守恒定律用的数值法PDF电子书下载

1 Introduction 1

1.1 Conservation laws 1

1.2 Applications 2

1.3 Mathematical difficulties 8

1.4 Numerical difficulties 9

1.5 Some references 12

2 The Derivation of Conservation Laws 14

2.1 Integral and differential forms 14

2.2 Scalar equations 16

2.3 Diffusion 17

3 Scalar Conservation Laws 19

3.1 The linear advection equation 19

3.1.1 Domain of dependence 20

3.1.2 Nonsmooth data 21

3.2 Burgers' equation 23

3.3 Shock formation 25

3.4 Weak solutions 27

3.5 The Riemann Problem 28

3.6 Shock speed 31

3.7 Manipulating conservation laws 34

3.8 Entropy conditions 36

3.8.1 Entropy functions 37

4 Some Scalar Examples 41

4.1 Traffic flow 41

4.1.1 Characteristics and "sound speed" 44

4.2 Two phase flow 48

5 Some Nonlinear Systems 51

5.1 The Euler equations 51

5.1.1 Ideal gas 53

5.1.2 Entropy 54

5.2 Isentropic flow 55

5.3 Isothermal flow 56

5.4 The shallow water equations 56

6 Linear Hyperbolic Systems 58

6.1 Characteristic variables 58

6.2 Simple waves 60

6.3 The wave equation 60

6.4 Linearization of nonlinear systems 61

6.4.1 Sound waves 63

6.5 The Riemann Problem 64

6.5.1 The phase plane 67

7 Shocks and the Hugoniot Locus 70

7.1 The Hugoniot locus 70

7.2 Solution of the Riemann problem 73

7.2.1 Riemann problems with no solution 75

7.3 Genuine nonlinearity 75

7.4 The Lax entropy condition 76

7.5 Linear degeneracy 78

7.6 The Riemann problem 79

8 Rarefaction Waves and Integral Curves 81

8.1 Integral curves 81

8.2 Rarefaction waves 82

8.3 General solution of the Riemann problem 86

8.4 Shock collisions 88

9 The Riemann problem for the Euler equations 89

9.1 Contact discontinuities 89

9.2 Solution to the Riemann problem 91

Ⅱ Numerical Methods 95

10 Numerical Methods for Linear Equations 97

10.1 The global error and convergence 102

10.2 Norms 103

10.3 Local truncation error 104

10.4 Stability 106

10.5 The Lax Equivalence Theorem 107

10.6 The CFL condition 110

10.7 Upwind methods 112

11 Computing Discontinuous Solutions 114

11.1 Modified equations 117

11.1.1 First order methods and diffusion 118

11.1.2 Second order methods and dispersion 119

11.2 Accuracy 121

12 Conservative Methods for Nonlinear Problems 122

12.1 Conservative methods 124

12.2 Consistency 126

12.3 Discrete conservation 128

12.4 The Lax-Wendroff Theorem 129

12.5 The entropy condition 133

13 Godunov's Method 136

13.1 The Courant-Isaacson-Rees method 137

13.2 Godunov's method 138

13.3 Linear systems 140

13.4 The entropy condition 142

13.5 Scalar conservation laws 143

14 Approximate Riemann Solvers 146

14.1 General theory 147

14.1.1 The entropy condition 148

14.1.2 Modified conservation laws 149

14.2 Roe's approximate Riemann solver 149

14.2.1 The numerical flux function for Roe's solver 150

14.2.2 A sonic entropy fix 151

14.2.3 The scalar case 153

14.2.4 A Roe matrix for isothermal flow 156

15 Nonlinear Stability 158

15.1 Convergence notions 158

15.2 Compactness 159

15.3 Total variation stability 162

15.4 Total variation diminishing methods 165

15.5 Monotonicity preserving methods 165

15.6 l1-contracting numerical methods 166

15.7 Monotone methods 169

16 High Resolution Methods 173

16.1 Artificial Viscosity 173

16.2 Flux-limiter methods 176

16.2.1 Linear systems 182

16.3 Slope-limiter methods 183

16.3.1 Linear Systems 187

16.3.2 Nonlinear scalar equations 188

16.3.3 Nonlinear Systems 191

17 Semi-discrete Methods 193

17.1 Evolution equations for the cell averages 193

17.2 Spatial accuracy 195

17.3 Reconstruction by primitive functions 196

17.4 ENO schemes 198

18 Multidimensional Problems 200

18.1 Semi-discrete methods 201

18.2 Splitting methods 202

18.3 TVD Methods 206

18.4 Multidimensional approaches 206

Bibliography 208