This algorithm begins by choosing the usual polar-spherical topology for the angular coordinates
, and rewriting the apparent horizon equation (16
) in the form
Next we expand in spherical harmonics (5
). Because the left hand side of Equation (21
) is just the
flat-space angular Laplacian of
, which has the
as orthogonal eigenfunctions, multiplying both
sides of Equation (21
) by
(the complex conjugate of
) and integrating over all solid angles gives
Based on this, Nakamura, Kojima, and Oohara [113] proposed the following functional-iteration
algorithm for solving Equation (21
):
Gundlach [80] observed that the subtraction and inversion of the flat-space angular Laplacian operator
in this algorithm is actually a standard technique for solving nonlinear elliptic PDEs by spectral methods. I
discuss this observation and its implications further in Section 8.7.4.
Nakamura, Kojima, and Oohara [113] report that their algorithm works well, but Nakao (cited as
personal communication in [146]) has argued that it tends to become inefficient (and possibly inaccurate)
for large
(high angular resolution) because the
fail to be numerically orthogonal due to the
finite resolution of the numerical grid. I know of no other published work addressing Nakao’s
criticism.
Kemball and Bishop [93] investigated the behavior of the Nakamura–Kojima–Oohara algorithm and found
that its (only) major weakness seems to be that the
-update equation (23
) “may have multiple roots
or minima even in the presence of a single marginally outer trapped surface, and all should be tried for
convergence”.
Kemball and Bishop [93] suggested and tested several modifications to improve the algorithm’s
convergence behavior. They verified that (either in its original form or with their modifications) the
algorithm’s rate of convergence (number of iterations to reach a given error level) is roughly independent of
the degree
of spherical-harmonic expansion used. They also give an analysis that the
algorithm’s cost is
, and its accuracy
, i.e. the cost is
.
This accuracy is surprisingly low: Exponential convergence with
is typical of spectral
algorithms and would be expected here. I do not know of any published work which addresses this
discrepancy.
Lin and Novak [104] have developed a variant of the Nakamura–Kojima–Oohara algorithm which avoids the
need for a separate search for
at each iteration: Write the apparent horizon equation (16
) in the form
Lin and Novak [104] showed that all the spherical-harmonic coefficients
(including
) can
then be found by iteratively solving the equation
Lin and Novak [104] find that this algorithm gives robust convergence and is quite fast, particularly at
modest accuracy levels. For example, running on a 2 GHz processor, their implementation takes
,
,
,
, and
seconds to find the apparent horizon in a test slice to a relative error
(measured in the horizon area) of
,
,
,
, and
respectively37.
This implementation is freely available as part of the
Lorene toolkit for spectral computations in
numerical relativity (see Table 2).
Despite what appears to be fairly good numerical behavior and reasonable ease of implementation, the
original Nakamura–Kojima–Oohara algorithm has not been widely used apart from later work by its
original developers (see, for example, [115, 114]). Kemball and Bishop [93] have proposed and tested several
modifications to the basic Nakamura–Kojima–Oohara algorithm. Lin and Novak [104] have develped
a variant of the Nakamura–Kojima–Oohara algorithm which avoids the need for a separate
search for the coefficient at each iteration. Their implementation of this variant is freely
available as part of the
Lorene toolkit for spectral computations in numerical relativity (see
Table 2).
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