Elements of Numerical Relativity and Relativistic Hydrodynamics : From Einstein' s Equations to Astrophysical Simulations

by Bona, C.; Palenzuela-Luque, C.; Bona-Casas, C.

• ISBN13: 9783642011634
• ISBN10: 3642011632
• Edition: 2nd
• Format: Hardcover
• Copyright: 2009-08-01
• Publisher: Springer Verlag

Summary

Many large-scale projects for detecting gravitational radiation are currently being developed, all with the aim of opening a new window onto the observable Universe. As a result, numerical relativity has recently become a major field of research, and Elements of Numerical Relativity and Relativistic Hydrodynamics is a valuable primer for both graduate students and non-specialist researchers wishing to enter the field.A revised and significantly enlarged edition of LNP 673 Elements of Numerical Relativity, this book starts with the most basic insights and aspects of numerical relativity before it develops coherent guidelines for the reliable and convenient selection of each of the following key aspects: evolution formalism; gauge, initial, and boundary conditions; and various numerical algorithms. And in addition to many revisions, it includes new, convenient damping terms for numerical implementations, a presentation of the recently-developed harmonic formalism, and an extensive, new chapter on matter space-times, containing a thorough introduction to relativistic hydrodynamics.While proper reference is given to advanced applications requiring large computational resources, most tests and applications in this book can be performed on a standard PC.

Table of Contents

The 4D Spacetime	p. 1
Spacetime geometry	p. 1
The metric	p. 1
General covariance	p. 2
Covariant derivatives	p. 3
Curvature	p. 5
Symmetries of the curvature tensor	p. 6
General covariant field equations	p. 7
The stress-energy tensor	p. 7
Einstein's field equations	p. 8
Structure of the field equations	p. 10
Einstein's equations solutions	p. 13
Symmetries: Lie derivatives	p. 13
Exact solutions	p. 15
Analytical and numerical approximations	p. 16
Harmonic formalism	p. 18
The relaxed system	p. 19
Analytical and numerical applications	p. 20
Harmonic coordinates	p. 22
References	p. 24
The Evolution Formalism	p. 25
Space plus time decomposition	p. 25
A prelude: Maxwell equations	p. 26
Spacetime synchronization	p. 27
The Eulerian observers	p. 30
Einstein's equations decomposition	p. 32
The 3+1 form of the field equations	p. 32
3+1 Covariance	p. 33
Generic space coordinates	p. 35
The evolution system	p. 38
Evolution and constraints	p. 38
Constraints conservation	p. 39
Evolution strategies	p. 40
Gravitational waves degrees of freedom	p. 42
Linearized field equations	p. 42
Plane-wave analysis	p. 43
Gravitational waves and gauge effects	p. 46
References	p. 47
Free Evolution	p. 49
The free evolution framework	p. 49
The ADM system	p. 49
Extended solution space	p. 50
Plane-wave analysis	p. 52
Robust stability test-bed	p. 55
Finite differences	p. 55
Numerical results	p. 58
Pseudo-hyperbolic systems	p. 60
Extra dynamical fields	p. 60
The BSSN system	p. 63
Plane-wave analysis	p. 65
Covariant formulations	p. 67
The Z4 formalism	p. 67
The generalized harmonic formalism	p. 69
Constraint-violation control	p. 70
The Z4 evolution system	p. 71
3+1 Decomposition	p. 71
Plane-wave analysis	p. 73
Symmetry breaking	p. 75
References	p. 77
First-Order Hyperbolic Systems	p. 79
First-order versions of second-order systems	p. 79
Introducting extra first-order quantities	p. 79
Ordering ambiguities	p. 81
First-order Z4 system (normal coordinates)	p. 82
Symmetry breaking: the KST system	p. 83
Hyperbolic systems	p. 86
Weak and strong hyperbolicity	p. 86
1D Energy estimates	p. 89
Symmetric-hyperbolic systems	p. 91
Generic space coordinates	p. 93
First-order fields	p. 93
Generalized harmonic formulations	p. 96
First-order Z4 formalism	p. 98
Boundary conditions	p. 100
Algebraic boundary conditions	p. 101
Energy methods	p. 103
Robust stability test	p. 106
References	p. 108
Numerical Methods	p. 109
Finite difference methods	p. 110
Accuracy and stability	p. 110
The method of lines	p. 112
Artificial dissipation	p. 113
The gauge waves test-bed	p. 115
Finite volume methods	p. 117
Systems of balance laws	p. 118
Weak solutions	p. 120
Flux formulae	p. 123
High-resolution methods	p. 126
The modified flux approach	p. 130
Simple CFD tests	p. 132
Advection equation	p. 133
Burgers equation	p. 135
Euler equations: Sod test	p. 137
MHD equations: Orszag-Tang vortex	p. 138
References	p. 141
Black Hole Simulations	p. 143
Black Hole initial data	p. 143
Conformal metric decomposition	p. 146
Singular initial data: punctured black holes	p. 147
Regular initial data	p. 149
Dynamical time slicing	p. 155
Singularity avoidance	p. 155
Limit surfaces	p. 157
Gauge pathologies	p. 159
Numerical Black Hole milestones	p. 160
Short-term simulations	p. 160
Long-term simulations	p. 166
Further developments	p. 167
References	p. 169
Matter Spacetimes	p. 171
Scalar fields	p. 172
The Klein-Gordon equation	p. 172
Boson stars initial data	p. 174
Evolution of a single boson star	p. 178
Electromagnetic fields	p. 179
Maxwell equations	p. 180
Electromagnetic potential	p. 181
The electromagnetic fields	p. 183
The electromagnetic stress-energy tensor	p. 185
Hydrodynamics	p. 185
Perfect fluids	p. 186
The equation of state	p. 189
Neutron stars	p. 192
Magnetohydrodynamics	p. 195
The MHD evolution equations	p. 196
Generalized Ohm's law	p. 197
Ideal MHD	p. 200
The force-free limit	p. 204
Further developments	p. 205
Boson stars collisions	p. 205
Neutron stars collisions	p. 207
References	p. 209
Index	p. 211
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