In more detail, elliptic-PDE algorithms assume that the horizon is a Strahlkörper about some local
coordinate origin, and choose an angular coordinate system and a finite-difference grid of points on
in the manner discussed in Section 2.2.
The most common choices are the usual polar-spherical coordinates and a uniform
“latitude/longitude” grid in these coordinates. Since these coordinates are “unwrapped” relative to the
actual
trial-horizon-surface topology, the horizon shape function
satisfies periodic boundary
conditions across the artificial grid boundary at
and
. The north and south
poles
and
are trickier, but Huq et al. [89
, 90
], Shibata and Uryƫ [147
], and
Schnetter [132
, 133
]39
“reaching across the pole” boundary conditions for these artificial grid boundaries.
Alternatively, Thornburg [156] avoids the
axis coordinate singularity of polar-spherical
coordinates by using an “inflated-cube” system of six angular patches to cover
. Here each
patch’s nominal grid is surrounded by a “ghost zone” of additional grid points where
is
determined by interpolation from the neighboring patches. The interpatch interpolation thus
serves to tie the patches together, enforcing the continuity and differentiability of
across
patch boundaries. Thornburg reports that this scheme works well but was quite complicated to
program.
Overall, the latitude/longitude grid seems to be the superior choice: it works well, is simple to program, and eases interoperation with other software.
The next step in the algorithm is to evaluate the expansion given by Equation (16
) on the angular grid
given a trial horizon surface shape function
on this same grid (6
).
Most researchers compute via 2-dimensional angular finite differencing of Equation (16
) on the trial
horizon surface. 2nd order angular finite differencing is most common, but Thornburg [156
] uses 4th order
angular finite differencing for increased accuracy.
With a latitude/longitude grid the
function in Equation (16
) is singular on
the
axis (at the north and south poles
and
) but can be regularized by applying
L’Hôpital’s rule. Schnetter [132
, 133
] observes that using a Cartesian basis for all tensors greatly aids in
this regularization.
Huq et al. [89, 90
] choose, instead, to use a completely different computation technique for
, based
on 3-dimensional Cartesian finite differencing:
Comparing the different ways of evaluating , 2-dimensional angular finite differencing of
Equation (16
) seems to me to be both simpler (easier to program) and likely more efficient than
3-dimensional Cartesian finite differencing of Equation (27
).
A variety of algorithms are possible for actually solving the nonlinear elliptic PDE (16) (or (27
) for the
Huq et al. [89
, 90
] horizon finder).
The most common choice is to use some variant of Newton’s method. That is, finite differencing
Equation (16) or (27
) (as appropriate) gives a system of
nonlinear algebraic equations for the
horizon shape function
at the
angular grid points; these can be solved by Newton’s method in
dimensions. (As explained by Thornburg [153
, Section VIII.C], this is usually equivalent to
applying the Newton–Kantorovich algorithm [37, Appendix C] to the original nonlinear elliptic PDE (16
)
or (27
).)
Newton’s method converges very quickly once the trial horizon surface is sufficiently close to a solution
(a MOTS). However, for a less accurate initial guess, Newton’s method may converge very slowly or even
fail to converge at all. There is no usable way of determining a priori just how large the radius of
convergence of the iteration will be, but in practice to
of the horizon radius is often a reasonable
estimate40.
Thornburg [153] described the use of various “line search” modifications to Newton’s method to improve
its radius and robustness of convergence, and reported that even fairly simple modifications of this sort
roughly doubled the radius of convergence.
Schnetter [132, 133
] used the PETSc general-purpose elliptic-solver library [22, 23, 24] to solve the
equations. This offers a wide variety of Newton-like algorithms already implemented in a highly optimized
form.
Rather than Newton’s method or one of its variants, Shibata et al. [146, 147
] use a functional-iteration
algorithm directly on the nonlinear elliptic PDE (16
). This seems likely to be less efficient than Newton’s
method but avoids having to compute and manipulate the Jacobian matrix.
Newton’s method, and all its variants, require an explicit computation of the Jacobian matrix
where the indices Notice that includes contributions both from the direct dependence of
on
,
, and
, and also from the indirect dependence of
on
through the position-dependence of the
geometry variables
,
, and
(since
depends on the geometry variables at the horizon
surface position, and this position is determined by
). Thornburg [153
] discusses this indirect
dependence in detail.
There are two basic ways to compute the Jacobian matrix.
This algorithm is easy to program but somewhat inefficient. It is used by a number of researchers
including Schnetter [132, 133
], and Huq et al. [89
, 90
].
This algorithm is best illustrated by an example which is simpler than the full apparent horizon
equation: Consider the flat-space Laplacian in standard polar-spherical coordinates,
All the algorithms described in Section 8.5.3 for treating nonlinear elliptic PDEs require
solving a sequence of linear systems of equations in
unknowns.
is
typically on the order of a few thousand, and the Jacobian matrices in question are sparse
due to the locality of the angular finite differencing (see Section 8.5.4). Thus, for reasonable
efficiency, it is essential to use linear solvers that exploit this sparsity. Unfortunately, many such
algorithms/codes only handle symmetric positive-definite matrices while, due to the angular boundary
conditions43
(see Section 8.5.1), the Jacobian matrices that arise in apparent horizon finding are generally neither of
these.
The numerical solution of large sparse linear systems is a whole subfield of numerical analysis. See, for example, Duff, Erisman, and Reid [65], and Saad [130] for extensive discussions44. In practice, a numerical relativist is unlikely to write her own linear solver but, rather, will use an existing subroutine (library).
Kershaw’s [94] ILUCG iterative solver is often used; this is only moderately efficient, but is quite easy to
program45.
Schnetter [132
, 133
] reports good results with an ILU-preconditioned GMRES solver from the PETSc
library. Thornburg [156
] experimented with both an ILUCG solver and a direct sparse
decomposition solver (Davis’ UMFPACK library [57, 58, 56, 55, 54]), and found each to
be more efficient in some situations; overall, he found the UMFPACK solver to be the best
choice.
As an example of the results obtained with this type of apparent horizon finder, Figure 10 shows the
numerically-computed apparent horizons (actually, MOTSs) at two times in a head-on binary black
hole collision. (The physical system being simulated here is very similar to that simulated by
Matzner et al. [108], a view of whose event horizon is shown in Figure 5
.)
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As another example, Figure 11 shows the time dependence of the irreducible masses of apparent
horizons found in a (spiraling) binary black hole collision, simulated at several different grid
resolutions, as found by both AHFinderDirect and another
Cactus apparent horizon finder,
AHFinder46.
For this evolution, the two apparent horizon finders give irreducible masses which agree to within about 2%
for the individual horizons and 0.5% for the common horizon.
|
As a final example, Figure 8 shows the numerically-computed event and apparent horizons in the
collapse of a rapidly rotating neutron star to a Kerr black hole. (The event horizons were computed using
the EHFinder code described in Section 5.3.3.)
Elliptic-PDE apparent horizon finders have been developed by many researchers, including
Eardley [67]47,
Cook [50, 52, 51], and Thornburg [153] in axisymmetry, and Shibata et al. [146, 147], Huq et al. [89, 90],
Schnetter [132
, 133
], and Thornburg [156
] in fully generic slices.
Elliptic-PDE algorithms are (or can be implemented to be) among the fastest
horizon finding algorithms. For example, running on a 1.7 GHz processor, Thornburg’s
AHFinderDirect [156] averaged 1.7 s per horizon finding, as compared with 61 s for
an alternate “fast-flow” apparent horizon finder AHFinder (discussed in more detail in
Section 8.7)48.
However, achieving maximum performance comes at some cost in implementation effort (e.g. the “symbolic
differentiation” Jacobian computation discussed in Section 8.5.4).
Elliptic-PDE algorithms are probably somewhat more robust in their convergence (i.e. they
have a slightly larger radius of convergence) than other types of local algorithms, particularly
if the “line search” modifications of Newton’s method described by Thornburg [153] are
implemented49.
Their typical radius of convergence is on the order of
of the horizon radius, but cases are known where
it is much smaller. For example, Schnetter, Herrmann, and Pollney [135
] report that (with no “line
search” modifications) it is only about 10% for some slices in a binary black hole coalescence
simulation.
Schnetter’s TGRapparentHorizon2D [132, 133] and Thornburg’s AHFinderDirect [156
] are
both elliptic-PDE apparent horizon finders implemented as freely available modules (“thorns”) in the
Cactus computational toolkit (see Table 2). Both work with either the PUGH unigrid driver or
the
Carpet mesh-refinement driver for
Cactus. TGRapparentHorizon2D is no longer
maintained, but AHFinderDirect is actively supported and is now used by many different research
groups50.
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