Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction

cover.jpg......Page 1
Front Matter......Page 2
References......Page 0
Preface......Page 4
Acknowledgements......Page 7
Preface to the Second Edition......Page 9
Table of Contents......Page 10
1. The Equations of Fluid Dynamics......Page 19
1.1 The Euler Equations......Page 20
1.1.1 Conservation-Law Form......Page 21
1.1.2 Other Compact Forms......Page 22
1.2.1 Units of Measure......Page 23
1.2.2 Equations of State (EOS)......Page 24
1.2.3 Other Variables and Relations......Page 25
1.2.4 Ideal Gases......Page 29
1.2.5 Covolume and van der Waal Gases......Page 31
1.3 Viscous Stresses......Page 33
1.4 Heat Conduction......Page 35
1.5 Integral Form of the Equations......Page 36
1.5.1 Time Derivatives......Page 37
1.5.2 Conservation of Mass......Page 38
1.5.3 Conservation of Momentum......Page 39
1.5.4 Conservation of Energy......Page 41
1.6.1 Summary of the Equations......Page 43
1.6.2 Flow with Area Variation......Page 45
1.6.3 Axi-Symmetric Flows......Page 46
1.6.5 Plain One-Dimensional Flow......Page 47
1.6.6 Steady Compressible Flow......Page 50
1.6.8 Free-Surface Gravity Flow......Page 51
1.6.9 The Shallow Water Equations......Page 53
1.6.10 Incompressible Viscous Flow......Page 56
1.6.11 The Artificial Compressibility Equations......Page 57
2.1 Quasi-Linear Equations: Basic Concepts......Page 59
2.2.1 Characteristics and the General Solution......Page 65
2.2.2 The Riemann Problem......Page 67
2.3 Linear Hyperbolic Systems......Page 68
2.3.1 Diagonalisation and Characteristic Variables......Page 69
2.3.2 The General Initial-Value Problem......Page 70
2.3.3 The Riemann Problem......Page 73
2.3.4 The Riemann Problem for Linearised Gas Dynamics......Page 76
2.3.5 Some Useful Definitions......Page 77
2.4 Conservation Laws......Page 78
2.4.1 Integral Forms of Conservation Laws......Page 79
2.4.2 Non-Linearities and Shock Formation......Page 83
2.4.3 Characteristic Fields......Page 94
2.4.4 Elementary-Wave Solutions of the Riemann Problem......Page 101
3.1.1 Conservative Formulation......Page 104
3.1.2 Non-Conservative Formulations......Page 108
3.1.3 Elementary Wave Solutions of the Riemann Problem......Page 111
3.2 Multi-Dimensional Euler Equations......Page 119
3.2.1 Two-Dimensional Equations in Conservative Form......Page 120
3.2.2 Three-Dimensional Equations in Conservative Form......Page 124
3.2.3 Three-Dimensional Primitive Variable Formulation......Page 125
3.2.4 The Split Three-Dimensional Riemann Problem......Page 127
3.3 Conservative versus Non-Conservative Formulations......Page 128
4. The Riemann Problem for the Euler Equations......Page 131
4.1 Solution Strategy......Page 132
4.2 Equations for Pressure and Particle Velocity......Page 135
4.2.1 Function f_L for a Left Shock......Page 136
4.2.2 Function f_L for Left Rarefaction......Page 138
4.2.3 Function f_R for a Right Shock......Page 139
4.2.4 Function f_R for a Right Rarefaction......Page 140
4.3.1 Behaviour of the Pressure Function......Page 141
4.3.2 Iterative Scheme for Finding the Pressure......Page 143
4.3.3 Numerical Tests......Page 145
4.4 The Complete Solution......Page 149
4.5 Sampling the Solution......Page 152
4.5.2 Right Side of Contact: S = x/t > u_*......Page 153
4.6 The Riemann Problem in the Presence of Vacuum......Page 154
4.6.1 Case 1: Vacuum Right State......Page 156
4.6.2 Case 2: Vacuum Left State......Page 157
4.6.3 Case 3: Generation of Vacuum......Page 158
4.7 The Riemann Problem for Covolume Gases......Page 159
4.7.1 Solution for Pressure and Particle Velocity......Page 160
4.7.3 The Complete Solution......Page 163
4.7.4 Solution inside Rarefactions......Page 164
4.8 The Split Multi-Dimensional Case......Page 165
4.9 FORTRAN Program for Exact Riemann Solver......Page 167
5.1 Discretisation: Introductory Concepts......Page 179
5.1.1 Approximation to Derivatives......Page 180
5.1.2 Finite Difference Approximation to a PDE......Page 181
5.2 Selected Difference Schemes......Page 183
5.2.1 The First Order Upwind Scheme......Page 184
5.2.2 Other Well-Known Schemes......Page 188
5.3 Conservative Methods......Page 190
5.3.1 Basic Definitions......Page 191
5.3.2 Godunov's First-Order Upwind Method......Page 193
5.3.3 Godunov's Method for Burgers's Equation......Page 196
5.3.4 Conservative Form of Difference Schemes......Page 200
5.4 Upwind Schemes for Linear Systems......Page 203
5.4.1 The CIR Scheme......Page 204
5.4.2 Godunov's Method......Page 206
5.5.1 Linear Advection......Page 209
5.5.2 The Inviscid Burgers Equation......Page 211
5.6 FORTRAN Program for Godunov's Method......Page 212
6.1 Bases of Godunov's Method......Page 229
6.2 The Godunov Scheme......Page 232
6.3 Godunov's Method for the Euler Equations......Page 234
6.3.1 Evaluation of the Intercell Fluxes......Page 235
6.3.2 Time Step Size......Page 237
6.3.3 Boundary Conditions......Page 238
6.4 Numerical Results and Discussion......Page 241
6.4.1 Numerical Results for Godunov's Method......Page 242
6.4.2 Numerical Results from Other Methods......Page 244
7.1 Introduction......Page 252
7.2 RCM on a Non-Staggered Grid......Page 253
7.2.1 The Scheme for Non-Linear Systems......Page 254
7.2.2 Boundary Conditions and the Time Step Size......Page 258
7.3.1 Review of the Lax-Friedrichs Scheme......Page 259
7.3.2 The Scheme......Page 260
7.4 The RCM on a Staggered Grid......Page 261
7.4.1 The Scheme for Non-Linear Systems......Page 262
7.4.2 A Deterministic First-Order Centred Scheme (FORCE)......Page 263
7.4.3 Analysis of the FORCE Scheme......Page 264
7.5.1 Van der Corput Pseudo-Random Numbers......Page 265
7.5.2 Statistical Properties......Page 267
7.5.3 Propagation of a Single Shock......Page 268
7.6 Numerical Results......Page 270
7.7 Concluding Remarks......Page 272
8.1 Introduction......Page 280
8.2.1 Upwind Differencing......Page 281
8.2.2 The FVS Approach......Page 283
8.3 FVS for the Isothermal Equations......Page 285
8.3.1 Split Fluxes......Page 286
8.3.2 FVS Numerical Schemes......Page 287
8.4.1 Recalling the Equations......Page 288
8.4.2 The Steger-Warming Splitting......Page 290
8.4.3 The van Leer Splitting......Page 291
8.4.4 The Liou-Steffen Scheme......Page 293
8.5.2 Results for Test 1......Page 295
8.5.4 Results for Test 3......Page 296
8.5.6 Results for Test 5......Page 297
9.1 Introduction......Page 307
9.2 The Riemann Problem and the Godunov Flux......Page 308
9.2.2 Sonic Rarefactions......Page 310
9.3 Primitive Variable Riemann Solvers (PVRS)......Page 311
9.4.1 A Two-Rarefaction Riemann Solver (TRRS)......Page 315
9.4.2 A Two-Shock Riemann Solver (TSRS)......Page 317
9.5.1 An Adaptive Iterative Riemann Solver (AIRS)......Page 318
9.5.2 An Adaptive Noniterative Riemann Solver (ANRS)......Page 319
9.6 Numerical Results......Page 320
10. The HLL and HLLC Riemann Solvers......Page 328
10.1 The Riemann Problem and the Godunov Flux......Page 329
10.2 The Riemann Problem and Integral Relations......Page 330
10.3 The HLL Approximate Riemann Solver......Page 332
10.4 The HLLC Approximate Riemann Solver......Page 334
10.5 Wave-Speed Estimates......Page 336
10.5.1 Direct Wave Speed Estimates......Page 337
10.5.2 Pressure-Velocity Based Wave Speed Estimates......Page 338
10.6.2 The Rusanov Flux......Page 340
10.7 Contact Waves and Passive Scalars......Page 341
10.8 Numerical Results......Page 342
10.9 Closing Remarks and Extensions......Page 344
11. The Riemann Solver of Roe......Page 353
11.1.1 The Exact Riemann Problem and the Godunov Flux......Page 354
11.1.2 Approximate Conservation Laws......Page 355
11.1.3 The Approximate Riemann Problem and the Intercell Flux......Page 357
11.2 The Original Roe Method......Page 359
11.2.1 The Isothermal Equations......Page 360
11.2.2 The Euler Equations......Page 362
11.3.1 The Approach......Page 366
11.3.2 The Isothermal Equations......Page 367
11.3.3 The Euler Equations......Page 371
11.4.1 The Entropy Problem......Page 374
11.4.2 The Harten-Hyman Entropy Fix......Page 375
11.4.3 The Speeds u_* , a_* L, a_* R......Page 378
11.5.1 The Tests......Page 379
11.5.2 The Results......Page 380
11.6 Extensions......Page 381
12. The Riemann Solver of Osher......Page 385
12.1.1 Mathematical Bases......Page 386
12.1.2 Osher's Numerical Flux......Page 388
12.1.3 Osher's Flux for the Single-Wave Case......Page 389
12.1.4 Osher's Flux for the Inviscid Burgers Equation......Page 391
12.1.5 Osher's Flux for the General Case......Page 392
12.2 Osher's Flux for the Isothermal Equations......Page 393
12.2.1 Osher's Flux with P-Ordering......Page 394
12.2.2 Osher's Flux with O-Ordering......Page 397
12.3 Osher's Scheme for the Euler Equations......Page 400
12.3.1 Osher's Flux with P-Ordering......Page 401
12.3.2 Osher's Flux with O-Ordering......Page 403
12.3.3 Remarks on Path Orderings......Page 408
12.3.4 The Split Three-Dimensional Case......Page 411
12.4 Numerical Results and Discussion......Page 412
12.5 Extensions......Page 413
13.1 Introduction......Page 420
13.2.1 Selected Schemes......Page 422
13.2.2 Accuracy......Page 424
13.2.3 Stability......Page 425
13.3.1 The Basic WAF Scheme......Page 427
13.3.2 Generalisations of the WAF Scheme......Page 430
13.4.1 Data Reconstruction......Page 433
13.4.2 The MUSCL-Hancock Method (MHM)......Page 436
13.4.3 The Piece-Wise Linear Method (PLM)......Page 439
13.4.4 The Generalised Riemann Problem (GRP) Method......Page 441
13.4.5 Slope-Limiter Centred (SLIC) Schemes......Page 443
13.4.7 Semi-Discrete Schemes......Page 446
13.4.8 Implicit Methods......Page 447
13.5.1 Monotone Schemes......Page 448
13.5.2 A Motivating Example......Page 451
13.5.3 Monotone Schemes and Godunov's Theorem......Page 453
13.5.4 Spurious Oscillations and High Resolution......Page 456
13.5.5 Data Compatibility......Page 457
14.1 Introduction......Page 459
14.2 CFL and Boundary Conditions......Page 460
14.3.1 The Original Version of WAF......Page 462
14.3.2 A Weighted Average State Version......Page 464
14.3.3 Rarefactions in State Riemann Solvers......Page 465
14.3.4 TVD Version of WAF Schemes......Page 467
14.3.6 Summary of the WAF Method......Page 469
14.4.1 The Basic Scheme......Page 470
14.4.2 A Variant of the Scheme......Page 472
14.4.3 TVD Version of the Scheme......Page 473
14.4.4 Summary of the MUSCL-Hancock Method......Page 476
14.5.1 Review of the FORCE Flux......Page 477
14.5.2 A Flux Limiter Centred (FLIC) Scheme......Page 478
14.5.3 A Slope Limiter Centred (SLIC) Scheme......Page 480
14.6.1 Formulation of the Equations and Primitive Schemes......Page 481
14.6.2 A WAF-Type Primitive Variable Scheme......Page 483
14.6.3 A MUSCL-Hancock Primitive Scheme......Page 486
14.6.4 Adaptive Primitive-Conservative Schemes......Page 488
14.7.1 Upwind TVD Methods......Page 489
14.7.2 Centred TVD Methods......Page 490
15.1 Introduction......Page 496
15.2 Splitting for a Model Equation......Page 497
15.3.1 Model Equations......Page 500
15.3.2 Schemes for Systems......Page 501
15.4.1 First-Order Systems of ODEs......Page 502
15.4.2 Numerical Methods......Page 504
15.4.3 Implementation Details for Split Schemes......Page 505
15.5 Concluding Remarks......Page 506
16.1 Introduction......Page 508
16.2.1 Splitting for a Model Problem......Page 509
16.2.2 Splitting Schemes for Two-Dimensional Systems......Page 510
16.2.3 Splitting Schemes for Three-Dimensional Systems......Page 512
16.3.1 Handling the Sweeps by a Single Subroutine......Page 514
16.3.3 The Intercell Flux and the TVD Condition......Page 516
16.4.1 Introductory Concepts......Page 520
16.4.2 Accuracy and Stability of Multidimensional Schemes......Page 523
16.5 A MUSCL-Hancock Finite Volume Scheme......Page 526
16.6.1 Two-Dimensional Linear Advection......Page 528
16.6.2 Three-Dimensional Linear Advection......Page 532
16.6.3 Schemes for Two-Dimensional Nonlinear Systems......Page 535
16.6.4 Schemes for Three-Dimensional Nonlinear Systems......Page 538
16.7.1 Introduction......Page 539
16.7.2 General Domains and Coordinate Transformation......Page 540
16.7.3 The Finite Volume Method for Non-Cartesian Domains......Page 542
17.1 Explosions and Implosions......Page 549
17.1.1 Explosion Test in Two-Space Dimensions......Page 550
17.1.2 Implosion Test in Two Dimensions......Page 553
17.1.3 Explosion Test in Three Space Dimensions......Page 554
17.2 Shock Wave Reflection from Wedges......Page 555
17.2.1 Mach Number M_s = 1.7 and Phi= 25 Degrees......Page 556
17.2.2 Mach Number M_s = 1.2 and Phi = 30 Degrees......Page 559
18.1 Summary......Page 561
18.2.2 Steady Supersonic Euler Equations......Page 562
18.2.5 Compressible Materials......Page 563
18.2.8 Magnetohydrodynamics (MHD)......Page 564
18.3 NUMERICA......Page 565
References......Page 567
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