694 Index
Numerical methods (cont.)
hyperbolic equations
methods, see Alternating direction implicit scheme;
Crank–Nicholson scheme; Forward-time
centered-space scheme; Iterative
Crank–Nicholson scheme; Lax scheme; Method
of lines; Predictor-corrector schemes; Upwind
differencing scheme
model advective equation, 200
wave equation, 184–185
Monte Carlo methods, 224
numerical viscosity, 203
ordinary differential equations
Euler method, 208
Runge–Kutta method, 209, 224
parabolic equations
diffusion equation, 184
methods, see Crank–Nicholson scheme;
Forward-time centered-space scheme
spectral methods, see also Chebychev polynomials;
Gibbs phenomena
collocation points, 217
comparison with finite difference methods, 222,
224–225
concept, 213–214
exponential convergence, 222
Galerkin method, 217
packages, see LORENE
pseudo-spectral method, 217
simple example, 214–217
tau method, 217
Oppenheimer–Snyder (OS) collapse
as numerical test, 137, 279–280, 301, 302
brief review, 18–22
horizons, see Apparent horizons; Event horizons;
Trapped Surfaces
magnetized, 161–163
thermal, 144, 146–147
Oppenheimer–Volkoff (OV) equilibrium stars
as numerical test, 137, 226
brief review, 15–18
Buchdahl limit, 16
maximum mass, 17, see also Chandrasekhar limit
onset of radial instability, 17, see also Turning-point
criterion
Outer-trapped surfaces, 236–237
Parallel transport, 3, 5, 599, see also Covariant derivative
Particle methods, see Collisionless matter
PETSc, 198
Phase-space distribution function, see Distribution
function
Phase-space methods, see Collisionless matter
Polar slicing
and radial gauge, 116
singularity avoidance, 240–241, 271, 276
spherical symmetry, 270–286
Polytropes, see also Binary black hole–neutron star initial
data; Binary neutron star initial data; Equations of
state; Hydrodynamics; Neutron stars;
Oppenheimer–Volkoff equilibrium models;
Rotating star equilibrium models; Supermassive
stars
brief review, 17–18
examples
nonrelativistic, degenerate fermions, 17, 466–468
thermal radiation pressure, 17
ultrarelativistic, degenerate fermions, 17
n = 3 polytropes, 17, 301, 465–466, 476–478,
487–491, 502–504
polytropic gas constant, 17
polytropic index, 17
scaling to arbitrary masses, 17, 463
summary table of maximum masses, 467
Population I stars, 476, 502, 504
Population III stars, 445, 476, 502, 504
Post-Newtonian formalism
binaries
binding energy, 620
gravitational wave luminosities, 620
orbital phase, 621–622
wave amplitudes, 622–628
effective one-body (EOB) approach, 425–426,628
expansions, 617–618
Pad
´
e approximant, 628
perturbation amplitudes, 617
Predictor-corrector schemes, 207
Price’s theorem, 178
ψ
4
, see Newman–Penrose formalism
Pulsars, 325–331, 459, 468, 471, 507, 533, 556,
578
Puncture method, see also Moving-puncture method
fixed puncture in binary black hole evolution,
432–433
initial data, see Binary black hole initial data; Binary
black hole–neutron star initial data
Pythagorean theorem, 45, 46
Quadrupole approximation, see also Gravitational waves
applications
binary black hole–neutron stars, 578
binary neutron stars, 530, 531, 533, 536, 538–539,
541, 551, 555
rotating stars, 503
brief review, 8
concept, 315–318
post-Newtonian formalism, 618, 619, 620
Quasars, 285, 329, 446, 563
Quasi-isotropic gauge, 115–117
Quasi-Kinnersly frame, see Newman–Penrose formalism
Quasilocal horizon mass, 251
Quintessence, 7, 176, see also Scalar fields
r-mode instability, 327, 469
r-process nuclei, 534, 535