PART ONE. PRELIMINARIES
1 Introduction 3
1.1 General 3
1.1.1 Historical Background 3
1.1.2 Organization of Text 4
1.2 One-Dimensional Computations by Finite Difference Methods 6
1.3 One-Dimensional Computations by Finite Element Methods 7
1.4 One-Dimensional Computations by Finite Volume Methods 11
1.4.1 FVM via FDM 11
1.4.2 FVM via FEM 13
1.5 Neumann Boundary Conditions 13
1.5.1 FDM 14
1.5.2 FEM 15
1.5.3 FVM via FDM 15
1.5.4 FVM via FEM 16
1.6 Example Problems 17
1.6.1 Dirichlet Boundary Conditions 17
1.6.2 Neumann Boundary Conditions 20
1.7 Summary 24
References 26
2 Governing Equations 29
2.1 Classification of Partial Differential Equations 29
2.2 Navier-Stokes System of Equations 33
2.3 Boundary Conditions 38
2.4 Summary 41
References 42
PART TWO. FINITE DIFFERENCE METHODS
3 Derivation of Finite Difference Equations 45
3.1 Simple Methods 45
3.2 General Methods 46
3.3 Higher Order Derivatives 50
vii
viii CONTENTS
3.4 Multidimensional Finite Difference Formulas 53
3.5 Mixed Derivatives 57
3.6 Nonuniform Mesh 59
3.7 Higher Order Accuracy Schemes 60
3.8 Accuracy of Finite Difference Solutions 61
3.9 Summary 62
References 62
4 Solution Methods of Finite Difference Equations 63
4.1 Elliptic Equations 63
4.1.1 Finite Difference Formulations 63
4.1.2 Iterative Solution Methods 65
4.1.3 Direct Method with Gaussian Elimination 67
4.2 Parabolic Equations 67
4.2.1 Explicit Schemes and von Neumann Stability Analysis 68
4.2.2 Implicit Schemes 71
4.2.3 Alternating Direction Implicit (ADI) Schemes 72
4.2.4 Approximate Factorization 73
4.2.5 Fractional Step Methods 75
4.2.6 Three Dimensions 75
4.2.7 Direct Method with Tridiagonal Matrix Algorithm 76
4.3 Hyperbolic Equations 77
4.3.1 Explicit Schemes and Von Neumann Stability Analysis 77
4.3.2 Implicit Schemes 81
4.3.3 Multistep (Splitting, Predictor-Corrector) Methods 81
4.3.4 Nonlinear Problems 83
4.3.5 Second Order One-DimensionalWave Equations 87
4.4 Burgers’ Equation 87
4.4.1 Explicit and Implicit Schemes 88
4.4.2 Runge-***** Method 90
4.5 Algebraic Equation Solvers and Sources of Errors 91
4.5.1 Solution Methods 91
4.5.2 Evaluation of Sources of Errors 91
4.6 Coordinate Transformation for Arbitrary Geometries 94
4.6.1 Determination of Jacobians and Transformed Equations 94
4.6.2 Application of Neumann Boundary Conditions 97
4.6.3 Solution by MacCormack Method 98
4.7 Example Problems 98
4.7.1 Elliptic Equation (Heat Conduction) 98
4.7.2 Parabolic Equation (Couette Flow) 100
4.7.3 Hyperbolic Equation (First OrderWave Equation) 101
4.7.4 Hyperbolic Equation (Second OrderWave Equation) 103
4.7.5 NonlinearWave Equation 104
4.8 Summary 105
References 105
5 Incompressible Viscous Flows via Finite Difference Methods 106
5.1 General 106
5.2 Artificial Compressibility Method 107
CONTENTS ix
5.3 Pressure Correction Methods 108
5.3.1 Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) 108
5.3.2 Pressure Implicit with Splitting of Operators 112
5.3.3 Marker-and-Cell (MAC) Method 115
5.4 Vortex Methods 115
5.5 Summary 118
References 119
6 Compressible Flows via Finite Difference Methods 120
6.1 Potential Equation 121
6.1.1 Governing Equations 121
6.1.2 Subsonic Potential Flows 123
6.1.3 Transonic Potential Flows 123
6.2 Euler Equations 129
6.2.1 Mathematical Properties of Euler Equations 130
6.2.1.1 Quasilinearization of Euler Equations 130
6.2.1.2 Eigenvalues and Compatibility Relations 132
6.2.1.3 Characteristic Variables 134
6.2.2 Central Schemes with Combined Space-Time Discretization 136
6.2.2.1 Lax-Friedrichs First Order Scheme 138
6.2.2.2 Lax-Wendroff Second Order Scheme 138
6.2.2.3 Lax-Wendroff Method with Artificial Viscosity 139
6.2.2.4 Explicit MacCormack Method 140
6.2.3 Central Schemes with Independent Space-Time Discretization 141
6.2.4 First Order Upwind Schemes 142
6.2.4.1 Flux Vector Splitting Method 142
6.2.4.2 Godunov Methods 145
6.2.5 Second Order Upwind Schemes with Low Resolution 148
6.2.6 Second Order Upwind Schemes with High Resolution
(TVD Schemes) 150
6.2.7 Essentially Nonoscillatory Scheme 163
6.2.8 Flux-Corrected Transport Schemes 165
6.3 Navier-Stokes System of Equations 166
6.3.1 Explicit Schemes 167
6.3.2 Implicit Schemes 169
6.3.3 PISO Scheme for Compressible Flows 175
6.4 Preconditioning Process for Compressible and Incompressible
Flows 178
6.4.1 General 178
6.4.2 Preconditioning Matrix 179
6.5 Flowfield-Dependent Variation Methods 180
6.5.1 Basic Theory 180
6.5.2 Flowfield-Dependent Variation Parameters 183
6.5.3 FDV Equations 185
6.5.4 Interpretation of Flowfield-Dependent Variation Parameters 187
6.5.5 Shock-Capturing Mechanism 188
6.5.6 Transitions and Interactions between Compressible
and Incompressible Flows 191
x CONTENTS
6.5.7 Transitions and Interactions between Laminar
and Turbulent Flows 193
6.6 Other Methods 195
6.6.1 Artificial Viscosity Flux Limiters 195
6.6.2 Fully Implicit High Order Accurate Schemes 196
6.6.3 Point Implicit Methods 197
6.7 Boundary Conditions 197
6.7.1 Euler Equations 197
6.7.1.1 One-Dimensional Boundary Conditions 197
6.7.1.2 Multi-Dimensional Boundary Conditions 204
6.7.1.3 Nonreflecting Boundary Conditions 204
6.7.2 Navier-Stokes System of Equations 205
6.8 Example Problems 207
6.8.1 Solution of Euler Equations 207
6.8.2 Triple ShockWave Boundary Layer Interactions Using
FDV Theory 208
6.9 Summary 213
References 214
7 Finite Volume Methods via Finite Difference Methods 218
7.1 General 218
7.2 Two-Dimensional Problems 219
7.2.1 Node-Centered Control Volume 219
7.2.2 Cell-Centered Control Volume 223
7.2.3 Cell-Centered Average Scheme 225
7.3 Three-Dimensional Problems 227
7.3.1 3-D Geometry Data Structure 227
7.3.2 Three-Dimensional FVM Equations 232
7.4 FVM-FDV Formulation 234
7.5 Example Problems 239
7.6 Summary 239
References 239
PART THREE. FINITE ELEMENT METHODS
8 Introduction to Finite Element Methods 243
8.1 General 243
8.2 Finite Element Formulations 245
8.3 Definitions of Errors 254
8.4 Summary 259
References 260
9 Finite Element Interpolation Functions 262
9.1 General 262
9.2 One-Dimensional Elements 264
9.2.1 Conventional Elements 264
9.2.2 Lagrange Polynomial Elements 269
9.2.3 Hermite Polynomial Elements 271
9.3 Two-Dimensional Elements 273
9.3.1 Triangular Elements 273
CONTENTS xi
9.3.2 Rectangular Elements 284
9.4 Three-Dimensional Elements 298
9.4.1 Tetrahedral Elements 298
9.4.2 Triangular Prism Elements 302
9.4.3 Hexahedral Isoparametric Elements 303
9.5 Axisymmetric Ring Elements 305
9.6 Lagrange and Hermite Families and Convergence Criteria 306
9.7 Summary 308
References 308
10 Linear Problems 309
10.1 Steady-State Problems – Standard Galerkin Methods 309
10.1.1 Two-Dimensional Elliptic Equations 309
10.1.2 Boundary Conditions in Two Dimensions 315
10.1.3 Solution Procedure 320
10.1.4 Stokes Flow Problems 324
10.2 Transient Problems – Generalized Galerkin Methods 327
10.2.1 Parabolic Equations 327
10.2.2 Hyperbolic Equations 332
10.2.3 Multivariable Problems 334
10.2.4 Axisymmetric Transient Heat Conduction 335
10.3 Solutions of Finite Element Equations 337
10.3.1 Conjugate Gradient Methods (CGM) 337
10.3.2 Element-by-Element (EBE) Solutions of FEM Equations 340
10.4 Example Problems 342
10.4.1 Solution of Poisson Equation with Isoparametric Elements 342
10.4.2 Parabolic Partial Differential Equation in Two Dimensions 343
10.5 Summary 346
References 346
11 Nonlinear Problems/Convection-Dominated Flows 347
11.1 Boundary and Initial Conditions 347
11.1.1 Incompressible Flows 348
11.1.2 Compressible Flows 353
11.2 Generalized Galerkin Methods and Taylor-Galerkin Methods 355
11.2.1 Linearized Burgers’ Equations 355
11.2.2 Two-Step Explicit Scheme 358
11.2.3 Relationship between FEM and FDM 362
11.2.4 Conversion of Implicit Scheme into Explicit Scheme 365
11.2.5 Taylor-Galerkin Methods for Nonlinear Burgers’ Equations 366
11.3 Numerical Diffusion Test Functions 367
11.3.1 Derivation of Numerical Diffusion Test Functions 368
11.3.2 Stability and Accuracy of Numerical Diffusion Test Functions 369
11.3.3 Discontinuity-Capturing Scheme 376
11.4 Generalized Petrov-Galerkin (GPG) Methods 377
11.4.1 Generalized Petrov-Galerkin Methods for Unsteady
Problems 377
11.4.2 Space-Time Galerkin/Least Squares Methods 378
xii CONTENTS
11.5 Solutions of Nonlinear and Time-Dependent Equations
and Element-by-Element Approach 380
11.5.1 Newton-Raphson Methods 380
11.5.2 Element-by-Element Solution Scheme for Nonlinear
Time Dependent FEM Equations 381
11.5.3 Generalized Minimal Residual Algorithm 384
11.6 Example Problems 391
11.6.1 NonlinearWave Equation (Convection Equation) 391
11.6.2 Pure Convection in Two Dimensions 391
11.6.3 Solution of 2-D Burgers’ Equation 394
11.7 Summary 396
References 396
12 Incompressible Viscous Flows via Finite Element Methods 399
12.1 Primitive Variable Methods 399
12.1.1 Mixed Methods 399
12.1.2 Penalty Methods 400
12.1.3 Pressure Correction Methods 401
12.1.4 Generalized Petrov-Galerkin Methods 402
12.1.5 Operator Splitting Methods 403
12.1.6 Semi-Implicit Pressure Correction 405
12.2 Vortex Methods 406
12.2.1 Three-Dimensional Analysis 407
12.2.2 Two-Dimensional Analysis 410
12.2.3 Physical Instability in Two-Dimensional
Incompressible Flows 411
12.3 Example Problems 413
12.4 Summary 416
References 416
13 Compressible Flows via Finite Element Methods 418
13.1 Governing Equations 418
13.2 Taylor-Galerkin Methods and Generalized Galerkin Methods 422
13.2.1 Taylor-Galerkin Methods 422
13.2.2 Taylor-Galerkin Methods with Operator Splitting 425
13.2.3 Generalized Galerkin Methods 427
13.3 Generalized Petrov-Galerkin Methods 428
13.3.1 Navier-Stokes System of Equations in Various Variable Forms 428
13.3.2 The GPG with Conservation Variables 431
13.3.3 The GPG with Entropy Variables 433
13.3.4 The GPG with Primitive Variables 434
13.4 Characteristic Galerkin Methods 435
13.5 Discontinuous Galerkin Methods or Combined FEM/FDM/FVM
Methods 438
13.6 Flowfield-Dependent Variation Methods 440
13.6.1 Basic Formulation 440
13.6.2 Interpretation of FDV Parameters Associated with Jacobians 443
13.6.3 Numerical Diffusion 445
CONTENTS xiii
13.6.4 Transitions and Interactions between Compressible
and Incompressible Flows and between Laminar
and Turbulent Flows 446
13.6.5 Finite Element Formulation of FDV Equations 447
13.6.6 Boundary Conditions 449
13.7 Example Problems 452
13.8 Summary 459
References 460
14 Miscellaneous Weighted Residual Methods 462
14.1 Spectral Element Methods 462
14.1.1 Spectral Functions 463
14.1.2 Spectral Element Formulations by Legendre Polynomials 467
14.1.3 Two-Dimensional Problems 471
14.1.4 Three-Dimensional Problems 475
14.2 Least Squares Methods 478
14.2.1 LSM Formulation for the Navier-Stokes System of Equations 478
14.2.2 FDV-LSM Formulation 480
14.2.3 Optimal Control Method 480
14.3 Finite Point Method (FPM) 481
14.4 Example Problems 483
14.4.1 Sharp Fin Induced ShockWave Boundary Layer Interactions 483
14.4.2 Asymmetric Double Fin Induced Shock Wave Boundary Layer
Interaction 486
14.5 Summary 489
References 489
15 Finite Volume Methods via Finite Element Methods 491
15.1 General 491
15.2 Formulations of Finite Volume Equations 492
15.2.1 Burgers’ Equations 492
15.2.2 Incompressible and Compressible Flows 500
15.2.3 Three-Dimensional Problems 502
15.3 Example Problems 503
15.4 Summary 507
References 508
16 Relationships between Finite Differences and Finite Elements
and Other Methods 509
16.1 Simple Comparisons between FDM and FEM 510
16.2 Relationships between FDM and FDV 514
16.3 Relationships between FEM and FDV 518
16.4 Other Methods 522
16.4.1 Boundary Element Methods 522
16.4.2 Coupled Eulerian-Lagrangian Methods 525
16.4.3 Particle-in-Cell (PIC) Method 528
16.4.4 Monte Carlo Methods (MCM) 528
16.5 Summary 530
References 530
xiv CONTENTS
PART FOUR. AUTOMATIC GRID GENERATION, ADAPTIVE METHODS,
AND COMPUTING TECHNIQUES
17 Structured Grid Generation 533
17.1 Algebraic Methods 533
17.1.1 Unidirectional Interpolation 533
17.1.2 Multidirectional Interpolation 537
17.1.2.1 Domain Vertex Method 537
17.1.2.2 Transfinite Interpolation Methods (TFI) 545
17.2 PDE Mapping Methods 551
17.2.1 Elliptic Grid Generator 551
17.2.1.1 Derivation of Governing Equations 551
17.2.1.2 Control Functions 557
17.2.2 Hyperbolic Grid Generator 558
17.2.2.1 Cell Area (Jacobian) Method 560
17.2.2.2 Arc-Length Method 561
17.2.3 Parabolic Grid Generator 562
17.3 Surface Grid Generation 562
17.3.1 Elliptic PDE Methods 563
17.3.1.1 Differential Geometry 563
17.3.1.2 Surface Grid Generation 567
17.3.2 Algebraic Methods 569
17.3.2.1 Points and Curves 569
17.3.2.2 Elementary and Global Surfaces 573
17.3.2.3 Surface Mesh Generation 574
17.4 Multiblock Structured Grid Generation 577
17.5 Summary 580
References 580
18 Unstructured Grid Generation 581
18.1 Delaunay-Voronoi Methods 581
18.1.1 Watson Algorithm 582
18.1.2 Bowyer Algorithm 587
18.1.3 Automatic Point Generation Scheme 590
18.3 Combined DVM and AFM 596
18.4 Three-Dimensional Applications 597
18.4.1 DVM in 3-D 597
18.4.2 AFM in 3-D 598
18.4.3 Curved Surface Grid Generation 599
18.4.4 Example Problems 599
18.5 Other Approaches 600
18.5.1 AFM Modified for Quadrilaterals 601
18.5.2 Iterative Paving Method 603
18.5.3 Quadtree and Octree Method 604
18.6 Summary 605
References 605
CONTENTS xv
19.1.1 Control Function Methods 607
19.1.1.1 Basic Theory 607
19.1.1.2 Weight Functions in One Dimension 609
19.1.1.3 Weight Function in Multidimensions 611
19.1.2 Variational Methods 612
19.1.2.1 Variational Formulation 612
19.1.2.2 Smoothness Orthogonality and Concentration 613
19.1.3 Multiblock Adaptive Structured Grid Generation 617
19.2.1 Mesh Refinement Methods (h-Methods) 618
19.2.1.1 Error Indicators 618
19.2.1.3 Three-Dimensional Hexahedral Element 624
19.2.2 Mesh Movement Methods (r-Methods) 629
19.2.3 Combined Mesh Refinement and Mesh Movement Methods
(hr-Methods) 630
19.2.4 Mesh Enrichment Methods (p-Method) 634
19.2.5 Combined Mesh Refinement and Mesh Enrichment Methods
(hp-Methods) 635
19.2.6 Unstructured Finite Difference Mesh Refinements 640
19.3 Summary 642
References 642
20 Computing Techniques 644
20.1 Domain Decomposition Methods 644
20.1.1 Multiplicative Schwarz Procedure 645
20.2 Multigrid Methods 651
20.2.1 General 651
20.2.2 Multigrid Solution Procedure on Structured Grids 651
20.2.3 Multigrid Solution Procedure on Unstructured Grids 655
20.3 Parallel Processing 656
20.3.1 General 656
20.3.2 Development of Parallel Algorithms 657
20.3.3 Parallel Processing with Domain Decomposition and Multigrid
Methods 661
20.4 Example Problems 666
20.4.1 Solution of Poisson Equation with Domain Decomposition
Parallel Processing 666
20.4.2 Solution of Navier-Stokes System of Equations
20.5 Summary 673
References 674
PART FIVE. APPLICATIONS
21 Applications to Turbulence 679
21.1 General 679
xvi CONTENTS
21.2 Governing Equations 680
21.3 Turbulence Models 683
21.3.1 Zero-Equation Models 683
21.3.2 One-Equation Models 686
21.3.3 Two-Equation Models 686
21.3.4 Second Order Closure Models (Reynolds Stress Models) 690
21.3.5 Algebraic Reynolds Stress Models 692
21.3.6 Compressibility Effects 693
21.4 Large Eddy Simulation 696
21.4.1 Filtering, Subgrid Scale Stresses, and Energy Spectra 696
21.4.2 The LES Governing Equations for Compressible Flows 699
21.4.3 Subgrid Scale Modeling 699
21.5 Direct Numerical Simulation 703
21.5.1 General 703
21.5.2 Various Approaches to DNS 704
21.6 Solution Methods and Initial and Boundary Conditions 705
21.7 Applications 706
21.7.1 Turbulence Models for Reynolds Averaged Navier-Stokes
(RANS) 706
21.7.2 Large Eddy Simulation (LES) 708
21.7.3 Direct Numerical Simulation (DNS) for Compressible Flows 716
21.8 Summary 718
References 721
22 Applications to Chemically Reactive Flows and Combustion 724
22.1 General 724
22.2 Governing Equations in Reactive Flows 725
22.2.1 Conservation of Mass for Mixture and Chemical Species 725
22.2.2 Conservation of Momentum 729
22.2.3 Conservation of Energy 730
22.2.4 Conservation Form of Navier-Stokes System of Equations
in Reactive Flows 732
22.2.5 Two-Phase Reactive Flows (Spray Combustion) 736
22.2.6 Boundary and Initial Conditions 738
22.3 Chemical Equilibrium Computations 740
22.3.1 Solution Methods of Stiff Chemical Equilibrium Equations 740
22.3.2 Applications to Chemical Kinetics Calculations 744
22.4 Chemistry-Turbulence Interaction Models 745
22.4.1 Favre-Averaged Diffusion Flames 745
22.4.2 Probability Density Functions 748
22.4.3 Modeling for Energy and Species Equations
in Reactive Flows 753
22.4.4 SGS Combustion Models for LES 754
22.5 Hypersonic Reactive Flows 756
22.5.1 General 756
22.5.2 Vibrational and Electronic Energy in Nonequilibrium 758
22.6 Example Problems 765
22.6.1 Supersonic Inviscid Reactive Flows (Premixed Hydrogen-Air) 765
CONTENTS xvii
22.6.2 Turbulent Reactive Flow Analysis with Various RANS Models 770
22.6.3 PDF Models for Turbulent Diffusion Combustion Analysis 775
22.6.4 Spectral Element Method for Spatially Developing Mixing Layer 778
22.6.5 Spray Combustion Analysis with Eulerian-Lagrangian
Formulation 778
22.6.6 LES and DNS Analyses for Turbulent Reactive Flows 782
22.6.7 Hypersonic Nonequilibrium Reactive Flows with Vibrational
and Electronic Energies 788
22.7 Summary 792
References 792
23 Applications to Acoustics 796
23.1 Introduction 796
23.2 Pressure Mode Acoustics 798
23.2.1 Basic Equations 798
23.2.2 Kirchhoff’s Method with Stationary Surfaces 799
23.2.3 Kirchhoff’s Method with Subsonic Surfaces 800
23.2.4 Kirchhoff’s Method with Supersonic Surfaces 800
23.3 Vorticity Mode Acoustics 801
23.3.1 Lighthill’s Acoustic Analogy 801
23.3.2 FfowcsWilliams-Hawkings Equation 802
23.4 Entropy Mode Acoustics 803
23.4.1 Entropy Energy Governing Equations 803
23.4.2 Entropy Controlled Instability (ECI) Analysis 804
23.4.3 Unstable EntropyWaves 806
23.5 Example Problems 808
23.5.1 Pressure Mode Acoustics 808
23.5.2 Vorticity Mode Acoustics 822
23.5.3 Entropy Mode Acoustics 829
23.6 Summary 837
References 838
24 Applications to Combined Mode Radiative Heat Transfer 841
24.1 General 841
24.2 Radiative Heat Transfer in Nonparticipating Media 845
24.2.1 Diffuse Interchange in an Enclosure 845
24.2.2 View Factors 848
24.2.4 Solution Methods for Integrodifferential Radiative Heat Transfer
Equation 863
24.3 Radiative Heat Transfer in Participating Media 864
24.3.1 Combined Conduction and Radiation 864
24.3.2 Combined Conduction, Convection, and Radiation 871
24.3.3 Three-Dimensional Radiative Heat Flux Integral Formulation 882
24.4 Example Problems 886
24.4.1 Nonparticipating Media 886
24.4.2 Solution of Radiative Heat Transfer Equation in Nonparticipating
Media 888
24.4.3 Participating Media with Conduction and Radiation 892
xviii CONTENTS
24.4.4 Participating Media with Conduction, Convection,
24.4.5 Three-Dimensional Radiative Heat Flux Integration
Formulation 896
24.5 Summary 900
References 901
25 Applications to Multiphase Flows 902
25.1 General 902
25.2 Volume of Fluid Formulation with Continuum Surface Force 904
25.2.1 Navier-Stokes System of Equations 904
25.2.2 Surface Tension 906
25.2.3 Surface and Volume Forces 908
25.2.4 Implementation of Volume Force 910
25.2.5 Computational Strategies 911
25.3 Fluid-Particle Mixture Flows 913
25.3.1 Laminar Flows in Fluid-Particle Mixture with Rigid Body Motions
of Solids 913
25.3.2 Turbulent Flows in Fluid-Particle Mixture 916
25.3.3 Reactive Turbulent Flows in Fluid-Particle Mixture 917
25.4 Example Problems 920
25.4.1 Laminar Flows in Fluid-Particle Mixture 920
25.4.2 Turbulent Flows in Fluid-Particle Mixture 921
25.4.3 Reactive Turbulent Flows in Fluid-Particle Mixture 922
25.5 Summary 924
References 924
26 Applications to Electromagnetic Flows 927
26.1 Magnetohydrodynamics 927
26.2 Rarefied Gas Dynamics 931
26.2.1 Basic Equations 931
26.2.2 Finite Element Solution of Boltzmann Equation 933
26.3 Semiconductor Plasma Processing 936
26.3.1 Introduction 936
26.3.2 Charged Particle Kinetics in Plasma Discharge 939
26.3.3 Discharge Modeling with Moment Equations 943
26.3.4 Reactor Model for ChemicalVapor Deposition (CVD) Gas Flow 945
26.4 Applications 946
26.4.1 Applications to Magnetohydrodynamic Flows in Corona Mass
Ejection 946
26.4.2 Applications to Plasma Processing in Semiconductors 946
26.5 Summary 951
References 953
27 Applications to Relativistic Astrophysical Flows 955
27.1 General 955
27.2 Governing Equations in Relativistic Fluid Dynamics 956
27.2.1 Relativistic Hydrodynamics Equations in Ideal Flows 956
27.2.2 Relativistic Hydrodynamics Equations in Nonideal Flows 958
27.2.3 Pseudo-Newtonian Approximations with Gravitational Effects 963
CONTENTS xix
27.3 Example Problems 964
27.3.1 Relativistic Shock Tube 964
27.3.2 Black Hole Accretion 965
27.3.3 Three-Dimensional Relativistic Hydrodynamics 966
27.3.4 Flowfield Dependent Variation (FDV) Method for Relativistic
Astrophysical Flows 967
27.4 Summary 973
References 974
APPENDIXES