Relativistic hydrodynamics:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
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Oxford
Oxford Univ. Press
2013
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| Edition: | 1. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. [682] - 720 |
| Physical Description: | XV, 735 S. Ill., graph. Darst. |
| ISBN: | 9780198528906 |
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| 245 | 1 | 0 | |a Relativistic hydrodynamics |c Luciano Rezzolla ; Olindo Zanotti |
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| 264 | 1 | |a Oxford |b Oxford Univ. Press |c 2013 | |
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Record in the Search Index
| _version_ | 1804150547960299520 |
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| adam_text | Titel: Relativistic hydrodynamics
Autor: Rezzolla, Luciano
Jahr: 2013
Contents
PART I THE PHYSICS OF RELATIVISTIC HYDRODYNAMICS
1 A Brief Review of General Relativity 3
1.1 Why this chapter? 3
1.2 The concept of spacetime 4
1.3 Spacetime as a manifold 5
1.3.1 Coordinates 6
1.3.2 Curves and paths 9
1.3.3 Tangent vectors 9
1.3.4 Gradients of a function 12
1.3.5 A geometrical view of vectors and covectors 13
1.3.6 Tensors 15
1.3.7 Tensor algebra 17
1.3.8 The most important tensor: the metric 20
1.3.9 Splitting a tensor through a vector 23
1.4 Flat spacetime: special relativity 24
1.5 Curved spacetimes: general relativity 29
1.5.1 Lie derivative 31
1.5.2 Covariant derivative and Christoffel symbols 34
1.5.3 Symmetries and Killing vector fields 37
1.5.4 Geodesic equation 38
1.5.5 The Riemann tensor 41
1.6 Einstein equations 46
1.7 Spacetimes of astrophysical relevance 48
1.7.1 Non-rotating black holes: the Schwarzschild solution 49
1.7.2 Rotating black holes: the Kerr solution 54
1.7.3 The Friedmann-Robertson-Walker metric 59
1.8 Gravitational radiation 61
1.9 Further reading 66
1.10 Problems 67
2 A Kinetic-Theory Description of Fluids 68
2.1 On the fluid approximation 68
2.2 Newtonian kinetic theory 70
2.2.1 The Boltzmann equation 70
2.2.2 The H -theorem 75
2.2.3 The moment equations 78
2.2.4 The Maxwell-Boltzmann equilibrium distribution 81
2.2.5 The zero-order approximation: perfect fluids 84
2.2.6 The first-order approximation: non-perfect fluids 87
2.3 Relativistic kinetic theory 89
2.3.1 The relativistic Boltzmann equation 90
2.3.2 Relativistic transport fluxes 91
2.3.3 The relativistic if-theorem 93
2.3.4 The relativistic moment equations 95
2.3.5 The general-relativistic hydrodynamic equations 96
2.3.6 Relativistic equilibrium distributions 97
2.3.7 The laws of thermodynamics 101
2.4 Equations of state 103
2.4.1 Degenerate relativistic fluid 109
2.4.2 Non-degenerate relativistic fluid 110
2.4.3 Non-degenerate non-relativistic fluid 111
2.4.4 Ultrarelativistic fluid 112
2.4.5 Degenerate Fermi fluid 114
2.4.6 Ideal fluid 115
2.4.7 Polytropic fluid 118
2.4.8 Radiation fluid 123
2.4.9 Dark-energy fluid 126
2.4.10 Newtonian and relativistic barotropic fluids 128
2.5 Further reading 131
2.6 Problems 132
Relativistic Perfect Fluids 133
3.1 Kinematic properties of fluids 133
3.1.1 Kinematic shear, expansion and vorticity 133
3.1.2 Evolution laws of the kinematic quantities 137
3.2 Mass current and energy-momentum of perfect fluids 138
3.3 Hydrodynamics equations of perfect fluids 143
3.4 Perfect fluids and symmetries 145
3.5 The Newtonian limit of the hydrodynamic equations 147
3.6 Stationary flows 150
3.6.1 Bernoulli s theorem 150
3.6.2 Relativistic Bernoulli theorem 152
3.7 Irrotational flows 152
3.7.1 Newtonian irrotational flows 152
3.7.2 Kelvin-Helmholtz theorem 154
3.7.3 Relativistic vorticity 155
3.7.4 Relativistic irrotational flows 157
3.7.5 Relativistic Kelvin-Helmholtz theorem 159
3.8 Isentropic flows 162
3.9 A velocity-potential approach to relativistic hydrodynamics 164
3.10 A variational principle for relativistic hydrodynamics 168
3.11 Perfect multifluids 175
3.11.1 Coupled multifluids 175
3.11.2 Interacting multifluids 179
3.12 Further reading 187
3.13 Problems 188
Linear and Nonlinear Hydrodynamic Waves 190
4.1 Hyperbolic systems of partial differential equations 190
4.1.1 Quasi-linear formulation 190
4.1.2 Conservative formulation 195
4.2 Linear and nonlinear behaviour 198
4.2.1 Characteristic equations for linear systems 198
4.2.2 Riemann invariants 200
4.2.3 Characteristic curves and caustics 203
4.2.4 Domain of determinacy and region of influence 206
4.3 Linear hydrodynamic waves 208
4.3.1 Soundwaves 208
4.4 Nonlinear hydrodynamic waves 209
4.4.1 Simple waves and discontinuous waves 209
4.4.2 Rarefaction waves 211
4.4.3 Shock waves 214
4.4.4 Contact discontinuities 222
4.5 The Riemann problem 223
4.6 Solution of the one-dimensional Riemann problem 227
4.6.1 Limiting relative velocities 228
4.7 Solution of the multidimensional Riemann problem 233
4.7.1 Jumps across a shock wave 235
4.7.2 Jumps across a rarefaction wave 236
4.7.3 Limiting relative velocities 237
4.7.4 Relativistic effects in multidimensional Riemann problems 239
4.7.5 Shock-detection techniques 243
4.8 Stability of shock waves 245
4.9 General-relativistic discontinuities 249
4.10 Further reading 255
4.11 Problems 256
Reaction Fronts: Detonations and Deflagrations 258
5.1 Basic properties of reaction fronts 258
5.2 Reaction adiabat 259
5.3 Relativistic detonations 262
5.4 Relativistic deflagrations 266
5.5 Stability of reaction fronts 268
5.5.1 Stability of detonations 271
5.5.2 Stability of deflagrations 281
5.6 Further reading 283
5.7 Problems 284
Relativistic Non-Perfect Fluids 285
6.1 On the four-velocity of a non-perfect fluid 285
6.2 The energy-momentum tensor of non-perfect fluids 287
6.3 Hydrodynamic equations of non-perfect fluids 290
6.3.1 The general form of the momentum and energy equations 290
6.3.2 The equilibrium state 291
6.4 Classical Irreversible Thermodynamics (first-order theories) 291
6.4.1 The constitutive equations 292
6.4.2 The Newtonian limit: Navier-Stokes and heat conduction 294
6.5 The importance of a causal theory 296
6.5.1 Parabolic versus hyperbolic 296
6.5.2 Non-causality of Classical Irreversible Thermodynamics 298
6.6 Extended Irreversible Thermodynamics (second-order theories) 299
6.6.1 The Israel-Stewart formulation 300
6.6.2 Characteristic speeds of the Israel-Stewart formulation 303
6.6.3 Divergence-type theories 306
6.7 Concluding remarks 312
6.8 Further reading 314
6.9 Problems 315
PART II NUMERICAL RELATIVISTIC HYDRODYNAMICS
Formulations of the Einstein-Euler Equations 319
7.1 The 3+1 decomposition of spacetime 319
7.2 Formulations of the Einstein equations 324
7.2.1 Spherically symmetric Lagrangian formulations 325
7.2.2 The ADM formulation 329
7.2.3 Conformal traceless formulations 335
7.2.4 Gauge conditions in 3+1 formulations 343
7.2.5 The generalised harmonic formulation 346
7.2.6 Constraint equations, initial data and constrained evolution 350
7.3 Formulations of the hydrodynamic equations 360
7.3.1 The Wilson formulation 360
7.3.2 The importance of conservative formulations 362
7.3.3 The 3+1 Valencia formulation 364
7.3.4 The covariant formulation 374
7.3.5 The light-cone formulation 375
7.3.6 The discontinuous Galerkin formulation 377
7.4 Further reading 383
7.5 Problems 384
Numerical Relativistic Hydrodynamics: Finite-Difference Methods 386
8.1 The discretisation process 387
8.2 Numerical errors 390
8.2.1 Consistency, convergence and stability 394
8.3 Finite-difference methods 396
8.3.1 Analysis of the numerical stability 396
8.3.2 The upwind scheme 399
8.3.3 The FTCS scheme 401
8.3.4 The Lax-Friedrichs scheme 402
8.3.5 The leapfrog scheme 404
8.3.6 The Lax-Wendroff scheme 405
8.3.7 Kreiss-Oliger dissipation 407
8.4 Artificial-viscosity approaches 409
8.5 Further reading 412
8.6 Problems 413
9 Numerical Relativistic Hydrodynamics: HRSC Methods 414
9.1 Conservative schemes 414
9.1.1 Rankine-Hugoniot conditions 414
9.1.2 Finite-volume conservative numerical schemes 416
9.1.3 Finite-difference conservative numerical schemes 418
9.2 Upwind methods 420
9.2.1 Monotone methods 420
9.2.2 Total variation diminishing methods 421
9.2.3 Godunov methods 423
9.3 Reconstruction techniques 427
9.3.1 Slope-limiter methods 428
9.3.2 The piecewise-parabolic method 430
9.3.3 Reconstruction in characteristic variables 434
9.4 Approximate Riemann solvers 436
9.4.1 Incomplete Riemann solvers 436
9.4.2 Complete Riemann solvers 439
9.5 The method of lines 447
9.5.1 Explicit Runge-Kutta methods 448
9.5.2 Implicit-explicit Runge-Kutta methods 449
9.6 Central numerical schemes 452
9.6.1 Staggered central schemes 453
9.6.2 Non-staggered central schemes 455
9.7 Further reading 457
9.8 Problems 458
10 Numerical Relativistic Hydrodynamics: High-Order Methods 459
10.1 Why high-order numerical methods? 459
10.2 ENO and WENO methods for hyperbolic conservation laws 460
10.2.1 Finite-volume ENO schemes 461
10.2.2 Finite-volume WENO schemes 466
10.2.3 Finite-difference ENO schemes 468
10.2.4 Finite-difference WENO schemes 472
10.3 Discontinuous Galerkin methods 472
10.3.1 The essence of DG methods 472
10.3.2 An example: a RKDG scheme in spherical symmetry 475
10.4 The ADER approach 479
10.4.1 The original formulation 479
10.4.2 The local spacetime DG scheme 482
10.5 Extension to multidimensional problems 485
10.5.1 Finite-difference multidimensional schemes 486
10.5.2 Finite-volume multidimensional schemes 487
10.6 Further reading 489
10.7 Problems 490
PART III APPLICATIONS OF RELATIVISTIC HYDRODYNAMICS
11 Relativistic Hydrodynamics of Non-Selfgravitating Fluids 493
11.1 Similar and self-similar flows 494
11.1.1 One-dimensional self-similar flows 495
11.1.2 Self-similar hydrodynamics of a bubble 501
11.1.3 Self-similar hydrodynamics of a drop 504
11.2 Relativistic blast waves 507
11.3 Spherical flows onto and out of a compact object 513
11.4 Spherical accretion onto a black hole 516
11.5 Non-spherical accretion onto a moving black hole 524
11.5.1 Accreting potential flows 525
11.5.2 Bondi-Hoyle-Lyttleton flows 529
11.6 Fluids in circular motion around a black hole 537
11.6.1 von Zeipel cylinders 537
11.7 Geometrically thick tori 541
11.7.1 The runaway instability 548
11.7.2 On the sound speed in polytropic tori 549
11.7.3 Thick tori in Schwarzschild-ae Sitter spacetimes 551
11.8 Relativistic accreting discs 553
11.8.1 Rest-mass conservation 558
11.8.2 Radial momentum conservation 559
11.8.3 Angular momentum conservation 560
11.8.4 Hydrostatic vertical equilibrium 562
11.8.5 Energy conservation 563
11.9 Relativistic jets 565
11.9.1 Apparently superluminal jets 566
11.9.2 Hydrodynamic acceleration mechanisms 569
11.9.3 Numerical modelling of relativistic jets 577
11.10 Relativistic heavy-ion collisions 580
11.10.1 Basic concepts 580
11.10.2 One-dimensional Bjorken flow 5 84
11.10.3 Cylindrically symmetric flows 585
11.11 Further reading 589
11.12 Problems 590
12 Relativistic Hydrodynamics of Selfgravitating Fluids 593
12.1 Spherical stars 593
12.2 Gravastars 599
12.3 Rotating stars 604
12.3.1 Uniformly rotating stars 604
12.3.2 Differentially rotating stars 608
12.4 Collapse of a compact star to a black hole 612
12.4.1 Dust collapse: the Oppenheimer-Snyder solution 612
12.4.2 Fluid collapse 617
12.5 Dynamics of binary neutron stars 623
12.5.1 Broadbrush picture 623
12.5.2 Dynamics of equal-mass binaries 627
12.5.3 Dynamics of unequal-mass binaries 637
12.6 Dynamics of black-hole-neutron-star binaries 647
12.6.1 Broadbrush picture 648
12.7 Further reading 657
12.8 Problems 658
Appendix A Geometrised System of Units 659
Appendix B Notable Thermodynamic Expressions 661
B.l Thermodynamic quantities and potentials 661
B.2 Maxwell relations 663
Appendix C Notable Tensors 665
C. 1 Relativistic expressions 665
C.2 Newtonian expressions 667
Appendix D Common Practices in Numerical Relativistic Hydrodynamics 668
D. 1 Conversion from conserved to primitive variables 668
D. 1.1 Analytic equations of state 668
D.1.2 Tabulated equations of state 671
D.2 Treatment of atmospheres 673
D.3 Guaranteeing the positivity of pressure 674
D.4 Domain excision 675
Appendix E Numerical Building Blocks 678
E.l TVD slope limiters 678
E.2 Basic Riemann solvers 679
E.3 Reference one-dimensional pseudo-code 680
References 682
Index 721
|
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| isbn | 9780198528906 |
| language | English |
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| spelling | Rezzolla, Luciano Verfasser aut Relativistic hydrodynamics Luciano Rezzolla ; Olindo Zanotti 1. ed. Oxford Oxford Univ. Press 2013 XV, 735 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. [682] - 720 Relativistische Hydrodynamik (DE-588)4177684-7 gnd rswk-swf Relativistische Hydrodynamik (DE-588)4177684-7 s DE-604 Zanotti, Olindo Verfasser (DE-588)1046252496 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026122755&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rezzolla, Luciano Zanotti, Olindo Relativistic hydrodynamics Relativistische Hydrodynamik (DE-588)4177684-7 gnd |
| subject_GND | (DE-588)4177684-7 |
| title | Relativistic hydrodynamics |
| title_auth | Relativistic hydrodynamics |
| title_exact_search | Relativistic hydrodynamics |
| title_full | Relativistic hydrodynamics Luciano Rezzolla ; Olindo Zanotti |
| title_fullStr | Relativistic hydrodynamics Luciano Rezzolla ; Olindo Zanotti |
| title_full_unstemmed | Relativistic hydrodynamics Luciano Rezzolla ; Olindo Zanotti |
| title_short | Relativistic hydrodynamics |
| title_sort | relativistic hydrodynamics |
| topic | Relativistische Hydrodynamik (DE-588)4177684-7 gnd |
| topic_facet | Relativistische Hydrodynamik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026122755&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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