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

Cover\r......Page 1
Riemann Solvers and Numerical Methodsfor Fluid Dynamics, Third Edition......Page 2
Preface to the First Edition......Page 6
Preface to the Third Edition......Page 10
Contents......Page 0
Preface to the Third Edition . . . . . . . . . . . . . . . . . XI......Page 13
4 The Riemann Problem for the Euler Equations ............115......Page 14
6 The Method of Godunov for Non–linear Systems ..........213......Page 15
9 Approximate–State Riemann Solvers ......................293......Page 16
11 The Riemann Solver of Roe ...............................345......Page 17
13 High–Order and TVD Methods for Scalar Equations ......413......Page 18
14 High–Order and TVD Schemes for Non–Linear Systems . . . 493......Page 19
17 Multidimensional Test Problems ...........................585......Page 20
20 The ADER Approach ......................................655......Page 21
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719......Page 22
1 The Equations of Fluid Dynamics......Page 23
1.1 The Euler Equations......Page 24
1.1.1 Conservation–Law Form......Page 25
1.1.2 Other Compact Forms......Page 26
1.2.1 Units of Measure......Page 27
1.2.2 Equations of State (EOS)......Page 28
1.2.3 Other Variables and Relations......Page 29
1.2.4 Ideal Gases......Page 33
1.2.5 Covolume and van der Waal Gases......Page 35
1.3 Viscous Stresses......Page 37
1.4 Heat Conduction......Page 39
1.5 Integral Form of the Equations......Page 40
1.5.1 Time Derivatives......Page 41
1.5.2 Conservation of Mass......Page 42
1.5.3 Conservation of Momentum......Page 43
1.5.4 Conservation of Energy......Page 45
1.6.1 Summary of the Equations......Page 47
1.6.2 Flow with Area Variation......Page 49
1.6.3 Axi–Symmetric Flows......Page 50
1.6.5 Plain One–Dimensional Flow......Page 51
1.6.6 Steady Compressible Flow......Page 53
1.6.8 Free–Surface Gravity Flow......Page 55
1.6.9 The Shallow Water Equations......Page 57
1.6.10 Incompressible Viscous Flow......Page 60
1.6.11 The Arti.cial Compressibility Equations......Page 61
2 Notions on Hyperbolic Partial Differential Equations......Page 63
2.1 Quasi–Linear Equations: Basic Concepts......Page 64
2.2.1 Characteristics and the General Solution......Page 69
2.2.2 The Riemann Problem......Page 71
2.3 Linear Hyperbolic Systems......Page 72
2.3.1 Diagonalisation and Characteristic Variables......Page 73
2.3.2 The General Initial–Value Problem......Page 74
2.3.4 The Riemann Problem for Linearised Gas Dynamics......Page 80
2.3.5 Some Useful De.nitions......Page 82
2.4 Conservation Laws......Page 83
2.4.1 Integral Forms of Conservation Laws......Page 84
2.4.2 Non–Linearities and Shock Formation......Page 88
2.4.3 Characteristic Fields......Page 99
2.4.4 Elementary–Wave Solutions of the Riemann Problem......Page 105
3.1.1 Conservative Formulation......Page 109
3.1.2 Non–Conservative Formulations......Page 113
3.1.3 Elementary Wave Solutions of the Riemann Problem......Page 116
3.2 Multi–Dimensional Euler Equations......Page 125
3.2.1 Two–Dimensional Equations in Conservative Form......Page 126
3.2.2 Three–Dimensional Equations in Conservative Form......Page 130
3.2.3 Three–Dimensional Primitive Variable Formulation......Page 131
3.2.4 The Split Three–Dimensional Riemann Problem......Page 133
3.3 Conservative Versus Non–Conservative Formulations......Page 134
4 The Riemann Problem for the Euler Equations......Page 137
4.1 Solution Strategy......Page 138
4.2 Equations for Pressure and Particle Velocity......Page 141
4.2.1 Function fL for a Left Shock......Page 142
4.2.2 Function fL for Left Rarefaction......Page 144
4.2.3 Function fR for a Right Shock......Page 145
4.2.4 Function fR for a Right Rarefaction......Page 146
4.3.1 Behaviour of the Pressure Function......Page 147
4.3.2 Iterative Scheme for Finding the Pressure......Page 149
4.3.3 Numerical Tests......Page 151
4.4 The Complete Solution......Page 154
4.5 Sampling the Solution......Page 158
4.5.2 Right Side of Contact: S = x/t = u*......Page 159
4.6 The Riemann Problem in the Presence of Vacuum......Page 161
4.6.1 Case 1: Vacuum Right State......Page 162
4.6.3 Case 3: Generation of Vacuum......Page 164
4.7 The Riemann Problem for Covolume Gases......Page 165
4.7.1 Solution for Pressure and Particle Velocity.......Page 166
4.7.3 The Complete Solution......Page 169
4.7.4 Solution Inside Rarefactions......Page 170
4.8 The Split Multi–Dimensional Case......Page 171
4.9 FORTRAN Program for Exact Riemann Solver......Page 173
5.1 Discretisation: Introductory Concepts......Page 185
5.1.1 Approximation to Derivatives......Page 186
5.1.2 Finite Di.erence Approximation to a PDE......Page 187
5.2.1 The First Order Upwind Scheme......Page 190
5.2.2 Other Well–Known Schemes......Page 194
5.3 Conservative Methods......Page 196
5.3.1 Basic De.nitions......Page 197
5.3.2 Godunov’s First–Order Upwind Method......Page 199
5.3.3 Godunov’s Method for Burgers’s Equation......Page 203
5.3.4 Conservative Form of Di.erence Schemes......Page 206
5.4 Upwind Schemes for Linear Systems......Page 209
5.4.1 The CIR Scheme......Page 210
5.4.2 Godunov’s Method......Page 212
5.5.1 Linear Advection......Page 216
5.6 FORTRAN Program for Godunov’s Method......Page 218
6.1 Bases of Godunov’s Method......Page 235
6.2 The Godunov Scheme......Page 238
6.3 Godunov’s Method for the Euler Equations......Page 240
6.3.1 Evaluation of the Intercell Fluxes......Page 241
6.3.2 Time Step Size......Page 243
6.3.3 Boundary Conditions......Page 244
6.4 Numerical Results and Discussion......Page 247
6.4.1 Numerical Results for Godunov’s Method......Page 248
6.4.2 Numerical Results from Other Methods......Page 250
7.1 Introduction......Page 258
7.2 RCM on a Non–Staggered Grid......Page 259
7.2.1 The Scheme for Non–Linear Systems......Page 260
7.2.2 Boundary Conditions and the Time Step Size......Page 264
7.3.1 Review of the Lax–Friedrichs Scheme......Page 265
7.3.2 The Scheme......Page 266
7.4.2 A Deterministic First–Order Centred Scheme (force)......Page 268
7.4.3 Analysis of the force Scheme......Page 270
7.5.1 Van der Corput Pseudo–Random Numbers......Page 271
7.5.2 Statistical Properties......Page 272
7.5.3 Propagation of a Single Shock......Page 274
7.6 Numerical Results......Page 276
7.7 Concluding Remarks......Page 277
8.1 Introduction......Page 286
8.2.1 Upwind Di.erencing......Page 287
8.2.2 The FVS Approach......Page 289
8.3 FVS for the Isothermal Equations......Page 291
8.3.1 Split Fluxes......Page 292
8.3.2 FVS Numerical Schemes......Page 293
8.4 FVS Applied to the Euler Equations......Page 294
8.4.1 Recalling the Equations......Page 295
8.4.2 The Steger–Warming Splitting......Page 297
8.4.3 The van Leer Splitting......Page 298
8.4.4 The Liou–Ste.en Scheme......Page 299
8.5.2 Results for Test 1......Page 301
8.5.4 Results for Test 3......Page 302
8.5.6 Results for Test 5......Page 303
9.1 Introduction......Page 313
9.2 The Riemann Problem and the Godunov Flux......Page 314
9.2.2 Sonic Rarefactions......Page 316
9.3 Primitive Variable Riemann Solvers (PVRS)......Page 317
9.4.1 A Two–Rarefaction Riemann Solver (TRRS)......Page 321
9.4.2 A Two–Shock Riemann Solver (TSRS)......Page 323
9.5.1 An Adaptive Iterative Riemann Solver (AIRS)......Page 324
9.5.2 An Adaptive Noniterative Riemann Solver (ANRS)......Page 325
9.6 Numerical Results......Page 326
10.1 Introduction......Page 334
10.2.1 The Godunov Flux......Page 336
10.2.2 Integral Relations......Page 337
10.3 The HLL Approximate Riemann Solver......Page 339
10.4.1 Useful Relations......Page 341
10.4.2 The HLLC Flux for the Euler Equations......Page 343
10.4.3 Multidimensional and Multicomponent Flow......Page 345
10.5 Wave–Speed Estimates......Page 346
10.5.1 Direct Wave Speed Estimates......Page 347
10.5.2 Pressure–Based Wave Speed Estimates......Page 348
10.6 Summary of HLLC Fluxes......Page 350
10.7 Contact Waves and Passive Scalars......Page 352
10.8 Numerical Results......Page 353
10.9 Closing Remarks......Page 355
11 The Riemann Solver of Roe......Page 364
11.1.1 The Exact Riemann Problem and the Godunov Flux......Page 365
11.1.2 Approximate Conservation Laws......Page 366
11.1.3 The Approximate Riemann Problem and the Intercell Flux......Page 368
11.2 The Original Roe Method......Page 370
11.2.1 The Isothermal Equations......Page 371
11.2.2 The Euler Equations......Page 373
11.3.1 The Approach......Page 377
11.3.2 The Isothermal Equations......Page 378
11.3.3 The Euler Equations......Page 382
11.4.1 The Entropy Problem......Page 385
11.4.2 The Harten–Hyman Entropy Fix......Page 386
11.4.3 The Speeds u*, a*L, a*R......Page 389
11.5.1 The Tests......Page 391
11.6 Extensions......Page 392
12 The Riemann Solver of Osher......Page 396
12.1.1 Mathematical Bases......Page 397
12.1.2 Osher’s Numerical Flux......Page 399
12.1.3 Osher’s Flux for the Single–Wave Case......Page 400
12.1.4 Osher’s Flux for the Inviscid Burgers Equation......Page 402
12.1.5 Osher’s Flux for the General Case......Page 403
12.2 Osher’s Flux for the Isothermal Equations......Page 404
12.2.1 Osher’s Flux with P–Ordering......Page 405
12.2.2 Osher’s Flux with O–Ordering......Page 408
12.3 Osher’s Scheme for the Euler Equations......Page 411
12.3.1 Osher’s Flux with P–Ordering......Page 412
12.3.2 Osher’s Flux with O–Ordering......Page 416
12.3.3 Remarks on Path Orderings......Page 421
12.3.4 The Split Three–Dimensional Case......Page 422
12.4 Numerical Results and Discussion......Page 423
12.5 Extensions......Page 425
13.1 Introduction......Page 431
13.2.1 Selected Schemes......Page 433
13.2.2 Accuracy......Page 435
13.2.3 Stability......Page 436
13.3.1 The Basic waf Scheme......Page 438
13.3.2 Generalisations of the waf Scheme......Page 441
13.4.1 Data Reconstruction......Page 444
13.4.2 The MUSCL–Hancock Method (MHM)......Page 447
13.4.3 The Piece–Wise Linear Method (PLM)......Page 450
13.4.4 The Generalised Riemann Problem (GRP) Method......Page 452
13.4.5 Slope–Limiter Centred (slic) Schemes......Page 454
13.4.8 Implicit Methods......Page 457
13.5.1 Monotone Schemes......Page 458
13.5.2 A Motivating Example......Page 461
13.5.3 Monotone Schemes and Godunov’s Theorem......Page 465
13.5.4 Spurious Oscillations and High Resolution......Page 466
13.5.5 Data Compatibility......Page 467
13.6 Total Variation Diminishing ( TVD) Methods......Page 469
13.6.1 The Total Variation......Page 470
13.6.2 TVD and Monotonicity Preserving Schemes......Page 471
13.7.1 TVD Version of the waf Method......Page 474
13.7.2 The General Flux–Limiter Approach......Page 482
13.7.3 TVD Upwind Flux Limiter Schemes......Page 487
13.7.4 TVD Centred Flux Limiter Schemes......Page 492
13.8.1 TVD Conditions......Page 498
Theorem 13.77. If the limited slopes .i are chosen according to......Page 499
13.8.3 Slope Limiters......Page 500
13.8.4 Limited Slopes Obtained from Flux Limiters......Page 502
13.9.2 TVD Schemes in the Presence of Di.usion Terms......Page 504
13.10 Numerical Results for Linear Advection......Page 505
14.1 Introduction......Page 511
14.2 CFL and Boundary Conditions......Page 513
14.3.1 The Original Version of waf......Page 514
14.3.2 A Weighted Average State Version......Page 516
14.3.3 Rarefactions in State Riemann Solvers......Page 517
14.3.4 TVD Version of waf Schemes......Page 519
14.3.6 Summary of the waf Method......Page 521
14.4.1 The Basic Scheme......Page 522
14.4.2 A Variant of the Scheme......Page 524
14.4.3 TVD Version of the Scheme......Page 525
14.4.4 Summary of the MUSCL–Hancock Method......Page 528
14.5 Centred TVD Schemes......Page 529
14.5.2 A Flux Limiter Centred (flic)Scheme......Page 530
14.5.3 A Slope Limiter Centred (slic)Scheme......Page 532
14.6.1 Formulation of the Equations and Primitive Schemes......Page 533
14.6.2 A waf–Type Primitive Variable Scheme......Page 535
14.6.3 A MUSCL–Hancock Primitive Scheme......Page 538
14.6.4 Adaptive Primitive–Conservative Schemes......Page 540
14.7.1 Upwind TVD Methods......Page 541
14.7.2 Centred TVD Methods......Page 542
15.1 Introduction......Page 548
15.2 Splitting for a Model Equation......Page 550
15.3.1 Model Equations......Page 552
15.3.2 Schemes for Systems......Page 553
15.4.1 First–Order Systems of ODEs......Page 554
15.4.2 Numerical Methods......Page 556
15.4.3 Implementation Details for Split Schemes......Page 557
15.5 Concluding Remarks......Page 558
16.1 Introduction......Page 560
16.2.1 Splitting for a Model Problem......Page 561
16.2.2 Splitting Schemes for Two–Dimensional Systems......Page 562
16.2.3 Splitting Schemes for Three–Dimensional Systems......Page 564
16.3.1 Handling the Sweeps by a Single Subroutine......Page 566
16.3.2 Choice of Time Step Size......Page 568
16.3.3 The Intercell Flux and the tvd Condition......Page 569
16.4.1 Introductory Concepts......Page 572
16.4.2 Accuracy and Stability of Multidimensional Schemes Accuracy Theorems......Page 575
16.5 A Muscl–Hancock Finite Volume Scheme......Page 578
16.6 WAF–Type Finite Volume Schemes......Page 580
16.6.1 Two–Dimensional Linear Advection......Page 581
16.6.2 Three–Dimensional Linear Advection......Page 584
16.6.3 Schemes for Two–Dimensional Nonlinear Systems......Page 587
16.6.4 Schemes for Three–Dimensional Nonlinear Systems......Page 590
16.7.1 Introduction......Page 591
16.7.2 General Domains and Coordinate Transformation......Page 592
16.7.3 The Finite Volume Method for Non–Cartesian Domains......Page 595
17 Multidimensional Test Problems......Page 602
17.1 Explosions and Implosions......Page 603
17.1.1 Explosion Test in Two–Space Dimensions......Page 604
17.1.2 Explosion Test in Three Space Dimensions......Page 607
17.2 Shock Wave Re.ection from a Wedge......Page 608
18.1 Introduction......Page 614
18.2.1 FORCE and Related Fluxes......Page 617
18.2.2 Monotonicity and Numerical Viscosity......Page 619
18.3.1 The Two–Dimensional Case......Page 622
18.3.2 The Three–Dimensional Case......Page 626
18.4.1 One–Dimensional Interpretation......Page 627
18.4.2 Some Numerical Experiments......Page 628
18.4.3 Analysis in Multiple Space Dimensions......Page 630
18.5 FORCE Schemes on General Meshes......Page 634
18.7 Concluding Remarks......Page 638
19.1 Introduction......Page 641
19.2 Statement of the Problem......Page 645
19.3 The Cauchy–Kowalewski Theorem......Page 647
19.3.1 Series Expansions and Analytic Functions......Page 648
19.3.3 The Cauchy–Kowalewski Method......Page 649
19.4 A Method of Solution......Page 651
19.4.1 The Leading Term......Page 652
19.4.2 Higher–Order Terms......Page 653
19.4.3 Source Terms......Page 656
19.5 Examples......Page 658
19.5.1 The Linear Advection Equation......Page 659
19.5.2 Linear Advection with a Source Term......Page 661
19.5.3 Non–Linear Equation with a Source Term......Page 662
19.5.4 The Burgers Equation with a Source Term......Page 664
19.6 Other Solvers......Page 667
19.7 Concluding Remarks......Page 669
20.1 Introduction......Page 670
20.2.1 The Framework......Page 672
20.2.2 The Numerical Flux......Page 673
20.2.3 The Numerical Source......Page 674
20.2.4 Reconstruction......Page 675
20.3.1 Numerical Flux and Numerical Source......Page 678
20.3.2 The Scheme......Page 681
20.4.1 The Numerical Flux......Page 682
20.4.3 Summary......Page 683
20.5.1 Long–Time Advection of Smooth Pro.les......Page 684
20.5.2 Convergence Rates......Page 687
20.6 Concluding Remarks......Page 688
21.1 Summary of Numerical Aspects......Page 693
21.2 Potential Applications......Page 695
21.3 Current Research Topics......Page 699
21.4 The NUMERICA Library......Page 700
References......Page 701
Index......Page 733