver-Relaxation Method (LSOR) 1655.3.6 The Alternating Direction Implicit Method (AD!) 165

5.4 Applications 1675.5 Summary Objectives 1745.6 Problems 175

Chapter Four:Stability Analysis 113

4.1 Introductory Remarks 1134.2 Discrete Perturbation Stability Analysis 1144.3 Von Neumann Stability Analysis 1244.4 Multidimensional Problems 1374.5 Error Analysis 1414.6 Modified Equation 1434.7 Artificial Viscosity 1464.8 Summary Objectives 1484.9 Problems 149

iiiContents

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