Preface to the First Edition .................................. VII
Preface to the Third Edition ................................... XI
1 The Equations of Fluid Dynamics .............................. 1
1.1 The Euler Equations ..................................... 2
1.1.1 Conservation-Law Form ............................ 3
1.1.2 Other Compact Forms .............................. 4
1.2 Thermodynamic Considerations ............................ 5
1.2.1 Units of Measure ................................. 5
1.2.2 Equations of State (EOS) ......................... 6
1.2.3 Other Variables and Relations .................... 7
1.2.4 Ideal Gases ..................................... 11
1.2.5 Covolume and van der Waal Gases ................. 13
1.3 Viscous Stresses ....................................... 15
1.4 Heat Conduction ........................................ 17
1.5 Integral Form of the Equations ......................... 18
1.5.1 Time Derivatives ................................ 19
1.5.2 Conservation of Mass ............................ 20
1.5.3 Conservation of Momentum ........................ 21
1.5.4 Conservation of Energy .......................... 23
1.6 Submodels .............................................. 25
1.6.1 Summary of the Equations ........................ 25
1.6.2 Flow with Area Variation ........................ 27
1.6.3 Axi-Symmetric Flows ............................. 28
1.6.4 Cylindrical and Spherical Symmetry .............. 29
1.6.5 Plain One-Dimensional Flow ...................... 29
1.6.6 Steady Compressible Flow ........................ 31
1.6.7 Viscous Compressible Flow ....................... 33
1.6.8 Free-Surface Gravity Flow ....................... 33
1.6.9 The Shallow Water Equations ..................... 35
1.6.1 Incompressible Viscous Flow ..................... 38
1.6.2 The Artificial Compressibility Equations ........ 39
2 Notions on Hyperbolic Partial Differential Equations ........ 41
2.1 Quasi-Linear Equations: Basic Concepts ................. 42
2.2 The Linear Advection Equation .......................... 47
2.2.1 Characteristics and the General Solution ........ 47
2.2.2 The Riemann Problem ............................. 49
2.3 Linear Hyperbolic Systems .............................. 50
2.3.1 Diagonalisation and Characteristic Variables .... 51
2.3.2 The General Initial-Value Problem ............... 52
2.3.3 The Riemann Problem ............................. 55
2.3.4 The Riemann Problem for Linearised Gas
Dynamics ........................................ 58
2.3.5 Some Useful Definitions ......................... 60
2.4 Conservation Laws ...................................... 61
2.4.1 Integral Forms of Conservation Laws ............. 62
2.4.2 Non-Linearities and Shock Formation ............. 66
2.4.3 Characteristic Fields ........................... 77
2.4.4 Elementary-Wave Solutions of the Riemann
Problem ......................................... 83
3 Some Properties of the Euler Equations ...................... 87
3.1 The One-Dimensional Euler Equations .................... 87
3.1.1 Conservative Formulation ........................ 87
3.1.2 Non-Conservative Formulations ................... 91
3.1.3 Elementary Wave Solutions of the Riemann
Problem ......................................... 94
3.2 Multi-Dimensional Euler Equations ..................... 103
3.2.1 Two-Dimensional Equations in Conservative
Form ........................................... 104
3.2.2 Three-Dimensional Equations in Conservative
Form .......................................... 108
3.2.3 Three-Dimensional Primitive Variable
Formulation .................................... 109
3.2.4 The Split Three-Dimensional Riemann Problem .... 111
3.3 Conservative Versus Non-Conservative Formulations ..... 112
4 The Riemann Problem for the Euler Equations ................ 115
4.1 Solution Strategy ..................................... 116
4.2 Equations for Pressure and Particle Velocity .......... 119
4.2.1 Function ƒL for a Left Shock ................... 120
4.2.2 Function ƒL for Left Rarefaction ............... 122
4.2.3 Function ƒR for a Right Shock .................. 123
4.2.4 Function ƒR for a Right Rarefaction ............ 124
4.3 Numerical Solution for Pressure ....................... 125
4.3.1 Behaviour of the Pressure Function ............. 125
4.3.2 Iterative Scheme for Finding the Pressure ...... 127
4.3.3 Numerical Tests ................................ 129
4.4 The Complete Solution ................................. 132
4.5 Sampling the Solution ................................. 136
4.5.1 Left Side of Contact: S = x/t ≤ u* ............. 137
4.5.2 Right Side of Contact: S = x/t ≥ u* ............ 137
4.6 The Riemann Problem in the Presence of Vacuum ......... 139
4.6.1 Case 1: Vacuum Right State ..................... 140
4.6.2 Case 2: Vacuum Left State ...................... 142
4.6.3 Case 3: Generation of Vacuum ................... 142
4.7 The Riemann Problem for Covolume Gases ................ 143
4.7.1 Solution for Pressure and Particle Velocity .... 144
4.7.2 Numerical Solution for Pressure ................ 147
4.7.3 The Complete Solution .......................... 147
4.7.4 Solution Inside Rarefactions ................... 148
4.8 The Split Multi-Dimensional Case ...................... 149
4.9 FORTRAN Program for Exact Riemann Solver .............. 151
5 Notions on Numerical Methods ............................... 163
5.1 Discretisation: Introductory Concepts ................. 163
5.1.1 Approximation to Derivatives ................... 164
5.1.2 Finite Difference Approximation to a PDE ....... 165
5.2 Selected Difference Schemes ........................... 168
5.2.1 The First Order Upwind Scheme .................. 168
5.2.2 Other Weil-Known Schemes ....................... 172
5.3 Conservative Methods .................................. 174
5.3.1 Basic Definitions .............................. 175
5.3.2 Godunov's First-Order Upwind Method ............ 177
5.3.3 Godunov's Method for Burgers's Equation ........ 181
5.3.4 Conservative Form of Difference Schemes ........ 184
5.4 Upwind Schemes for Linear Systems ..................... 187
5.4.1 The CIR Scheme ................................. 188
5.4.2 Godunov's Method ............................... 190
5.5 Sample Numerical Results .............................. 194
5.5.1 Linear Advection ............................... 194
5.5.2 The Inviscid Burgers Equation .................. 196
5.6 FORTRAN Program for Godunov's Method .................. 196
6 The Method of Godunov for Non-linear Systems ............... 213
6.1 Bases of Godunov's Method ............................. 213
6.2 The Godunov Scheme .................................... 216
6.3 Godunov's Method for the Euler Equations .............. 218
6.3.1 Evaluation of the Intercell Fluxes ............. 219
6.3.2 Time Step Size ................................. 221
6.3.3 Boundary Conditions ............................ 222
6.4 Numerical Results and Discussion ...................... 225
6.4.1 Numerical Results for Godunov's Method ......... 226
6.4.2 Numerical Results from Other Methods ........... 228
7 Random Choice and Related Methods .......................... 237
7.1 Introduction .......................................... 237
7.2 RCM on a Non-Staggered Grid ........................... 238
7.2.1 The Scheme for Non-Linear Systems .............. 239
7.2.2 Boundary Conditions and the Time Step Size ..... 243
7.3 A Random Choice Scheme of the Lax-Friedrichs Type ..... 244
7.3.1 Review of the Lax-Friedrichs Scheme ............ 244
7.3.2 The Scheme ..................................... 245
7.4 The RCM on a Staggered Grid ........................... 247
7.4.1 The Scheme for Non-Linear Systems .............. 247
7.4.2 A Deterministic First-Order Centred Scheme
(force) ........................................ 247
7.4.3 Analysis of the force Scheme ................... 249
7.5 Random Numbers ........................................ 250
7.5.1 Van der Corput Pseudo-Random Numbers ........... 250
7.5.2 Statistical Properties ......................... 251
7.5.3 Propagation of a Single Shock .................. 253
7.6 Numerical Results ..................................... 255
7.7 Concluding Remarks .................................... 256
8 Flux Vector Splitting Methods .............................. 265
8.1 Introduction .......................................... 265
8.2 The Flux Vector Splitting Approach .................... 266
8.2.1 Upwind Differencing ............................ 266
8.2.2 The FVS Approach ............................... 268
8.3 FVS for the Isothermal Equations ...................... 270
8.3.1 Split Fluxes ................................... 271
8.3.2 FVS Numerical Schemes .......................... 272
8.4 FVS Applied to the Euler Equations .................... 273
8.4.1 Recalling the Equations ........................ 274
8.4.2 The Steger-Warming Splitting ................... 276
8.4.3 The van Leer Splitting ......................... 277
8.4.4 The Liou-Steffen Scheme ........................ 278
8.5 Numerical Results ..................................... 280
8.5.1 Tests .......................................... 280
8.5.2 Results for Test 1 ............................. 280
8.5.3 Results for Test 2 ............................. 281
8.5.4 Results for Test 3 ............................. 281
8.5.5 Results for Test 4 ............................. 282
8.5.6 Results for Test 5 ............................. 282
9 Approximate—State Riemann Solvers .......................... 293
9.1 Introduction .......................................... 293
9.2 The Riemann Problem and the Godunov Flux .............. 294
9.2.1 Tangential Velocity Components ................. 296
9.2.2 Sonic Rarefactions ............................. 296
9.3 Primitive Variable Riemann Solvers (PVRS) ............. 297
9.4 Approximations Based on the Exact Solver .............. 301
9.4.1 A Two-Rarefaction Riemann Solver (TRRS) ........ 301
9.4.2 A Two-Shock Riemann Solver (TSRS) .............. 303
9.5 Adaptive Riemann Solvers .............................. 304
9.5.1 An Adaptive Iterative Riemann Solver (AIRS) .... 304
9.5.2 An Adaptive Noniterative Riemann Solver
(ANRS) ......................................... 305
9.6 Numerical Results ..................................... 306
10 The HLL and HLLC Riemann Solvers ........................... 315
10.1 Introduction .......................................... 315
10.2 The Riemann Problem ................................... 317
10.2.1 The Godunov Flux ............................... 317
10.2.2 Integral Relations ............................. 318
10.3 The HLL Approximate Riemann Solver .................... 320
10.4 The HLLC Approximate Riemann Solver ................... 322
10.4.1 Useful Relations ............................... 322
10.4.2 The HLLC Flux for the Euler Equations .......... 324
10.4.3 Multidimensional and Multicomponent Flow ....... 326
10.5 Wave-Speed Estimates .................................. 327
10.5.1 Direct Wave Speed Estimates .................... 328
10.5.2 Pressure-Based Wave Speed Estimates ............ 329
10.6 Summary of HLLC Fluxes ................................ 331
10.7 Contact Waves and Passive Scalars ..................... 333
10.8 Numerical Results ..................................... 334
10.9 Closing Remarks ....................................... 336
11 The Riemann Solver of Roe .................................. 345
11.1 Bases of the Roe Approach ............................. 346
11.1.1 The Exact Riemann Problem and the Godunov
Flux ........................................... 346
11.1.2 Approximate Conservation Laws .................. 347
11.1.3 The Approximate Riemann Problem and the
Intercell Flux ................................. 349
11.2 The Original Roe Method ............................... 351
11.2.1 The Isothermal Equations ....................... 352
11.2.2 The Euler Equations ............................ 354
11.3 The Roe-Pike Method ................................... 358
11.3.1 The Approach ................................... 358
11.3.2 The Isothermal Equations ....................... 359
11.3.3 The Euler Equations ............................ 363
11.4 An Entropy Fix ........................................ 366
11.4.1 The Entropy Problem ............................ 366
11.4.2 The Harten-Hyman Entropy Fix ................... 367
11.4.3 The Speeds u*, a*L, а*R ........................ 370
11.5 Numerical Results and Discussion ...................... 372
11.5.1 The Tests ...................................... 372
11.5.2 The Results .................................... 373
11.6 Extensions ............................................ 373
12 The Riemann Solver of Osher ................................ 377
12.1 Osher's Scheme for a General System ................... 378
12.1.1 Mathematical Bases ............................. 378
12.1.2 Osher's Numerical Flux ......................... 380
12.1.3 Osher's Flux for the Single-Wave Case .......... 381
12.1.4 Osher's Flux for the Inviscid Burgers
Equation ....................................... 383
12.1.5 Osher's Flux for the General Case .............. 384
12.2 Osher's Flux for the Isothermal Equations ............. 385
12.2.1 Osher's Flux with P-Ordering ................... 386
12.2.2 Osher's Flux with O-Ordering ................... 389
12.3 Osher's Scheme for the Euler Equations ................ 392
12.3.1 Osher's Flux with P-Ordering ................... 393
12.3.2 Osher's Flux with O-Ordering ................... 397
12.3.3 Remarks on Path Orderings ...................... 402
12.3.4 The Split Three-Dimensional Case ............... 403
12.4 Numerical Results and Discussion ...................... 404
12.5 Extensions ............................................ 406
13 High-Order and TVD Methods for Scalar Equations ............ 413
13.1 Introduction .......................................... 413
13.2 Basic Properties of Selected Schemes .................. 415
13.2.1 Selected Schemes ............................... 415
13.2.2 Accuracy ....................................... 417
13.2.3 Stability ...................................... 418
13.3 WAF-Type High Order Schemes ........................... 420
13.3.1 The Basic waf Scheme ........................... 420
13.3.2 Generalisations of the waf Scheme .............. 423
13.4 MUSCL-Type High-Order Methods ......................... 426
13.4.1 Data Reconstruction ............................ 426
13.4.2 The MUSCL-Hancock Method (MHM) ................. 429
13.4.3 The Piece-Wise Linear Method (PLM) ............. 432
13.4.4 The Generalised Riemann Problem (GRP) Method ... 434
13.4.5 Slope-Limiter Centred (SLIC) Schemes ........... 436
13.4.6 Other Approaches ............................... 439
13.4.7 Semi-Discrete Schemes .......................... 439
13.4.8 Implicit Methods ............................... 439
13.5 Monotone Schemes and Accuracy ......................... 440
13.5.1 Monotone Schemes ............................... 440
13.5.2 A Motivating Example ........................... 443
13.5.3 Monotone Schemes and Godunov's Theorem ......... 447
13.5.4 Spurious Oscillations and High Resolution ...... 448
13.5.5 Data Compatibility ............................. 449
13.6 Total Variation Diminishing (TVD) Methods ............. 451
13.6.1 The Total Variation ............................ 452
13.6.2 TVD and Monotonicity Preserving Schemes ........ 453
13.7 Flux Limiter Methods .................................. 456
13.7.1 TVD Version of the waf Method ................. 456
13.7.2 The General Flux-Limiter Approach .............. 464
13.7.3 TVD Upwind Flux Limiter Schemes ................ 469
13.7.4 TVD Centred Flux Limiter Schemes ............... 474
13.8 Slope Limiter Methods ................................. 480
13.8.1 TVD Conditions ................................. 480
13.8.2 Construction of TVD Slopes ..................... 481
13.8.3 Slope Limiters ................................. 482
13.8.4 Limited Slopes Obtained from Flux Limiters ..... 484
13.9 Extensions of TVD Methods ............................. 486
13.9.1 TVD Schemes in the Presence of Source Terms .... 486
13.9.2 TVD Schemes in the Presence of Diffusion
Terms .......................................... 486
13.10 Numerical Results for Linear Advection ............... 487
14 High-Order and TVD Schemes for Non-Linear Systems .......... 493
14.1 Introduction ......................................... 493
14.2 CFL and Boundary Conditions .......................... 495
14.3 Weighted Average Flux (WAF) Schemes .................. 496
14.3.1 The Original Version of WAF .................... 496
14.3.2 A Weighted Average State Version ............... 498
14.3.3 Rarefactions in State Riemann Solvers .......... 499
14.3.4 TVD Version of waf Schemes ..................... 501
14.3.5 Riemann Solvers ................................ 503
14.3.6 Summary of the waf Method ...................... 503
14.4 The MUSCL-Hancock Scheme .............................. 504
14.4.1 The Basic Scheme ............................... 504
14.4.2 A Variant of the Scheme ........................ 506
14.4.3 TVD Version of the Scheme ...................... 507
14.4.4 Summary of the MUSCL-Hancock Method ............ 510
14.5 Centred TVD Schemes ................................... 511
14.5.1 Review of the force Flux ....................... 512
14.5.2 A Flux Limiter Centred (FLIC) Scheme ........... 512
14.5.3 A Slope Limiter Centred (SLIC) Scheme .......... 514
14.6 Primitive-Variable Schemes ............................ 515
14.6.1 Formulation of the Equations and Primitive
Schemes ........................................ 515
14.6.2 A WAF-Type Primitive Variable Scheme ........... 517
14.6.3 A MUSCL-Hancock Primitive Scheme ............... 520
14.6.4 Adaptive Primitive-Conservative Schemes ........ 522
14.7 Some Numerical Results ................................ 523
14.7.1 Upwind TVD Methods ............................. 523
14.7.2 Centred TVD Methods ............................ 524
15 Splitting Schemes for PDEs with Source Terms ............... 531
15.1 Introduction .......................................... 531
15.2 Splitting for a Model Equation ........................ 533
15.3 Numerical Methods Based on Splitting .................. 535
15.3.1 Model Equations ................................ 535
15.3.2 Schemes for Systems ............................ 536
15.4 Remarks on ODE Solvers ................................ 537
15.4.1 First-Order Systems of ODEs .................... 537
15.4.2 Numerical Methods .............................. 539
15.4.3 Implementation Details for Split Schemes ....... 540
15.5 Concluding Remarks .................................... 541
16 Methods for Multi-Dimensional PDEs ......................... 543
16.1 Introduction .......................................... 543
16.2 Dimensional Splitting ................................. 544
16.2.1 Splitting for a Model Problem .................. 544
16.2.2 Splitting Schemes for Two-Dimensional Systems .. 545
16.2.3 Splitting Schemes for Three-Dimensional
Systems ........................................ 547
16.3 Practical Implementation of Splitting Schemes in
Three Dimensions ...................................... 549
16.3.1 Handling the Sweeps by a Single Subroutine ..... 549
16.3.2 Choice of Time Step Size ....................... 551
16.3.3 The Intercell Flux and the tvd Condition ....... 552
16.4 Unsplit Finite Volume Methods ......................... 555
16.4.1 Introductory Concepts .......................... 555
16.4.2 Accuracy and Stability of Multidimensional
Schemes ........................................ 558
16.5 A MusCL-Hancock Finite Volume Scheme .................. 561
16.6 WAF-Type Finite Volume Schemes ........................ 563
16.6.1 Two-Dimensional Linear Advection ............... 564
16.6.2 Three-Dimensional Linear Advection ............. 567
16.6.3 Schemes for Two-Dimensional Nonlinear
Systems ........................................ 570
16.6.4 Schemes for Three-Dimensional Nonlinear
Systems ........................................ 573
16.7 Non-Cartesian Geometries .............................. 574
16.7.1 Introduction ................................... 574
16.7.2 General Domains and Coordinate
Transformation ................................. 575
16.7.3 The Finite Volume Method for Non-Cartesian
Domains ........................................ 578
17 Multidimensional Test Problems ............................. 585
17.1 Explosions and Implosions ............................. 586
17.1.1 Explosion Test in Two-Space Dimensions ......... 587
17.1.2 Explosion Test in Three Space Dimensions ....... 590
17.2 Shock Wave Reflection from a Wedge .................... 591
18 FORCE Fluxes in Multiple Space Dimensions .................. 597
18.1 Introduction .......................................... 597
18.2 Review of FORCE in One Space Dimension ................ 600
18.2.1 FORCE and Related Fluxes ....................... 600
18.2.2 Monotonicity and Numerical Viscosity ........... 602
18.3 FORCE Schemes on Cartesian Meshes ..................... 605
18.3.1 The Two-Dimensional Case ....................... 605
18.3.2 The Three-Dimensional Case ..................... 609
18.4 Properties of the FORCE Schemes ....................... 610
18.4.1 One-Dimensional Interpretation ................. 610
18.4.2 Some Numerical Experiments ..................... 611
18.4.3 Analysis in Multiple Space Dimensions .......... 613
18.5 FORCE Schemes on General Meshes ....................... 617
18.6 Sample Numerical Results .............................. 621
18.7 Concluding Remarks .................................... 621
19 The Generalized Riemann Problem ............................ 625
19.1 Introduction .......................................... 625
19.2 Statement of the Problem .............................. 629
19.3 The Cauchy-Kowalewski Theorem ......................... 631
19.3.1 Series Expansions and Analytic Functions ....... 632
19.3.2 Illustration of the Cauchy-Kowalewski
Theorem ........................................ 633
19.3.3 The Cauchy-Kowalewski Method ................... 633
19.4 A Method of Solution .................................. 635
19.4.1 The Leading Term ............................... 636
19.4.2 Higher-Order Terms ............................. 637
19.4.3 Source Terms ................................... 640
19.4.4 Summary: Numerical Flux and Numerical Source ... 640
19.4.5 Some remarks ................................... 642
19.5 Examples .............................................. 642
19.5.1 The Linear Advection Equation .................. 643
19.5.2 Linear Advection with a Source Term ............ 645
19.5.3 Non-Linear Equation with a Source Term ......... 646
19.5.4 The Burgers Equation with a Source Term ........ 648
19.6 Other Solvers ......................................... 651
19.7 Concluding Remarks .................................... 653
20 The ADER Approach .......................................... 655
20.1 Introduction .......................................... 655
20.2 The Finite Volume Method .............................. 657
20.2.1 The Framework .................................. 657
20.2.2 The Numerical Flux ............................. 658
20.2.3 The Numerical Source ........................... 659
20.2.4 Reconstruction ................................. 660
20.3 Second-Order Scheme for a Model Equation .............. 663
20.3.1 Numerical Flux and Numerical Source ............ 663
20.3.2 The Scheme ..................................... 666
20.4 Schemes of Arbitrary Accuracy ......................... 667
20.4.1 The Numerical Flux ............................. 667
20.4.2 The Numerical Source ........................... 668
20.4.3 Summary ........................................ 668
20.5 Sample Numerical Results .............................. 669
20.5.1 Long-Time Advection of Smooth Profiles ......... 669
20.5.2 Convergence Rates .............................. 672
20.6 Concluding Remarks .................................... 673
21 Concluding Remarks ......................................... 679
21.1 Summary of Numerical Aspects .......................... 679
21.2 Potential Applications ................................ 681
21.3 Current Research Topics ............................... 685
21.4 The NUMERICA Library .................................. 686
References .................................................... 687
Index ......................................................... 719
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