Godunov-type Schemes (eBook)

Front Cover 1
Godunov-type schemes: an introduction for engineers 4
Copyright Page 5
Preface 6
Acknowledgements 10
Contents 12
Preface 6
Acknowledgements 10
Notation 18
VariabIes 18
Operators 24
Subscripts and superscripts 24
Others 25
Chapter 1. Scalar conservation laws 26
1.1 Definitions and basic notions 26
1.2 The Riemann problem 43
1.3 A linear conservation law: the advection equation 50
1.4 A convex conservation law: the Burgers equation 54
1.5 A concave conservation law: the LWR model 57
1.6 A non-convex conservation law: the Buckley-Leverett equation 63
1.7 Extension to multiple dimensions 71
Chapter 2. Hyperbolic systems of conservation laws 76
2.1 Definitions 77
2.2 A linear system: the water hammer equations 87
2.3 Two-phase flow in pipes 93
2.4 A 2x2 model for traffic flow 98
2.5 The open channel flow equations with solute transport 107
2.6 The shallow water equations in two dimensions 114
Chapter 3. An outline of Godunov-type schemes 118
3.1 The six steps of Godunov-type algorithms 119
3.2 Lagrangian schemes 128
3.3 Multidimensional problems 130
3.4 Stability constraints 139
Chapter 4. The Godunov method for scalar laws in one dimension 142
4.1 The linear advection equation 143
4.2 Application to the inviscid Burgers equation 150
4.3 Application to the LWR model 162
4.4 Application to the Buckley-Leverett equation 171
Chapter 5. The Godunov method for systems of conservation laws 180
5.1 Application to the water hammer equations 180
5.2 Application to the simplified model for two-phase flow in pipes 197
5.3 Application to a 2x2 traffic flow model 212
5.4 Application to the open channel flow equations 225
Chapter 6. Higher-order schemes 250
6.1 Principle of higher-order schemes 251
6.2 The MUSCUPLM schemes 269
6.3 The PPM scheme 277
6.4 The DPM scheme 290
6.5 Boundary conditions for higher-order schemes 299
6.6 Application example 303
Chapter 7. Multidimensional schemes 316
7.1 Multidimensional hyperbolic systems of conservation laws 317
7.2 Alternate directions 329
7.3 The finite volume approach 334
7.4 Wave splitting 343
7.5 Computational examples 359
7.6 Higher-order multidimensional schemes 367
Chapter 8. Large-time-step algorithms 370
8.1 Front tracking algorithms 372
8.2 Implicit/explicit methods 381
8.3 The time-line reconstruction method 391
8.4 Computational examples 407
Chapter 9. Concluding remarks 412
Appendix A. Notions in mathematics 414
A.1 Linear algebra 414
A.2 Accuracy/consistency, stability, convergence 425
Appendix B. Riemann solvers 440
B.1 Exact Riemann solvers 440
B.2 The HLL Riemann solver 442
B.3 Roe’s Riemann solver 445
B.4 Approximate-state solvers 452
Appendix C. Sample codes 456
C.1 The code Linadv 456
C.2 The code ‘Burgers’ 460
C.3 The code ‘LWR’ 463
C.4 The code ‘BL’ 467
C.5 The code ‘WatHam’ 470
C.6 The code ‘2phase’ 473
C.7 The code ‘Traffic’ 477
C.8 The code ‘Channel’ 481
C.9 The code ‘Sh2D’ 486
C.10 The code ‘Large’ 491
References 496
Index 506