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

Cover......Page 1
Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction - Third edition......Page 4
3540252029......Page 5
Preface to the First Edition......Page 8
Preface to the Third Edition......Page 12
Contents......Page 16
1. The Equations of Fluid Dynamics......Page 26
1.1 The Euler Equations......Page 27
1.1.1 Conservation-Law Form......Page 28
1.1.2 Other Compact Forms......Page 29
1.2.1 Units of Measure......Page 30
1.2.2 Equations of State (EOS)......Page 31
1.2.3 Other Variables and Relations......Page 32
1.2.4 Ideal Gases......Page 36
1.2.5 Covolume and van der Waal Gases......Page 38
1.3 Viscous Stresses......Page 40
1.4 Heat Conduction......Page 42
1.5 Integral Form of the Equations......Page 43
1.5.1 Time Derivatives......Page 44
1.5.2 Conservation of Mass......Page 45
1.5.3 Conservation of Momentum......Page 46
1.5.4 Conservation of Energy......Page 48
1.6.1 Summary of the Equations......Page 50
1.6.2 Flow with Area Variation......Page 52
1.6.3 Axi-Symmetric Flows......Page 53
1.6.5 Plain One-Dimensional Flow......Page 54
1.6.6 Steady Compressible Flow......Page 56
1.6.8 Free-Surface Gravity Flow......Page 58
1.6.9 The Shallow Water Equations......Page 60
1.6.10 Incompressible Viscous Flow......Page 63
1.6.11 The Artificial Compressibility Equations......Page 64
2. Notions on Hyperbolic Partial Differential Equations......Page 66
2.1 Quasi-Linear Equations: Basic Concepts......Page 67
2.2.1 Characteristics and the General Solution......Page 72
2.2.2 The Riemann Problem......Page 74
2.3 Linear Hyperbolic Systems......Page 75
2.3.1 Diagonalisation and Characteristic Variables......Page 76
2.3.2 The General Initial-Value Problem......Page 77
2.3.3 The Riemann Problem......Page 80
2.3.4 The Riemann Problem for Linearised Gas Dynamics......Page 83
2.3.5 Some Useful Definitions......Page 85
2.4 Conservation Laws......Page 86
2.4.1 Integral Forms of Conservation Laws......Page 87
2.4.2 Non-Linearities and Shock Formation......Page 91
2.4.3 Characteristic Fields......Page 102
2.4.4 Elementary-Wave Solutions of the Riemann Problem......Page 108
3.1.1 Conservative Formulation......Page 112
3.1.2 Non-Conservative Formulations......Page 116
3.1.3 Elementary Wave Solutions of the Riemann Problem......Page 119
3.2 Multi--Dimensional Euler Equations......Page 128
3.2.1 Two-Dimensional Equations in Conservative Form......Page 129
3.2.2 Three-Dimensional Equations in Conservative Form......Page 133
3.2.3 Three-Dimensional Primitive Variable Formulation......Page 134
3.2.4 The Split Three-Dimensional Riemann Problem......Page 136
3.3 Conservative Versus Non-Conservative Formulations......Page 137
4. The Riemann Problem for the Euler Equations......Page 140
4.1 Solution Strategy......Page 141
4.2 Equations for Pressure and Particle Velocity......Page 144
4.2.1 Function f[sub(L)] for a Left Shock......Page 145
4.2.2 Function f[sub(L)] for Left Rarefaction......Page 147
4.2.3 Function f[sub(R)] for a Right Shock......Page 148
4.2.4 Function f[sub(R)] for a Right Rarefaction......Page 149
4.3.1 Behaviour of the Pressure Function......Page 150
4.3.2 Iterative Scheme for Finding the Pressure......Page 152
4.3.3 Numerical Tests......Page 154
4.4 The Complete Solution......Page 157
4.5 Sampling the Solution......Page 161
4.5.2 Right Side of Contact: S=x/t ≤ u[sub(*)]......Page 162
4.6 The Riemann Problem in the Presence of Vacuum......Page 164
4.6.1 Case 1: Vacuum Right State......Page 165
4.6.3 Case 3: Generation of Vacuum......Page 167
4.7 The Riemann Problem for Covolume Gases......Page 168
4.7.1 Solution for Pressure and Particle Velocity......Page 169
4.7.3 The Complete Solution......Page 172
4.7.4 Solution Inside Rarefactions......Page 173
4.8 The Split Multi-Dimensional Case......Page 174
4.9 FORTRAN Program for Exact Riemann Solver......Page 176
5.1 Discretisation: Introductory Concepts......Page 188
5.1.1 Approximation to Derivatives......Page 189
5.1.2 Finite Difference Approximation to a PDE......Page 190
5.2.1 The First Order Upwind Scheme......Page 193
5.2.2 Other Well-Known Schemes......Page 197
5.3 Conservative Methods......Page 199
5.3.1 Basic Definitions......Page 200
5.3.2 Godunov's First-Order Upwind Method......Page 202
5.3.3 Godunov's Method for Burgers's Equation......Page 206
5.3.4 Conservative Form of Difference Schemes......Page 209
5.4 Upwind Schemes for Linear Systems......Page 212
5.4.1 The CIR Scheme......Page 213
5.4.2 Godunov's Method......Page 215
5.5.1 Linear Advection......Page 219
5.6 FORTRAN Program for Godunov's Method......Page 221
6.1 Bases of Godunov's Method......Page 238
6.2 The Godunov Scheme......Page 241
6.3 Godunov's Method for the Euler Equations......Page 243
6.3.1 Evaluation of the Intercell Fluxes......Page 244
6.3.2 Time Step Size......Page 246
6.3.3 Boundary Conditions......Page 247
6.4 Numerical Results and Discussion......Page 250
6.4.1 Numerical Results for Godunov's Method......Page 251
6.4.2 Numerical Results from Other Methods......Page 253
7.1 Introduction......Page 262
7.2 RCM on a Non-Staggered Grid......Page 263
7.2.1 The Scheme for Non-Linear Systems......Page 264
7.2.2 Boundary Conditions and the Time Step Size......Page 268
7.3.1 Review of the Lax-Friedrichs Scheme......Page 269
7.3.2 The Scheme......Page 270
7.4.2 A Deterministic First-Order Centred Scheme (force)......Page 272
7.4.3 Analysis of the force Scheme......Page 274
7.5.1 Van der Corput Pseudo-Random Numbers......Page 275
7.5.2 Statistical Properties......Page 276
7.5.3 Propagation of a Single Shock......Page 278
7.6 Numerical Results......Page 280
7.7 Concluding Remarks......Page 281
8.1 Introduction......Page 290
8.2.1 Upwind Differencing......Page 291
8.2.2 The FVS Approach......Page 293
8.3 FVS for the Isothermal Equations......Page 295
8.3.1 Split Fluxes......Page 296
8.3.2 FVS Numerical Schemes......Page 297
8.4 FVS Applied to the Euler Equations......Page 298
8.4.1 Recalling the Equations......Page 299
8.4.2 The Steger-Warming Splitting......Page 301
8.4.3 The van Leer Splitting......Page 302
8.4.4 The Liou-Steffen Scheme......Page 303
8.5.2 Results for Test 1......Page 305
8.5.4 Results for Test 3......Page 306
8.5.6 Results for Test 5......Page 307
9.1 Introduction......Page 318
9.2 The Riemann Problem and the Godunov Flux......Page 319
9.2.2 Sonic Rarefactions......Page 321
9.3 Primitive Variable Riemann Solvers (PVRS)......Page 322
9.4.1 A Two-Rarefaction Riemann Solver (TRRS)......Page 326
9.4.2 A Two-Shock Riemann Solver (TSRS)......Page 328
9.5.1 An Adaptive Iterative Riemann Solver (AIRS)......Page 329
9.5.2 An Adaptive Noniterative Riemann Solver (ANRS)......Page 330
9.6 Numerical Results......Page 331
10.1 Introduction......Page 340
10.2.1 The Godunov Flux......Page 342
10.2.2 Integral Relations......Page 343
10.3 The HLL Approximate Riemann Solver......Page 345
10.4.1 Useful Relations......Page 347
10.4.2 The HLLC Flux for the Euler Equations......Page 349
10.4.3 Multidimensional and Multicomponent Flow......Page 351
10.5 Wave-Speed Estimates......Page 352
10.5.1 Direct Wave Speed Estimates......Page 353
10.5.2 Pressure-Based Wave Speed Estimates......Page 354
10.6 Summary of HLLC Fluxes......Page 356
10.7 Contact Waves and Passive Scalars......Page 358
10.8 Numerical Results......Page 359
10.9 Closing Remarks......Page 361
11. The Riemann Solver of Roe......Page 370
11.1 The Exact Riemann Problem and the Godunov Flux......Page 371
11.2 Approximate Conservation Laws......Page 372
11.3 The Approximate Riemann Problem and the Intercell Flux......Page 374
11.2 The Original Roe Method......Page 376
11.2.1 The Isothermal Equations......Page 377
11.2.2 The Euler Equations......Page 379
11.3.1 The Approach......Page 383
11.3.2 The Isothermal Equations......Page 384
11.3.3 The Euler Equations......Page 388
11.4.1 The Entropy Problem......Page 391
11.4.2 The Harten-Hyman Entropy Fix......Page 392
11.4.3 The Speeds u[sub(*)], a[sub(*L)], a[sub(*R)]......Page 395
11.5.1 The Tests......Page 397
11.6 Extensions......Page 398
12. The Riemann Solver of Osher......Page 402
12.1.1 Mathematical Bases......Page 403
12.1.2 Osher's Numerical Flux......Page 405
12.1.3 Osher's Flux for the Single-Wave Case......Page 406
12.1.4 Osher's Flux for the Inviscid Burgers Equation......Page 408
12.1.5 Osher's Flux for the General Case......Page 409
12.2 Osher's Flux for the Isothermal Equations......Page 410
12.2.1 Osher's Flux with P-Ordering......Page 411
12.2.2 Osher's Flux with O-Ordering......Page 414
12.3 Osher's Scheme for the Euler Equations......Page 417
12.3.1 Osher's Flux with P-Ordering......Page 418
12.3.2 Osher's Flux with O-Ordering......Page 422
12.3.3 Remarks on Path Orderings......Page 427
12.3.4 The Split Three-Dimensional Case......Page 428
12.4 Numerical Results and Discussion......Page 429
12.5 Extensions......Page 431
13.1 Introduction......Page 438
13.2.1 Selected Schemes......Page 440
13.2.2 Accuracy......Page 442
13.2.3 Stability......Page 443
13.3.1 The Basic waf Scheme......Page 445
13.3.2 Generalisations of the waf Scheme......Page 448
13.4.1 Data Reconstruction......Page 451
13.4.2 The MUSCL-Hancock Method (MHM)......Page 454
13.4.3 The Piece-Wise Linear Method (PLM)......Page 457
13.4.4 The Generalised Riemann Problem (GRP) Method......Page 459
13.4.5 Slope-Limiter Centred (slic) Schemes......Page 461
13.4.8 Implicit Methods......Page 464
13.5.1 Monotone Schemes......Page 465
13.5.2 A Motivating Example......Page 468
13.5.3 Monotone Schemes and Godunov's Theorem......Page 472
13.5.4 Spurious Oscillations and High Resolution......Page 473
13.5.5 Data Compatibility......Page 474
13.6 Total Variation Diminishing (TVD) Methods......Page 476
13.6.1 The Total Variation......Page 477
13.6.2 TVD and Monotonicity Preserving Schemes......Page 478
13.7.1 TVD Version of the waf Method......Page 481
13.7.2 The General Flux-Limiter Approach......Page 489
13.7.3 TVD Upwind Flux Limiter Schemes......Page 494
13.7.4 TVD Centred Flux Limiter Schemes......Page 499
13.8.1 TVD Conditions......Page 505
13.8.2 Construction of TVD Slopes......Page 506
13.8.3 Slope Limiters......Page 507
13.8.4 Limited Slopes Obtained from Flux Limiters......Page 509
13.9.2 TVD Schemes in the Presence of Diffusion Terms......Page 511
13.10 Numerical Results for Linear Advection......Page 512
14.1 Introduction......Page 518
14.2 CFL and Boundary Conditions......Page 520
14.3.1 The Original Version of waf......Page 521
14.3.2 A Weighted Average State Version......Page 523
14.3.3 Rarefactions in State Riemann Solvers......Page 524
14.3.4 TVD Version of waf Schemes......Page 526
14.3.6 Summary of the waf Method......Page 528
14.4.1 The Basic Scheme......Page 529
14.4.2 A Variant of the Scheme......Page 531
14.4.3 TVD Version of the Scheme......Page 532
14.4.4 Summary of the MUSCL-Hancock Method......Page 535
14.5 Centred TVD Schemes......Page 536
14.5.2 A Flux Limiter Centred (FLIC) Scheme......Page 537
14.5.3 A Slope Limiter Centred (SLIC) Scheme......Page 539
14.6.1 Formulation of the Equations and Primitive Schemes......Page 540
14.6.2 A waf-Type Primitive Variable Scheme......Page 542
14.6.3 A MUSCL-Hancock Primitive Scheme......Page 545
14.6.4 Adaptive Primitive-Conservative Schemes......Page 547
14.7.1 Upwind TVD Methods......Page 548
14.7.2 Centred TVD Methods......Page 549
15.1 Introduction......Page 556
15.2 Splitting for a Model Equation......Page 558
15.3.1 Model Equations......Page 560
15.3.2 Schemes for Systems......Page 561
15.4.1 First-Order Systems of ODEs......Page 562
15.4.2 Numerical Methods......Page 564
15.4.3 Implementation Details for Split Schemes......Page 565
15.5 Concluding Remarks......Page 566
16.1 Introduction......Page 568
16.2.1 Splitting for a Model Problem......Page 569
16.2.2 Splitting Schemes for Two-Dimensional Systems......Page 570
16.2.3 Splitting Schemes for Three-Dimensional Systems......Page 572
16.3.1 Handling the Sweeps by a Single Subroutine......Page 574
16.3.2 Choice of Time Step Size......Page 576
16.3.3 The Intercell Flux and the TVD Condition......Page 577
16.4.1 Introductory Concepts......Page 580
16.4.2 Accuracy and Stability of Multidimensional Schemes......Page 583
16.5 A Muscl-Hancock Finite Volume Scheme......Page 586
16.6 WAF-Type Finite Volume Schemes......Page 588
16.6.1 Two-Dimensional Linear Advection......Page 589
16.6.2 Three-Dimensional Linear Advection......Page 592
16.6.3 Schemes for Two-Dimensional Nonlinear Systems......Page 595
16.6.4 Schemes for Three-Dimensional Nonlinear Systems......Page 598
16.7.1 Introduction......Page 599
16.7.2 General Domains and Coordinate Transformation......Page 600
16.7.3 The Finite Volume Method for Non-Cartesian Domains......Page 603
17. Multidimensional Test Problems......Page 610
17.1 Explosions and Implosions......Page 611
17.1.1 Explosion Test in Two-Space Dimensions......Page 612
17.1.2 Explosion Test in Three Space Dimensions......Page 615
17.2 Shock Wave Reflection from a Wedge......Page 616
18.1 Introduction......Page 622
18.2.1 FORCE and Related Fluxes......Page 625
18.2.2 Monotonicity and Numerical Viscosity......Page 627
18.3.1 The Two-Dimensional Case......Page 630
18.3.2 The Three-Dimensional Case......Page 634
18.4.1 One-Dimensional Interpretation......Page 635
18.4.2 Some Numerical Experiments......Page 636
18.4.3 Analysis in Multiple Space Dimensions......Page 638
18.5 FORCE Schemes on General Meshes......Page 642
18.7 Concluding Remarks......Page 646
19.1 Introduction......Page 650
19.2 Statement of the Problem......Page 654
19.3 The Cauchy-Kowalewski Theorem......Page 656
19.3.1 Series Expansions and Analytic Functions......Page 657
19.3.3 The Cauchy-Kowalewski Method......Page 658
19.4 A Method of Solution......Page 660
19.4.1 The Leading Term......Page 661
19.4.2 Higher-Order Terms......Page 662
19.4.4 Summary: Numerical Flux and Numerical Source......Page 665
19.5 Examples......Page 667
19.5.1 The Linear Advection Equation......Page 668
19.5.2 Linear Advection with a Source Term......Page 670
19.5.3 Non-Linear Equation with a Source Term......Page 671
19.5.4 The Burgers Equation with a Source Term......Page 673
19.6 Other Solvers......Page 676
19.7 Concluding Remarks......Page 678
20.1 Introduction......Page 680
20.2.1 The Framework......Page 682
20.2.2 The Numerical Flux......Page 683
20.2.3 The Numerical Source......Page 684
20.2.4 Reconstruction......Page 685
20.3.1 Numerical Flux and Numerical Source......Page 688
20.3.2 The Scheme......Page 691
20.4.1 The Numerical Flux......Page 692
20.4.3 Summary......Page 693
20.5.1 Long-Time Advection of Smooth Profiles......Page 694
20.5.2 Convergence Rates......Page 697
20.6 Concluding Remarks......Page 698
21.1 Summary of Numerical Aspects......Page 704
21.2 Potential Applications......Page 706
21.3 Current Research Topics......Page 710
21.4 The NUMERICA Library......Page 711
References......Page 712
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