Fundamentals of Computational Fluid Dynamics / Edition 1

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Overview

"This book is intended as a textbook for a first course in computational fluid dynamics and will be of interest to researchers and practitioners as well. It emphasizes fundamental concepts in developing, analyzing, and understanding numerical methods for the partial differential equations governing the physics of fluid flow. The linear convection and diffusion equations are used to illustrate concepts throughout. The chosen approach, in which the partial differential equations are reduced to ordinary differential equations, and finally to difference equations, gives the book its distinctiveness and provides a sound basis for a deep understanding of the fundamental concepts in computational fluid dynamics."--BOOK JACKET.
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Editorial Reviews

From the Publisher
From the reviews of the first edition:

"[The book] is much needed to fill a gap in the market for texts that try to cover some of the fundamental mathematical aspects of the subject. The book is aimed at graduate students and concentrates an analysing the properties of approximations produced by finite-difference and finite-volume methods. The emphasis throughout the book is on the study of the properties of the techniques as applied to model equations such as the linear convection and diffusion equations. [...] The main strengths of the book are that the theoretical aspects are treated in an elegant and simple manner, making it easy for the reader to appreciate the subtle links between the discrete and continuous operators and linear algebra. The mathematics is self-contained and not daunting. Most of the sections are well written and the section on ordinary differential equations and time marching methods is particularly good." (Contemporary Physics, 44/4, 2003)

"An introduction to finite volume methods for initial-boundary value problems for partial differential equations, developed with applications in CFD in mind. [...] The student who has mastered this material will be well equipped for further study and use of numerical methods in the computational disciplines, where one’s only guide is often analogy with simple cases. [...] I found the book pleasant to read, and good for students. The level is that of a course for students studying for a Masters degree in their final year. Teachers of similar courses will find the book useful. A good collection of exercises is included." (P. Wesseling, Structural and Multidisciplinary Optimization, 24/6, 2003)

"The book is well written and well organized. It can be easily adopted as a textbook for senior or graduate students studying numerical methods of fluid mechanics. Practice exercises are provided at the end of each chapter, some of them expecting the reader to write his own computer codes. This reviewer would regard Fundamentals of Computational Fluid Dynamics as essential to anyone planning to use CFD modeling." (T.A. Kowalewski, Applied Mechanics Review, 55/4, 2002)

"[...] emphasizes fundamental concepts in developing, analyzing, and understanding numerical methods for the partial differential equations governing the physics of fluid flow. [...] The chosen approach, in which the partial differential equations are reduced to ordinary differential equations, and finally to difference equations, gives the book its distinctiveness and provides a sound basis for a deep understanding of the fundamental concepts in computational fluid dynamics." (CFD Online, Oct. 21. 2002)

"The unaffected style adopted by the authors makes the book very readable and brings a surprising degree of freshness to the mature concepts that are its emphasis. For this reason, in addition to graduate students, the book may appeal to professionals who do not have formal training in CFD but who wish to learn more theory than is found in cookbook-oriented code manuals... [There is a] sharp focus on ideas and analysis rather than tips and techniques ..." (Datta V. Gaitonde, U.S. Air Force Research Laboratory, in the American Institute for Aeronautics and Astronautics Journal)

"... the book covers a good set of introductory material and includes some topics and insights not found in other books at this level, along with numerous exercises. In the hands of a knowledgeable instructor, it could form the basis for an excellent course and would be a useful supplement in general." (Randall J. LeVeque, University of Washington, in the SIAM Review)

"The objective of this book is to provide an elementary but exhaustive tutorial presentation on computational fluid dynamics (CFD), emphasizing the fundamental principles and surveying a variety of algorithms whose applications range from laminar incompressible flow to hypersonic turbulent flow. The book is aimed at students, who have had little or no experience in this field, both recent graduates as well as researchers … . In conclusion, it must be mentioned that this textbook includes superb exercises to close each chapter." (Andrzej Icha, Pure and Applied Geophysics, Vol. 161. 2004)

"The book contains a broad variety of numerical methods for partial differential equations arising in fluid dynamics. After a brief introduction of conservation laws, the authors discuss various finite volume and finite difference methods. Thereby, also higher-order schemes are introduced, and fundamental concepts of time marching methods are presented. The book is written as a textbook and provides valuable information for students, researchers and practitioners in science and engineering." (A. Meister, Zentralblatt MATH, Vol. 970, 2001)

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Product Details

  • ISBN-13: 9783540416074
  • Publisher: Springer Berlin Heidelberg
  • Publication date: 3/22/2004
  • Series: Scientific Computation Series
  • Edition description: 1st ed. 2001 Corr. 2nd printing 2003
  • Edition number: 1
  • Pages: 249
  • Product dimensions: 0.69 (w) x 6.14 (h) x 9.21 (d)

Table of Contents

1 Introduction 1
2 Conservation Laws and the Model Equations 7
2.1 Conservation Laws 7
2.2 The Navier-Stokes and Euler Equations 8
2.3 The Linear Convection Equation 11
2.4 The Diffusion Equation 13
2.5 Linear Hyperbolic Systems 15
3 Finite-Difference Approximations 19
3.1 Meshes and Finite-Difference Notation 19
3.2 Space Derivative Approximations 21
3.3 Finite-Difference Operators 22
3.4 Constructing Differencing Schemes of Any Order 28
3.5 Fourier Error Analysis 37
3.6 Difference Operators at Boundaries 41
4 The Semi-Discrete Approach 49
4.1 Reduction of PDE's to ODE's 50
4.2 Exact Solutions of Linear ODE's 51
4.3 Real Space and Eigenspace 58
4.4 The Representative Equation 64
5 Finite-Volume Methods 67
5.1 Basic Concepts 67
5.2 Model Equations in Integral Form 69
5.3 One-dimensional Examples 70
5.4 A Two-dimensional Example 76
6 Time-Marching Methods for ODE'S 81
6.1 Notation 82
6.2 Converting Time-Marching Methods to O[Delta]E's 83
6.3 Solution of Linear O[Delta]E's with Constant Coefficients 84
6.4 Solution of the Representative O[Delta]E's 87
6.5 The [lambda]-[sigma] Relation 89
6.6 Accuracy Measures of Time-Marching Methods 92
6.7 Linear Multistep Methods 96
6.8 Predictor-Corrector Methods 101
6.9 Runge-Kutta Methods 103
6.10 Implementation of Implicit Methods 105
7 Stability of Linear Systems 115
7.1 Dependence on the Eigensystem 115
7.2 Inherent Stability of ODE's 116
7.3 Numerical Stability of O[Delta]E's 118
7.4 Time-Space Stability and Convergence of O[Delta]E's 119
7.5 Numerical Stability Concepts in the Complex [sigma]-Plane 121
7.6 Numerical Stability Concepts in the Complex [lambda]h Plane 127
7.7 Fourier Stability Analysis 133
7.8 Consistency 135
8 Choosing a Time-Marching Method 141
8.1 Stiffness Definition for ODE's 141
8.2 Relation of Stiffness to Space Mesh Size 143
8.3 Practical Considerations for Comparing Methods 144
8.4 Comparing the Efficiency of Explicit Methods 145
8.5 Coping With Stiffness 149
8.6 Steady Problems 151
9 Relaxation Methods 153
9.1 Formulation of the Model Problem 154
9.2 Classical Relaxation 157
9.3 The ODE Approach to Classical Relaxation 159
9.4 Eigensystems of the Classical Methods 162
9.5 Nonstationary Processes 171
10 Multigrid 177
10.1 Motivation 177
10.2 The Basic Process 178
10.3 A Two-Grid Process 185
11 Numerical Dissipation 189
11.1 One-sided First-Derivative Space Differencing 189
11.2 The Modified Partial Differential Equation 190
11.3 The Lax-Wendroff Method 192
11.4 Upwind Schemes 195
11.5 Artificial Dissipation 199
12 Split and Factored Forms 203
12.1 The Concept 203
12.2 Factoring Physical Representations - Time Splitting 204
12.3 Factoring Space Matrix Operators in 2D 206
12.4 Second-Order Factored Implicit Methods 211
12.5 Importance of Factored Forms in Two and Three Dimensions 212
12.6 The Delta Form 213
13 Analysis of Split and Factored Forms 217
13.1 The Representative Equation for Circulant Operators 217
13.2 Example Analysis of Circulant Systems 218
13.3 The Representative Equation for Space-Split Operators 222
13.4 Example Analysis of the 2D Model Equation 225
13.5 Example Analysis of the 3D Model Equation 228
App. A Useful Relations from Linear Algebra 231
App. B Some Properties of Tridiagonal Matrices 237
App. C The Homogeneous Property of the Euler Equations 245
Index 247
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