《ComputationaI Aerodynamics and Fluid Dynamics：An Introduction》

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《ComputationaI Aerodynamics and Fluid Dynamics：An Introduction》
计算空气动力学和流体动力学导论
作者：Professor Jean-Jacques Chattot
University of California
Department of Mechanical
and Aeronautical Engineering
出版社：Springer
出版时间：2002年

《ComputationaI Aerodynamics and Fluid Dynamics：An Introduction》

《ComputationaI Aerodynamics and Fluid Dynamics：An Introduction》

《ComputationaI Aerodynamics and Fluid Dynamics：An Introduction》

《ComputationaI Aerodynamics and Fluid Dynamics：An Introduction》

目录
1. Introduction.............................................. 1
1.1 Motivation............................................ 1
1.2 Content............................................... 1
2. Basics of the Finite-Difference Method..... . . ... . . .... ... 5
2.1 Representation of a Function by Discrete Values. . . . . . . . . . . . 5
2.2 Representation of a First Derivative ...................... 6
2.3 Representation of a Second Derivative. . . . . . . . . . . . . . . . . . . . . 7
2.4 Geometric Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 Taylor Expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.6 Consistency and Accuracy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.7 Stability............................................... 12
2.8 Complements on Truncation Error. . . . . . . . . . . . . . . . . . . . . . .. 14
3. Application to the Integration
of Ordinary Differential Equations. . . . . . . . . . . . . . . . . . . . . . .. 19
3.1 Introduction........................................... 19
3.2 The Euler-Cauchy Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20
3.3 Improved Euler Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 21
3.4 The Runge-Kutta Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22
3.5 Integration of Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22
3.6 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23
4. Partial Differential Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27
4.1 Introduction........................................... 27
4.2 General Classification
and Notion of Characteristic Surface. . . . . . . . . . . . . . . . . . . . .. 28
4.3 Model Equations and Types ............................. 30
4.3.1 Linear Convection Equation. . . . . . . . . . . . . . . . . . . . . .. 30
4.3.2 The Wave Equation.. . . . .... . .... . . ... . . ... . . . ... 31
4.3.3 Laplace's Equation ............................... 32
4.3.4 The Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 33
4.3.5 Burgers' Equation (Inviscid) . . . . . . . . . . . . . . . . . . . . . .. 34
4.3.6 Other Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35
X Table of Contents
4.4 Conservation Laws and Jumps for a System of PDEs. . . . . . .. 35
4.4.1 Jump Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36
4.4.2 Examples....................................... 37
5. Integration of a Linear Hyperbolic Equation. . . . . . . . . . . . .. 41
5.1 Introduction........................................... 41
5.2 The Linear Convection Equation. . . . . . . . . . . . . . . . . . . . . . . .. 41
5.2.1 A Centered Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 41
5.2.2 An Upwind Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 42
5.2.3 The Lax Scheme ................................. 43
5.2.4 The Lax-Wendroff Scheme. . . . . . . . . . . . . . . . . . . . . . . .. 43
5.2.5 The MacCormack Scheme... .. . . ... . . . . .. . . . . . .... 44
5.3 The Wave Equation... . . ... . . . .... . . . . .... . . .... . . . ..... 45
5.3.1 Exact Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 46
5.3.2 Numerical Scheme, Consistency and Accuracy ....... 47
5.3.3 Numerical Implementation. . . . . . . . . . . . . . . . . . . . . . . .. 49
5.3.4 Stability........................................ 49
5.4 Implicit Scheme for the Wave Equation ................... 52
6. Integration of a Linear Parabolic Equation ............... 53
6.1 Introduction........................................... 53
6.2 Exact Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 54
6.3 A Simple Explicit Scheme ............................... 55
6.3.1 Consistency and Accuracy. . . . . . . . . . . . . . . . . . . . . . . .. 55
6.3.2 Numerical Implementation. . . . . . . . . . . . . . . . . . . . . . . .. 56
6.3.3 Stability........................................ 57
6.4 Simple Implicit Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58
6.5 Combined Method A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 59
6.6 Solution of a Linear System with Tridiagonal Matrix. . . . . . .. 59
7. Integration of a Linear Elliptic Equation. . . . . . . . . . . . . . . . .. 63
7.1 Introduction........................................... 63
7.2 Numerical Scheme, Consistency, Accuracy. . . . . . . . . . . . . . . .. 63
7.3 Matrix Formulation; Direct Solution. . . . . . . . . . . . . . . . . . . . .. 64
7.4 Outlook of Iterative Methods. . . . . . . . . . . . . . . . . . . . . . . . . . .. 65
7.4.1 The Method of Jacobi ............................ 67
7.4.2 The Gauss-Seidel Method. . . . . . . . . . . . . . . . . . . . . . . .. 68
7.4.3 The Successive Over-Relaxation Method (SOR) ...... 69
7.5 Other Iterative Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 71
7.5.1 The SLOR Method. . . . . ... . . . ... . . . .... . . . .... . .. 72
7.5.2 AD! Methods. . ... . . . .... . ..... . . . .... . . ..... . . .. 74
Table of Contents XI
8. Finite Difference Scheme
for a Convection-Diffusion Equation ...... . . . . . . . . . . . . . . .. 75
8.1 Introduction........................................... 75
8.2 FTCS Method .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 76
8.3 The Box and Modified Box Methods. . . . . . . . . . . . . . . . . . . . .. 79
8.4 A Mixed-Type Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 80
9. The Method of Murman and Cole ........................ 81
9.1 Introduction........................................... 81
9.2 The Model Problem .................................... 81
9.3 The Murman-Cole Scheme (1970) ..... . . . . . . . . . . . . . . . . . .. 85
9.4 The Four-Operator Scheme of Murman (1973) ............. 87
10. Treatment of Non-Linearities . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 93
10.1 Introduction....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 93
10.2 An Explicit Mixed-Type Scheme ......................... 93
10.3 An Implicit Mixed-Type Scheme ......................... 96
10.4 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 99
11. Application to a System of Equations ..................... 101
11.1 Introduction ........................................... 101
11.2 The Equations of Gas Dynamics .......................... 101
11.3 Jump Conditions ....................................... 103
11.4 The Riemann Problem .................................. 104
11.5 A Box-Scheme for the Equations of Gas Dynamics .......... 105
11.6 Some Results .......................................... 108
11. 7 The Bigger Picture ..................................... 111
Appendix ..................................................... 113
A. Problems ................................................ 113
B. Solutions to Problems ..................................... 137
References .................................................... 181
Index ......................................................... 183

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