1 | An algorithm’s actual “convergence region” (the set of all initial guesses for which the algorithm converges to the correct solution) may even be fractal in shape. For example, the Julia set is the convergence region of Newton’s method on a simple nonlinear algebraic equation. | |
2 | For convenience of exposition I use spherical harmonics here, but there are no essential differences if other basis sets are used. | |
3 | I discuss the choice of this angular grid in more detail in Section 8.5.1. | |
4 | There has been some controversy over whether, and if so how quickly, Regge calculus converges to the continuum Einstein equations. (See, for example, the debate between Brewin [40] and Miller [110], and the explicit convergence demonstration of Gentle and Miller [73].) However, Brewin and Gentle [41] seem to have resolved this: Regge calculus does, in fact, converge to the continuum solution, and this convergence is generically 2nd order in the resolution. | |
5 | See, for example, Choptuik [48], Pretorius [127], Schnetter et al. [134![]() |
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6 | Chruściel and Galloway [49] showed that if a “cloud of sand” falls into a large black hole, each “sand grain” generates a non-differentiable caustic in the event horizon. | |
7 | This is a statement about the types of spacetimes usually studied by numerical relativists, not a statement about the mathematical properties of the event horizon itself. | |
8 | I briefly review ODE integration algorithms and codes in Appendix B. | |
9 | In practice the differentiation can usefully be combined with the interpolation; I outline how this can be done in Section 7.5. | |
10 | This convergence is only true in a global sense: locally the event horizon has no special geometric properties, and the Riemann tensor components which govern geodesic convergence/divergence may have either sign. | |
11 | Diener [60![]() |
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12 | Walker [162![]() |
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13 | See [7, 103![]() |
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14 | Equivalently, Diener [60![]() ![]() |
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15 | They describe how Richardson extrapolation can be used to improve the accuracy of the solutions from ![]() ![]() |
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16 | Note that the surface must be smooth everywhere. If this condition were not imposed, then MOTSs would lose many of
their important properties. For example, even a standard ![]() |
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17 | Andersson and Metzger [6] have shown that MOTSs can only intersect if they are contained within an outer common MOTS. Szilágyi et al. [151] give a numerical example of such overlapping MOTSs found in a binary black hole collision. | |
18 | As an indication of the importance of the “closed” requirement, Hawking [81] observed that if we consider two spacelike-separated events in Minkowski spacetime, the intersection of their backwards light cones satisfies all the conditions of the MOTS definition, except that it is not closed. | |
19 | The proof is given by Hawking and Ellis [82, Proposition 9.2.8] and by Wald [160, Propositions 12.2.3 and 12.2.4]. | |
20 | Wald and Iyer [161] proved this by explicitly constructing a family of angularly anisotropic slices in Schwarzschild
spacetime which approach arbitrarily close to ![]() |
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21 | This world-tube is sometimes called “the apparent horizon”, but this is not standard terminology. In this review I always use the terminology that an MOTS or apparent horizon is a 2-surface contained in a (single) slice. | |
22 | Ashtekar and Galloway [17] have recently proved “a number of physically interesting constraints” on this slicing-dependence. | |
23 | The derivation of this condition is given by (for example) York [164], Gundlach [80![]() |
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24 | Notice that in order for the 3-divergence in Equation (15![]() ![]() ![]() |
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25 | Or, in the Huq et al. [89![]() ![]() |
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26 | If the underlying simulation uses spectral methods then the spectral series can be evaluated anywhere,
so no actual interpolation need be done, although the term “spectral interpolation” is still often used. See
Fornberg [70![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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27 | Conceptually, an interpolator generally works by locally fitting a fitting function (usually a low-degree polynomial) to the
data points in a neighborhood of the interpolation point, then evaluating the fitting function at the interpolation point. By
evaluating the derivative of the fitting function, the ![]() ![]() |
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28 | Thornburg [154![]() |
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29 | Note that ![]() ![]() ![]() ![]() |
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30 | See also the work of Bizoń, Malec, and Ó Murchadha [32] for an interesting analytical study giving necessary and sufficient conditions for apparent horizons to form in non-vacuum spherically symmetric spacetimes. | |
31 | Ascher, Mattheij, and Russel [15, Chapter 4] give a more detailed discussion of shooting methods. | |
32 | See, for example, Dennis and Schnabel [59], or Brent [39] for general surveys of general-purposes function-minimization algorithms and codes. | |
33 | There is a simple heuristic argument (see, for example, Press et al. [125![]() ![]() ![]() ![]() ![]() ![]() |
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34 | A simple counting argument suffices to show that any general-purpose function-minimization algorithm in ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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35 | In the context of an underlying simulation with spectral accuracy, Pfeiffer [122![]() ![]() |
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36 | AHFinder also includes a “fast flow” algorithm (Section 8.7). | |
37 | For comparison, the elliptic-PDE AHFinderDirect horizon finder (discussed in Section 8.5.6), running on a roughly
similar processor, takes about ![]() ![]() |
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38 | In theory this equation could also be solved by a spectral method on ![]() ![]() |
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39 | See [133![]() ![]() |
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40 | Thornburg [153![]() |
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41 | A very important optimization here is that ![]() ![]() |
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42 | Because of the one-sided finite differencing, the approximation (29![]() ![]() ![]() |
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43 | Or the interpatch interpolation conditions in Thornburg’s multiple-grid-patch scheme [156![]() |
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44 | Multigrid algorithms are also important here; these exploit the geometric structure of the underlying elliptic PDE. See
Briggs, Henson, and McCormick [42![]() ![]() |
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45 | Madderom’s Fortran subroutine DILUCG [107], which implements the method of [94], has been used by a number of numerical relativists for both this and other purposes. | |
46 | AHFinder incorporates both a minimization algorithm (Section 8.3) and a fast-flow algorithm (Section 8.7.4); these tests used the fast-flow algorithm. | |
47 | This paper does not say how the author finds apparent horizons, but [68, page 135] cites a preprint of this as treating the apparent-horizon equation as a 2-point (ODE) boundary value problem: Eardley uses a ‘beads on a string’ technique to solve the set of simultaneous equations, i.e., imagining the curve to be defined as a bead on each ray of constant angle. He solves for the positions on each ray at which the relation is satisfied everywhere. | |
48 | As another comparison, the Lorene apparent horizon finder (discussed in more detail in Section 8.4.2), running on a roughly similar processor, takes between 3 and 6 seconds to find apparent horizons to comparable accuracy. | |
49 | The convergence problems, which Thornburg [153] noted when high-spatial-frequency perturbations are present in the slice’s geometry, seem to be rare in practice. | |
50 | In addition, at least two different research groups have now ported, or are in the process of porting, AHFinderDirect to
their own (non-![]() |
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51 | Schnetter et al. [135![]() |
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52 | There is one complication here: Any local apparent horizon finding algorithm may fail if the initial guess is not good
enough, even if the desired surface is actually present. The solution is to use the constant-expansion surface for a slightly
larger expansion ![]() ![]() ![]() ![]() |
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53 | As far as I know this is the only case that has been considered for horizon pretracking. Extension to other types of apparent horizon finders might be a fruitful area for further research. | |
54 | This refers to the period before a common apparent horizon is found. Once a common apparent horizon is found, then pretracking can be disabled as the apparent horizon finder can easily “track” the apparent horizon’s motion from one time step to the next. With a binary search the number of iterations depends only weakly (logarithmically) on the pretracking algorithm’s accuracy tolerance. It might be possible to replace the binary search by a more sophisticated 1-dimensional search algorithm (I discuss such algorithms in Appendix A), potentially cutting the number of iterations substantially. This might be a fruitful area for further research. | |
55 | Alternatively, a flow algorithm could use the most recent previously-found apparent horizon as an initial guess. In this case the algorithm would have only local convergence (in particular, it would probably fail to find a new outermost MOTS that appeared well outside the previously-found MOTS). However, the algorithm would only need to flow the surface a small distance, so the algorithm should be fairly fast. | |
56 | Cited as Ref. [17] by [80![]() |
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57 | Linearizing the ![]() ![]() ![]() |
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58 | For a spatial resolution ![]() ![]() |
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59 | Richtmyer and Morton [129, Section 7.5] give a very clear presentation and analysis of the Du Fort–Frankel scheme. | |
60 | More precisely, Pasch [118![]() ![]() |
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61 | Teukolsky [152], and Leiler and Rezzolla [101] have analyzed ICN’s stability under various conditions. | |
62 | See [42, 158] for general introductions to multigrid algorithms for elliptic PDEs. | |
63 | AHFinder also includes a minimization algorithm (Section 8.3). | |
64 | The parabola generically has two roots, but normally only one of them lies between ![]() ![]() |
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65 | The numerical-analysis literature usually refers to this as the “initial value problem”. Unfortunately, in a relativity context
this terminology often causes confusion with the “initial data problem” of solving the ADM constraint equations. I use the
term “time-integration problem for ODEs” to (try to) avoid this confusion. In this appendix, sans-serif lower-case letters
![]() ![]() ![]() ![]() ![]() |
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66 | LSODA can also automatically detect stiff systems of ODEs and adjust its integration scheme so as to handle them efficiently. |
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