Abstract
Event and apparent horizons are key diagnostics for the presence and properties of black holes. In this article I review numerical algorithms and codes for finding event and apparent horizons in numericallycomputed spacetimes, focusing on calculations done using the 3 + 1 ADM formalism. The event horizon of an asymptoticallyflat spacetime is the boundary between those events from which a futurepointing null geodesic can reach future null infinity and those events from which no such geodesic exists. The event horizon is a (continuous) null surface in spacetime. The event horizon is defined nonlocally in time: it is a global property of the entire spacetime and must be found in a separate postprocessing phase after all (or at least the nonstationary part) of spacetime has been numerically computed.
There are three basic algorithms for finding event horizons, based on integrating null geodesics forwards in time, integrating null geodesics backwards in time, and integrating null surfaces backwards in time. The last of these is generally the most efficient and accurate.
In contrast to an event horizon, an apparent horizon is defined locally in time in a spacelike slice and depends only on data in that slice, so it can be (and usually is) found during the numerical computation of a spacetime. A marginally outer trapped surface (MOTS) in a slice is a smooth closed 2surface whose futurepointing outgoing null geodesics have zero expansion Θ. An apparent horizon is then defined as a MOTS not contained in any other MOTS. The MOTS condition is a nonlinear elliptic partial differential equation (PDE) for the surface shape, containing the ADM 3metric, its spatial derivatives, and the extrinsic curvature as coefficients. Most “apparent horizon” finders actually find MOTSs.
There are a large number of apparent horizon finding algorithms, with differing tradeoffs between speed, robustness, accuracy, and ease of programming. In axisymmetry, shooting algorithms work well and are fairly easy to program. In slices with no continuous symmetries, spectral integraliteration algorithms and ellipticPDE algorithms are fast and accurate, but require good initial guesses to converge. In many cases, Schnetter’s “pretracking” algorithm can greatly improve an ellipticPDE algorithm’s robustness. Flow algorithms are generally quite slow but can be very robust in their convergence. Minimization methods are slow and relatively inaccurate in the context of a finite differencing simulation, but in a spectral code they can be relatively faster and more robust.
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1 Part I Introduction
Systems with strong gravitational fields, particularly systems which may contain event horizons and/or apparent horizons, are a major focus of numerical relativity. The usual output of a numerical relativity simulation is some (approximate, discrete) representation of the spacetime geometry (the 4metric and possibly its derivatives) and any matter fields, but not any explicit information about the existence, precise location, or other properties of any event/apparent horizons. To gain this information, we must explicitly find the horizons from the numericallycomputed spacetime geometry. The subject of this review is numerical algorithms and codes for doing this, focusing on calculations done using the 3 + 1 ADM formalism [14, 163]. Baumgarte and Shapiro [27, Section 6] have also recently reviewed event and apparent horizon finding algorithms. The scope of this review is limited to the finding of event/apparent horizons and omits any but the briefest mention of the many uses of this information in gaining physical understanding of numericallycomputed spacetimes.
In this review I distinguish between a numerical algorithm (an abstract description of a mathematical computation; also often known as a “method” or “scheme”), and a computer code (a “horizon finder”, a specific piece of computer software which implements a horizon finding algorithm or algorithms). My main focus is on the algorithms, but I also mention specific codes where they are freely available to other researchers.
In this review I have tried to cover all the major horizon finding algorithms and codes, and to accurately credit the earliest publication of important ideas. However, in a field as large and active as numerical relativity, it is not unlikely that I have overlooked and/or misdescribed some important research. I apologise to anyone whose work I’ve slighted, and I ask readers to help make this a truly “living” review by sending me corrections, updates, and/or pointers to additional work (either their own or others) that I should discuss in future revisions of this review.
The general outline of this review is as follows: In the remainder of Part I, I define notation and terminology (Section 1), discuss how 2surfaces should be parameterized (Section 2), and outline some of the softwareengineering issues that arise in modern numerical relativity codes (Section 3). I then discuss numerical algorithms and codes for finding event horizons (Part II) and apparent horizons (Part III). Finally, in the appendices I briefly outline some of the excellent numerical algorithms/codes available for two standard problems in numerical analysis, the solution of a single nonlinear algebraic equation (Appendix A) and the time integration of a system of ordinary differential equations (Appendix B).
2 Notation and Terminology
Except as noted below, I generally follow the sign and notation conventions of Wald [160]. I assume that all spacetimes are globally hyperbolic, and for eventhorizon finding I further assume asymptotic flatness; in this latter context \({{\mathcal J}^ +}\) is future null infinity. I use the Penrose abstractindex notation, with summation over all repeated indices. 4indices abc range over all spacetime coordinates {x^{a}}, and 3indices ijk range over the spatial coordinates {x^{i}} in a spacelike slice t = constant. The spacetime coordinates are thus x^{a} = (t, x^{i}).
Indices uvw range over generic angular coordinates (θ, ϕ) on S^{2} or on a horizon surface. Note that these coordinates are conceptually distinct from the 3dimensional spatial coordinates x^{i}. Depending on the context, (θ, ϕ) may or may not have the usual polarspherical topology. Indices ijk label angular grid points on S^{2} or on a horizon surface. These are 2dimensional indices: a single such index uniquely specifies an angular grid point. δ_{IJ} is the Kronecker delta on the space of these indices or, equivalently, on surface grid points.
For any indices p and q, ∂_{p} and ∂_{pq} are the coordinate partial derivatives ∂/∂x^{p} and ∂^{2}/∂x^{p}∂x^{q} respectively; for any coordinates µ and ν, ∂_{u} and ∂_{µν} are the coordinate partial derivatives ∂/∂/µ and ∂^{2}/∂µ∂ν respectively. Δ is the flatspace angular Laplacian operator on S^{2}, while Δx refers to a finitedifference grid spacing in some variable x.
g_{ab} is the spacetime 4metric, and g^{ab} the inverse spacetime 4metric; these are used to raise and lower 4indices. \(\Gamma _{ab}^c\) are the 4Christoffel symbols. \({{\mathcal L}_\upsilon}\) is the Lie derivative along the 4vector field v^{a}.
I use the 3 + 1 “ADM” formalism first introduced by Arnowitt, Deser, and Misner [14]; York [163] gives a general overview of this formalism as it is used in numerical relativity. g_{ij} is the 3metric defined in a slice, and g^{ij} is the inverse 3metric; these are used to raise and lower 3indices. ∇_{i} is the associated 3covariant derivative operator, and \(\Gamma _{ij}^k\) are the 3Christoffel symbols. α and β^{i} are the 3 + 1 lapse function and shift vector respectively, so the spacetime line element is
As is common in 3 + 1 numerical relativity, I follow the sign convention of Misner, Thorne, and Wheeler [112] and York [163] in defining the extrinsic curvature of the slice as \({K_{ij}} =  {1 \over 2}{{\mathcal L}_n}{g_{ij}} =  {\nabla _i}{n_j}\), where n^{a} is the futurepointing unit normal to the slice. (In contrast, Wald [160] omits the minus signs from this definition.) \(K\equiv {K_i^i}\) is the trace of the extrinsic curvature K_{ij}. m_{ADM} is the ADM mass of an asymptotically flat slice.
I often write a differential operator as F = F (y, ∂_{u}y, ∂_{uv}y; g_{ij}, ∂_{k}g_{ij}, K_{ij}), where the “;” notation means that F is a (generally nonlinear) algebraic function of the variable y and its 1st and 2nd angular derivatives, and that F also depends on the coefficients g_{ij}, ∂_{k}g_{ij}, and K_{ij} at the apparent horizon position.
There are three common types of spacetimes/slices where numerical event or apparent horizon finding is of interest: sphericallysymmetric spacetimes/slices, axisymmetric spacetimes/slices, and spacetimes/slices with no continuous spatial symmetries (no spacelike Killing vectors). I refer to the latter as “fully generic” spacetimes/slices.
In this review I use the abbreviations “ODE” for ordinary differential equation, “PDE” for partial differential equation, “CE surface” for constantexpansion surface, and “MOTS” for marginally outer trapped surface. Names in Small Capitals refer to horizon finders and other computer software.
When discussing iterative numerical algorithms, it is often convenient to use the concept of an algorithm’s “radius of convergence”. Suppose the solution space within which the algorithm is iterating is S. Then given some norm ‖ · ‖ on S, the algorithm’s radius of convergence about a solution s ∈ S is defined as the smallest r > 0 such that the algorithm will converge to the correct solution s for any initial guess g with ‖g − s‖ ≤ r. We only rarely know the exact radius of convergence of an algorithm, but practical experience often provides a rough estimate^{Footnote 1}.
3 2Surface Parameterizations
3.1 Levelsetfunction parameterizations
The most general way to parameterize a 2surface in a slice is to define a scalar “levelset function” F on some neighborhood of the surface, with the surface itself then being defined as the level set
Assuming the surface to be orientable, it is conventional to choose F so that F > 0 (F < 0) outside (inside) the surface. The choice of levelset function for a given surface is nonunique, but in general this is not a problem.
This parameterization is valid for any surface topology including timedependent topologies. The 2surface itself can then be found by a standard isosurfacefinding algorithm such as the marchingcubes algorithm [105]. (This algorithm is widely used in computer graphics and is implemented in a number of widelyavailable software libraries.)
3.2 Strahlkörper parameterizations
Most apparent horizon finders, and some eventhorizon finders, assume that each connected component of the apparent (event) horizon has S^{2} topology. With the exception of toroidal event horizons (discussed in Section 4), this is generally a reasonable assumption.
To parameterize an S^{2} surface’s shape, it is common to further assume that we are given (or can compute) some “local coordinate origin” point inside the surface such that the surface’s 3coordinate shape relative to that point is a “Strahlkörper” (literally “ray body”, or more commonly “starshaped region”), defined by Minkowski [138, Page 108] as
a region in nD Euclidean space containing the origin and whose surface, as seen from the origin, exhibits only one point in any direction.
The Strahlkörper assumption is a significant restriction on the horizon’s coordinate shape (and the choice of the local coordinate origin). For example, it rules out the coordinate shape and local coordinate origin illustrated in Figure 1: a horizon with such a coordinate shape about the local coordinate origin could not be found by any horizon finder which assumes a Strahlkörper surface parameterization.
For eventhorizon finding, algorithms and codes are now available which allow an arbitrary horizon topology with no Strahlkörper assumption (see the discussion in Section 5.3.3 for details). For apparent horizon finding, the flow algorithms discussed in Section 8.7 theoretically allow any surface shape, although many implementations still make the Strahlkörper assumption. Removing this assumption for other apparent horizon finding algorithms might be a fruitful area for further research.
Given the Strahlkörper assumption, the surface can be explicitly parameterized as
where r is the Euclidean distance from the local coordinate origin to a surface point, (θ, ϕ) are generic angular coordinates on the horizon surface (or equivalently on S^{2}), and the “horizon shape function” h : S^{2} → ℜ^{+} is a positive realvalued function on the domain of angular coordinates defining the surface shape. Given the choice of local coordinate origin, there is clearly a onetoone mapping between Strahlkörper 2surfaces and horizon shape functions.
There are two common ways to discretize a horizon shape function:

Spectral representation
Here we expand the horizon shape function h in an infinite series in some (typically orthonormal) set of basis functions such as spherical harmonics Y_{ℓm} or symmetric tracefree tensors^{Footnote 2},
$$h(\theta, \phi) = \sum\limits_{\ell, m} {{a_{\ell m}}{Y_{\ell m}}(\theta, \phi).}$$(5)This series can then be truncated at some finite order ℓ_{max}, and the N_{coeff} = ℓ_{max}(ℓ_{max}+1) coefficients {a_{ℓm}} used to represent (discretely approximate) the horizon shape. For reasonable accuracy, ℓ_{max} is typically on the order of 8 to 12.

Finite difference representation
Here we choose some finite grid of angular coordinates {(θ_{K}, ϕ_{K})}, K = 1, 2, 3, …, N_{ang} on S^{2} (or equivalently on the surface)^{Footnote 3}, and represent (discretely approximate) the surface shape by the N_{ang} values
$$\{h({\theta _{\rm{K}}},{\phi _{\rm{K}}})\} \quad {\rm{K = 1,2,3,}} \ldots, {N_{{\rm{ang}}}}.$$(6)For reasonable accuracy, N_{ang} is typically on the order of a few thousand.
It is sometimes useful to explicitly construct a levelset function describing a given Strahlkörper. A common choice here is
3.3 Finiteelement parameterizations
Another way to parameterize a 2surface is via finite elements where the surface is modelled as a triangulated mesh, i.e. as a set of interlinked “vertices” (points in the slice, represented by their spatial coordinates {x^{i}}), “edges” (represented by ordered pairs of vertices), and faces. Typically only triangular faces are used (represented as oriented triples of vertices).
A key benefit of this representation is that it allows an arbitrary topology for the surface. However, determining the actual surface topology (e.g. testing for whether or not the surface selfintersects) is somewhat complicated.
This representation is similar to that of Regge calculus [128, 72]^{Footnote 4}, and can similarly be expected to show 2nd order convergence with the surface resolution.
4 SoftwareEngineering Issues
Historically, numerical relativists wrote their own codes from scratch. As these became more complex, many researchers changed to working on “group codes” with multiple contributors.
4.1 Software libraries and toolkits
More recently, particularly in work on fully generic spacetimes, where all three spatial dimensions must be treated numerically, there has been a strong trend towards the use of higherlevel software libraries and modular “computational toolkits” such as Cactus [74] (http://www.cactuscode.org). These have a substantial learning overhead, but can allow researchers to work much more productively by focusing more on numerical relativity instead of computerscience and softwareengineering issues such as parameterfile parsing, parallelization, I/O, etc.
A particularly important area for such software infrastructure is mesh refinement^{Footnote 5}. This is essential to much current numericalrelativity research but is moderately difficult to implement even in only one spatial dimension, and much harder in multiple spatial dimensions. There are now a number of software libraries providing multidimensional meshrefinement infrastructure (sometimes combined with parallelization), such as those listed in Table 1. The Cactus toolkit can be used in either unigrid or meshrefinement modes, the latter using a “meshrefinement driver” such as PAGH or Carpet [134, 131] (http://www.carpetcode.org).
In this review I point out event and apparent horizon finders which have been written in particular frameworks and comment on whether they work with mesh refinement.
4.2 Code reuse and sharing
Another important issue is that of code reuse and sharing. It is common for codes to be shared within a research group but relatively uncommon for them to be shared between different (competing) research groups. Even apart from concerns about competitive advantage, without a modular structure and clear documentation it is difficult to reuse another group’s code. The use of a common computational toolkit can greatly simplify such reuse.
If such reuse can be accomplished, it becomes much easier for other researchers to build on existing work rather than having to “reinvent the wheel”. As well as the obvious ease of reusing existing code that (hopefully!) already works and has been thoroughly debugged and tested, there is another — less obvious — benefit of code sharing: It greatly eases the replication of past work, which is essential as a foundation for new development. That is, without access to another researcher’s code, it can be surprisingly difficult to replicate her results because the success or failure of a numerical algorithm frequently depends on subtle implementation details not described in even the most complete of published papers.
Event and apparent horizon finders are excellent candidates for software reuse: Many numericalrelativity researchers can benefit from using them, and they have a relatively simple interface to an underlying numericalrelativity simulation. Even if a standard computational toolkit is not used, this relatively simple interface makes it fairly easy to port an event or apparent horizon finder to a different code.
Table 2 lists event and apparent horizon finders which are freely available to any researcher.
4.3 Using multiple event/apparent horizon finders
It is useful to have multiple event or apparent horizon finders available: Their strengths and weaknesses may complement each other, and the extent of agreement or disagreement between their results can help to estimate the numerical accuracy. For example, Figure 11 shows a comparison between the irreducible masses of apparent horizons in a binary black hole coalescence simulation (Alcubierre et al. [5], [Figure 4b]), as computed by two different apparent horizon finders in the Cactus toolkit, AHFinder and AHFinderDireot. In this case the two agree to within about 2% for the individual horizons and 0.5% for the common horizon.
5 Part II Finding Event Horizons
6 Introduction
The black hole region of an asymptoticallyflat spacetime is defined [81, 82] as the set of events from which no futurepointing null geodesic can reach future null infinity (\({{\mathcal J}^ +}\)). The event horizon is defined as the boundary of the black hole region. The event horizon is a null surface in spacetime with (in the words of Hawking and Ellis [82, Page 319]) “a number of nice properties” for studying the causal stucture of spacetime.
The event horizon is a global property of an entire spacetime and is defined nonlocally in time: The event horizon in a slice is defined in terms of (and cannot be computed without knowing) the full future development of that slice.
In practice, to find an event horizon in a numericallycomputed spacetime, we typically instrument a numerical evolution code to write out data files of the 4metric. After the evolution (or at least the strongfield region) has reached an approximatelystationary final state, we then compute a numerical approximation to the event horizon in a separate postprocessing pass, using the 4metric data files as inputs.
As a null surface, the event horizon is necessarily continuous. In theory it need not be anywhere differentiable^{Footnote 6}, but in practice this behavior rarely occurs^{Footnote 7}: The event horizon is generally smooth except for possibly a finite set of “cusps” where new generators join the surface; the surface normal has a jump discontinuity across each cusp. (The classic example of such a cusp is the “inseam” of the “pair of pants” event horizon illustrated in Figures 4 and 5.)
A black hole is defined as a connected component of the black hole region in a 3 + 1 slice. The boundary of a black hole (the event horizon) in a slice is a 2dimensional set of events. Usually this has 2sphere (S^{2}) topology. However, numerically simulating rotating dust collapse, Abrahams et al. [1] found that in some cases the event horizon in a slice may be toroidal in topology. Lehner et al. [99], and Husa and Winicour [91] have used null (characteristic) algorithms to give a general analysis of the event horizon’s topology in black hole collisions; they find that there is generically a (possibly brief) toroidal phase before the final 2spherical state is reached. Lehner et al. [100] later calculated movies showing this behavior for several asymmetric black hole collisions.
7 Algorithms and Codes for Finding Event Horizons
There are three basic eventhorizon finding algorithms:

Integrate null geodesics forwards in time (Section 5.1).

Integrate null geodesics backwards in time (Section 5.2).

Integrate null surfaces backwards in time (Section 5.3).
I describe these in detail in the following.
7.1 Integrating null geodesics forwards in time
The first generation of eventhorizon finders were based directly on Hawking’s original definition of an event horizon: an event \({\mathcal P}\) is within the black hole region of spacetime if and only if there is no futurepointing “escape route” null geodesic from \({\mathcal P}\) to \({{\mathcal J}^ +}\); the event horizon is the boundary of the black hole region.
That is, as described by Hughes et al. [88], we numerically integrate the null geodesic equation
(where λ is an affine parameter) forwards in time from a set of starting events and check which events have “escaping” geodesics. For analytical or semianalytical studies like that of Bishop [31], this is an excellent algorithm.
For numerical work it is straightforward to rewrite the null geodesic equation (8) as a coupled system of two firstorder equations, giving the time evolution of photon positions and 3momenta in terms of the 3 + 1 geometry variables α, β^{i}, g^{ij}, and their spatial derivatives. These can then be timeintegrated by standard numerical algorithms^{Footnote 8}. However, in practice several factors complicate this algorithm.
We typically only know the 3 + 1 geometry variables on a discrete lattice of spacetime grid points, and we only know the 3 + 1 geometry variables themselves, not their spatial derivatives. Therefore we must numerically differentiate the field variables, then interpolate the field variables and their spacetime derivatives to each integration point along each null geodesic. This is straightforward to implement^{Footnote 9}, but the numerical differentiation tends to amplify any numerical noise that may be present in the field variables.
Another complicating factor is that the numerical computations generally only span a finite region of spacetime, so it is not entirely obvious whether or not a given geodesic will eventually reach \({{\mathcal J}^ +}\). However, if the final numericallygenerated slice contains an apparent horizon, we can use this as an approximation: Any geodesic which is inside this apparent horizon will definitely not reach \({{\mathcal J}^ +}\), while any other geodesic may be assumed to eventually reach \({{\mathcal J}^ +}\) if its momentum is directed away from the apparent horizon. If the final slice (or at least its strongfield region) is approximately stationary, the error from this approximation should be small. I discuss this stationarity assumption further in Section 5.3.1.
7.1.1 Sphericallysymmetric spacetimes
In spherical symmetry this algorithm works well and has been used by a number of researchers. For example, Shapiro and Teukolsky [141, 142, 143, 144] used it to study event horizons in a variety of dynamical evolutions of spherically symmetric collapse systems. Figure 2 shows an example of the event and apparent horizons in one of these simulations.
7.1.2 Nonsphericallysymmetric spacetimes
In a nonsphericallysymmetric spacetime, several factors make this algorithm very inefficient:

Many trial events must be tried to accurately resolve the event horizon’s shape. (Hughes et al. [88] describe a 2stage adaptive numerical algorithm for choosing the trial events so as to accurately locate the event horizon as efficiently as possible.)

At each trial event we must try many different trialgeodesic starting directions to see if any of the geodesics escape to \({{\mathcal J}^ +}\) (or our numerical approximation to it). Hughes et al. [88] report needing only 48 geodesics per trial event in several nonrotating axisymmetric spacetimes, but about 750 geodesics per trial event in rotating axisymmetric spacetimes, with up to 3000 geodesics per trial event in some regions of the spacetimes.

Finally, each individual geodesic integration requires many (short) time steps for an accurate integration, particularly in the strongfield region near the event horizon.
Because of these limitations, for nonsphericallysymmetric spacetimes the “integrate null geodesics forwards” algorithm has generally been supplanted by the more efficient algorithms I describe in the following.
7.2 Integrating null geodesics backwards in time
It is wellknown that futurepointing outgoing null geodesics near the event horizon tend to diverge exponentially in time away from the event horizon. Figure 3 illustrates this behavior for Schwarzschild spacetime, but the behavior is actually quite generic.
Anninos et al. [7] and Libson et al. [103] observed that while this instability is a problem for the “integrate null geodesics forwards in time” algorithm (it forces that algorithm to take quite short time steps when integrating the geodesics), we can turn it to our advantage by integrating the geodesics backwards in time: The geodesics will now converge on to the horizon^{Footnote 10}.
This eventhorizon finding algorithm thus integrates a large number of such (futurepointing outgoing) null geodesics backwards in time, starting on the final numericallygenerated slice. As the backwards integration proceeds, even geodesics which started far from the event horizon will quickly converge to it. This can be seen, for example, in Figures 2 and 3.
Unfortunately, this convergence property holds only for outgoing geodesics. In spherical symmetry the distinction between outgoing and ingoing geodesics is trivial but, as described by Libson et al. [103],
[…] for the general 3D case, when the two tangential directions of the EH are also considered, the situation becomes more complicated. Here normal and tangential are meant in the 3D spatial, not spacetime, sense. Whether or not a trajectory can eventually be “attracted” to the EH, and how long it takes for it to become “attracted,” depends on the photon’s starting direction of motion. We note that even for a photon which is already exactly on the EH at a certain instant, if its velocity at that point has some component tangential to the EH surface as generated by, say, numerical inaccuracy in integration, the photon will move outside of the EH when traced backward in time. For a small tangential velocity, the photon will eventually return to the EH […but] the position to which it returns will not be the original position.
This kind of tangential drifting is undesirable not just because it introduces inaccuracy in the location of the EH, but more importantly, because it can lead to spurious dynamics of the “EH” thus found. Neighboring generators may cross, leading to numerically artificial caustic points […].
Libson et al. [103] also observed:
Another consequence of the second order nature of the geodesic equation is that not just the positions but also the directions must be specified in starting the backward integration. Neighboring photons must have their starting direction well correlated in order to avoid tangential drifting across one another.
Libson et al. [103] give examples of the numerical difficulties that can result from these difficulties and conclude that this eventhorizon finding algorithm
[…] is still quite demanding in finding an accurate history of the EH, although the difficulties are much milder than those arising from the instability of integrating forward in time.
Because of these difficulties, this algorithm has generally been supplanted by the “backwards surface” algorithm I describe next.
7.3 Integrating null surfaces backwards in time
Anninos et al. [7], Libson et al. [103], and Walker [162] introduced the important concept of explicitly (numerically) finding the event horizon as a null surface in spacetime. They observed that if we parameterize the event horizon with any levelset function F satisfying the basic levelset definition (3), then the condition for the surface F = 0 to be null is just
Applying a 3 + 1 decomposition to this then gives a quadratic equation which can be solved to find the time evolution of the levelset function,
Alternatively, assuming the event horizon in each slice to be a Strahlkörper in the manner of Section 2.2, we can define a suitable levelset function F by Equation (7). Substituting this definition into Equation (10) then gives an explicit evolution equation for the horizon shape function,
Surfaces near the event horizon share the same “attraction” property discussed in Section 5.2 for geodesics near the event horizon. Thus by integrating either surface representation (10) or (11) backwards in time, we can refine an initial guess into a very accurate approximation to the event horizon.
In contrast to the null geodesic equation (8), neither Equation (10) nor Equation (11) contain any derivatives of the 4metric (or equivalently the 3 + 1 geometry variables). This makes it much easier to integrate these latter equations accurately^{Footnote 11}. This formulation of the eventhorizon finding problem also completely eliminates the tangentialdrifting problem discussed in Section 5.2, since the levelset function only parameterizes motion normal to the surface.
7.3.1 Error bounds: Integrating a pair of surfaces
For a practical algorithm, it is useful to integrate a pair of trial null surfaces backwards: an “innerbound” one which starts (and thus always remains) inside the event horizon and an “outerbound” one which starts (and thus always remains) outside the event horizon. If the final slice contains an apparent horizon then any 2surface inside this can serve as our innerbound surface. However, choosing an outerbound surface is more difficult.
It is this desire for a reliable outer bound on the event horizon position that motivates our requirement (Section 4) for the final slice (or at least its strongfield region) to be approximately stationary: In the absence of timedependent equations of state or external perturbations entering the system, this requirement ensures that, for example, any surface substantially outside the apparent horizon can serve as an outerbound surface.
Assuming we have an inner and an outerbound surface on the final slice, the spacing between these two surfaces after some period of backwards integration then gives an error bound for the computed event horizon position. Equivalently, a necessary (and, if there are no other numerical problems, sufficient) condition for the eventhorizon finding algorithm to be accurate is that the backwards integration must have proceeded far enough for the spacing between the two trial surfaces to be “small”. For a reasonable definition of “small”, this typically takes at least 15m_{ADM} of backwards integration, with 20m_{ADM} or more providing much higher accuracy.
In some cases it is difficult to obtain a long enough span of numerical data for this backwards integration. For example, in some simulations of binary black hole collisions, the evolution becomes unstable and crashes soon after a common apparent horizon forms. This means that we cannot compute an accurate event horizon for the most interesting region of the spacetime, that which is close to the blackhole merger. There is no good solution to this problem except for the obvious one of developing a stable (or lessunstable) simulation that can be continued for a longer time.
7.3.2 Explicit Strahlkörper surface representation
The initial implementations of the “integrate null surface backwards” algorithm by Anninos et al. [7], Libson et al. [103], and Walker [162] were based on the explicit Strahlkörper surface integration formula (11), further restricted to axisymmetry^{Footnote 12}.
For a single black hole the coordinate choice is straightforward. For the twoblackhole case, the authors used topologically cylindrical coordinates (ρ, z, ϕ), where the two black holes collide along the axisymmetry (z) axis. Based on the symmetry of the problem, they then assumed that the event horizon shape could be written in the form
in each t = constant slice.
This spacetime’s event horizon has the nowclassic “pair of pants” shape, with a nondifferentiable cusp along the “inseam” (the z axis ρ = 0) where new generators join the surface. The authors tried two ways of treating this cusp numerically:

Since the cusp’s location is known a priori, it can be treated as a special case in the angular finite differencing, using onesided numerical derivatives as necessary.

Alternatively, in 1994 Thorne suggested calculating the union of the event horizon and all its null generators (including those which have not yet joined the surface)^{Footnote 13}. This “surface” has a complicated topology (it selfintersects along the cusp), but it is smooth everywhere. This is illustrated by Figure 4, which shows a crosssection of this surface in a single slice, for a headon binary black hole collision. For comparison, Figure 5 shows a perspective view of part of the event horizon and some of its generators, for a similar headon binary black hole collision.
Caveny et al. [44, 46] implemented the “integrate null surfaces backwards” algorithm for fully generic numericallycomputed spacetimes using the explicit Strahlkörper surface integration formula (11). To handle moving black holes, they recentered each black hole’s Strahlkörper parameterization (4) on the black hole’s coordinate centroid at each time step.
For singleblackhole test cases (Kerr spacetime in various coordinates), they report typical accuracies of a few percent in determining the event horizon position and area. For binaryblackhole test cases (KastorTraschen extremalcharge black hole coalescence with a cosmological constant), they detect black hole coalescence (which appears as a bifurcation in the backwards time integration) by the “necking off” of the surface. Figure 6 shows an example of their results.
7.3.3 Levelset parameterization
Caveny et al. [44, 45] and Diener [60] (independently) implemented the “integrate null surfaces backwards” algorithm for fully generic numericallycomputed spacetimes, using the levelset function integration formula (10). Here the levelset function F is initialized on the final slice of the evolution and evolved backwards in time using Equation (10) on (conceptually) the entire numerical grid. (In practice, only a smaller box containing the event horizon need be evolved.)
This surface parameterization has the advantage that the eventhorizon topology and (non) smoothness are completely unconstrained, allowing the numerical study of configurations such as toroidal event horizons (discussed in Section 4). It is also convenient that the levelset function F is defined on the same numerical grid as the spacetime geometry, so that no interpolation is needed for the evolution.
The major problem with this algorithm is that during the backwards evolution, spatial gradients in F tend to steepen into a jump discontinuity at the event horizon^{Footnote 14}, eventually causing numerical difficulty.
Caveny et al. [44, 45] deal with this problem by adding an artificial viscosity (i.e. diffusion) term to the levelset function evolution equation, smoothing out the jump discontinuity in F. That is, instead of Equation (10), they actually evolve F via
where rhs Eq. (10) is the right hand side of Equation (10) and ∇^{2} is a generic 2nd order linear (elliptic) spatial differential operator, and ε > 0 is a (small) dissipation constant. This scheme works, but the numerical viscosity does seem to lead to significant errors (several percent) in their computed eventhorizon positions and areas^{Footnote 15}, and even failure to converge to the correct solution for some test cases (e.g. rapidlyspinning Kerr black holes).
Alternatively, Diener [60] developed a technique of periodically reinitializing the levelset function to approximately the signed distance from the event horizon. To do this, he periodically evolves
in an unphysical “pseudotime” λ until an approximate steady state has been achieved. He reports that this works well in most circumstances but can significantly distort the computed event horizon if the F = 0 isosurface (the current approximation to the event horizon) is only a few grid points thick in any direction, as typically occurs just around the time of a topology change in the isosurface. He avoids this problem by estimating the minimum thickness of this isosurface and, if it is below a threshold, deferring the reinitialization.
In various tests on analytical data, Diener [60] found this eventhorizon finder, EHFinder, to be robust and highly accurate, typically locating the event horizon to much less than 1% of the 3dimensional grid spacing. As an example of results obtained with EHFinder, Figure 7 shows two views of the numericallycomputed event horizon for a spiraling binary black hole collision. As another example, Figure 8 shows the numericallycomputed event and apparent horizons in the collapse of a rapidly rotating neutron star to a Kerr black hole. (The apparent horizons were computed using the AHFinderDireot code described in Section 8.5.7.)
EHFinder is implemented as a freely available module (“thorn”) in the Cactus computational toolkit (see Table 2). It originally worked only with the PUGH unigrid driver, but work is ongoing [61] to enhance it to work with the Carpet meshrefinement driver [134, 131].
8 Summary of Algorithms/Codes for Finding Event Horizons
In spherical symmetry the “integrate null geodesics forwards” algorithm (Section 5.1) can be used, although the “integrate null geodesics backwards” and “integrate null surfaces backwards” algorithms (Sections 5.2 and 5.3 respectively) are more efficient.
In nonsphericallysymmetric spacetimes the “integrate null surfaces backwards” algorithm (Section 5.3) is clearly the best algorithm known: It is efficient, accurate, and fairly easy to implement. For generic spacetimes, Diener’s eventhorizon finder EHFinder [60] is particularly notable as a freely available implementation of this algorithm as a module (“thorn”) in the widelyused Cactus computational toolkit (see Table 2).
9 Part III Finding Apparent Horizons
10 Introduction
10.1 Definition
Given a (spacelike) 3 + 1 slice, a “trapped surface” is defined as a smooth closed 2surface in the slice whose futurepointing outgoing null geodesics have negative expansion Θ. The “trapped region” in the slice is then defined as the union of all trapped surfaces, and the “apparent horizon” is defined as the outer boundary of the trapped region.
While mathematically elegant, this definition is not convenient for numerically finding apparent horizons. Instead, an alternate definition can be used: A MOTS is defined as a smooth (differentiable) closed orientable 2surface in the slice whose futurepointing outgoing null geodesics have zero expansion Θ.^{Footnote 16} There may be multiple MOTSs in a slice, either nested within each other or intersecting^{Footnote 17}. An apparent horizon is then defined as an outermost MOTS in a slice, i.e. a MOTS not contained in any other MOTS. Kriele and Hayward [98] have shown that subject to certain technical conditions, this definition is equivalent to the “outer boundary of the trapped region” one.
Notice that the apparent horizon is defined locally in time (it can be computed using only Cauchy data on a spacelike slice), but (because of the requirement that it be closed) nonlocally in space^{Footnote 18}. Hawking and Ellis [82] discuss the general properties of MOTSs and apparent horizons in more detail.
Except for flow algorithms (Section 8.7), all numerical “apparent horizon” finding algorithms and codes actually find MOTSs, and hereinafter I generally follow the common (albeit sloppy) practice in numerical relativity of blurring the distinction between an MOTS and an apparent horizon.
10.2 General properties
Given certain technical assumptions (including energy conditions), the existence of any trapped surface (and hence any apparent horizon) implies that the slice contains a black hole^{Footnote 19}. (The converse of this statement is not true: An arbitrary (spacelike) slice through a black hole need not contain any apparent horizon^{Footnote 20}.) However, if an apparent horizon does exist, it necessarily coincides with, or is contained in, an event horizon. In a stationary spacetime the event and apparent horizons coincide.
It is this relation to the event horizon which makes apparent horizons valuable for numerical computation: An apparent horizon provides a useful approximation to the event horizon in a slice, but unlike the event horizon, an apparent horizon is defined locally in time and so can be computed “on the fly” during a numerical evolution.
Given a family of spacelike 3 + 1 slices which foliate part of spacetime, the union of the slices’ apparent horizons (assuming they exist) forms a worldtube^{Footnote 21}. This worldtube is necessarily either null or spacelike. If it is null, this worldtube is slicingindependent (choosing a different family of slices gives the same worldtube, at least so long as each slice still intersects the worldtube in a surface with 2sphere topology). However, if the worldtube is spacelike, it is slicingdependent: Choosing a different family of slices will in general give a different worldtube^{Footnote 22}.
10.3 Trapping, isolated, and dynamical horizons
Hayward [83] introduced the important concept of a “trapping horizon” (roughly speaking an apparent horizon worldtube where the expansion becomes negative if the surface is deformed in the inward null direction) along with several useful variants. Ashtekar, Beetle, and Fairhurst [16], and Ashtekar and Krishnan [18] later defined the related concepts of an “isolated horizon”, essentially an apparent horizon worldtube which is null, and a “dynamical horizon”, essentially an apparent horizon worldtube which is spacelike.
These worldtubes obey a variety of local and global conservation laws, and have many applications in analyzing numericallycomputed spacetimes. See the references cited above and also Dreyer et al. [63], Ashtekar and Krishnan [19, 20], Gourgoulhon and Jaramillo [76], Booth [36], and Schnetter, Krishnan, and Beyer [137] for further discussions, including applications to numerical relativity.
10.4 Description in terms of the 3 + 1 variables
In terms of the 3 + 1 variables, a MOTS (and thus an apparent horizon) satisfies the condition^{Footnote 23}
where s^{i} is the outwardpointing unit 3vector normal to the surface^{Footnote 24}. Assuming the Strahlkörper surface parameterization (4), Equation (15) can be rewritten in terms of angular 1st and 2nd derivatives of the horizon shape function h,
where Θ is a complicated nonlinear algebraic function of the arguments shown. (Shibata [146] and Thornburg [153, 156] give the Θ(h, ∂_{u}h, ∂_{uv}h) function explicitly.)
10.5 Geometry interpolation
Θ depends on the slice geometry variables g_{ij}, ∂_{k}g_{ij}, and K_{ij} at the horizon position^{Footnote 25}. In practice these variables are usually only known on the (3dimensional) numerical grid of the underlying numericalrelativity simulation^{Footnote 26}, so they must be interpolated to the horizon position and, more generally, to the position of each intermediateiterate trial shape the apparent horizon finding algorithm tries in the process of (hopefully) converging to the horizon position.
Moreover, usually the underlying simulation gives only g_{ij} and K_{ij}, so g_{ij} must be numerically differentiated to obtain ∂_{k}g_{ij}. As discussed by Thornburg [156, Section 6.1], it is somewhat more efficient to combine the numerical differentiation and interpolation operations, essentially doing the differentiation inside the interpolator^{Footnote 27}.
Thornburg [156, Section 6.1] argues that for an ellipticPDE algorithm (Section 8.5), for best convergence of the nonlinear elliptic solver the interpolated geometry variables should be smooth (differentiable) functions of the trial horizon surface position. He argues that that the usual Lagrange polynomial interpolation does not suffice here (in some cases his Newton’smethod iteration failed to converge) because this interpolation gives results which are only piecewise differentiable^{Footnote 28}. He uses Hermite polynomial interpolation to avoid this problem. Cook and Abrahams [51], and Pfeiffer et al. [124] use bicubic spline interpolation; most other researchers either do not describe their interpolation scheme or use Lagrange polynomial interpolation.
10.6 Criteria for assessing algorithms
Ideally, an apparent horizon finder should have several attributes:

Robust: The algorithm/code should find an (the) apparent horizon in a wide range of numericallycomputed slices, without requiring extensive tuning of initial guesses, iteration parameters, etc. This is often relatively easy to achieve for “tracking” the time evolution of an existing apparent horizon (where the most recent previouslyfound apparent horizon provides an excellent initial guess for the new apparent horizon position) but may be difficult for detecting the appearance of a new (outermost) apparent horizon in an evolution, or for initialdata or other studies where there is no “previous time step”.

Accurate: The algorithm/code should find an (the) apparent horizon to high accuracy and should not report spurious “solutions” (“solutions” which are not actually good approximations to apparent horizons or, at least, to MOTSs).

Efficient: The algorithm/code should be efficient in terms of its memory use and CPU time; in practice CPU time is generally the major constraint. It is often desirable to find apparent horizons at each time step (or, at least, at frequent intervals) during a numerical evolution. For this to be practical the apparent horizon finder must be very fast.
In practice, no apparent horizon finder is perfect in all these dimensions, so tradeoffs are inevitable, particularly when ease of programming is considered.
10.7 Local versus global algorithms
Apparent horizon finding algorithms can usefully be divided into two broad classes:

Local algorithms are those whose convergence is only guaranteed in some (functional) neighborhood of a solution. These algorithms require a “good” initial guess in order to find the apparent horizon. Most apparent horizon finding algorithms are local.

Global algorithms are those which can (in theory, ignoring finitestepsize and other numerical effects) converge to the apparent horizon independent of any initial guess. Flow algorithms (Section 8.7) are the only truely global algorithms. Zerofinding in spherical symmetry (Section 8.1) and shooting in axisymmetry (Section 8.2) are “almost global” algorithms: They require only 1dimensional searches, which (as discussed in Appendix A) can be programmed to be very robust and efficient. In many cases horizon pretracking (Section 8.6) can semiautomatically find an initial guess for a local algorithm, essentially making the local algorithm behave like an “almost global” one.
One might wonder why local algorithms are ever used, given the apparently superior robustness (guaranteed convergence independent of any initial guess) of global algorithms. There are two basic reasons:

In practice, local algorithms are much faster than global ones, particularly when “tracking” the time evolution of an existing apparent horizon.

Due to finitestepsize and other numerical effects, in practice even “global” algorithms may fail to converge to an apparent horizon. (That is, the algorithms may sometimes fail to find an apparent horizon even when one exists in the slice.)
11 Algorithms and Codes for Finding Apparent Horizons
Many researchers have studied the apparent horizon finding problem, and there are a large number of different apparent horizon finding algorithms and codes. Almost all of these require (assume) that any apparent horizon to be found is a Strahlkörper (Section 2) about some local coordinate origin; both finitedifference and spectral parameterizations of the Strahlkörper are common.
For slices with continuous symmetries, special algorithms are sometimes used:

ZeroFinding in Spherical Symmetry (Section 8.1)
In spherical symmetry the apparent horizon equation (16) becomes a 1dimensional nonlinear algebraic equation, which can be solved by zerofinding.

The Shooting Algorithm in Axisymmetry (Section 8.2)
In axisymmetry the apparent horizon equation (16) becomes a nonlinear 2point boundary value ODE, which can be solved by a shooting algorithm.
Alternatively, all the algorithms described below for generic slices are also applicable to axisymmetric slices and can take advantage of the axisymmetry to simplify the implementation and boost performance.
For fully generic slices, there are several broad categories of apparent horizon finding algorithms and codes:

Minimization Algorithms (Section 8.3)
These algorithms define a scalar norm on Θ over the space of possible trial surfaces. A generalpurpose scalarfunctionminimization routine is then used to search trialsurfaceshape space for a minimum of this norm (which should give Θ = 0).

Spectral IntegralIteration Algorithms (Section 8.4)
These algorithms expand the (Strahlkörper) apparent horizon shape function in a sphericalharmonic basis, use the orthogonality of spherical harmonics to write the apparent horizon equation as a set of integral equations for the spectral coefficients, and solve these equations using a functionaliteration algorithm.

EllipticPDE Algorithms (Section 8.5)
These algorithms write the apparent horizon equation (16) as a nonlinear elliptic (boundaryvalue) PDE for the horizon shape and solve this PDE using (typically) standard ellipticPDE numerical algorithms.

Horizon Pretracking (Section 8.6)
Horizon pretracking solves a slightly more general problem than apparent horizon finding: Roughly speaking, the determination of the smallest E ≥ 0 such that the equation Θ = E has a solution, and the determination of that solution. By monitoring the time evolution of E and of the surfaces satisfying this condition, we can determine — before it appears — approximately where (in space) and when (in time) a new MOTS will appear in a dynamic numericallyevolving spacetime. Horizon pretracking is implemented as a 1dimensional (binary) search using a slightlymodified ellipticPDE apparent horizon finding algorithm as a “subroutine”.

Flow Algorithms (Section 8.7)
These algorithms start with a large 2surface (larger than any possible apparent horizon in the slice) and shrink it inwards using an algorithm which ensures that the surface will stop shrinking when it coincides with the apparent horizon.
I describe the major algorithms and codes in these categories in detail in the following.
11.1 Zerofinding in spherical symmetry
In a spherically symmetric slice, any apparent horizon must also be spherically symmetric, so the apparent horizon equation (16) becomes a 1dimensional nonlinear algebraic equation Θ(h) = 0 for the horizon radius h. For example, adopting the usual (symmetryadapted) polarspherical spatial coordinates x^{i} = (r, θ, ϕ), we have [154, Equation (B7)]
Given the geometry variables g_{rr}, g_{θθ}, ∂_{r}g_{θθ}, and K_{θθ}, this equation may be easily and accurately solved using one of the zerofinding algorithms discussed in Appendix A^{Footnote 29}.
Zerofinding has been used by many researchers, including [141, 142, 143, 144, 119, 47, 139, 9, 154, 155]^{Footnote 30}. For example, the apparent horizons shown in Figure 2 were obtained using this algorithm. As another example, Figure 9 shows Θ(r) and h at various times in a (different) spherically symmetric collapse simulation.
11.2 The shooting algorithm in axisymmetry
In an axisymmetric spacetime we can use symmetryadapted coordinates (θ, ϕ), so (given the Strahlkörper assumption) without further loss of generality we can write the horizon shape function as h = h(θ). The apparent horizon equation (16) then becomes a nonlinear 2point boundaryvalue ODE for the horizon shape function h [146, Equation (1.1)]
where Θ(h) is a nonlinear 2nd order (ordinary) differential operator in h as shown.
Taking the angular coordinate θ to have the usual polarspherical topology, local smoothness of the apparent horizon gives the boundary conditions
where θ_{max} is π/2 if there is “bitant” reflection symmetry across the z = 0 plane, or π otherwise.
As well as the more general algorithms described in the following, this may be solved by a shooting algorithm^{Footnote 31}:

1.
Guess the value of h at one endpoint, say h(θ=0) ≡ h_{*}.

2.
Use this guessed value of h(θ=0) together with the boundary condition (19) there as initial data to integrate (“shoot”) the ODE (18) from θ=0 to the other endpoint θ=θ_{max}. This can be done easily and efficiently using one of the ODE codes described in Appendix B.

3.
If the numerically computed solution satisfies the other boundary condition (19) at θ=θ_{max} to within some tolerance, then the justcomputed h(θ) describes the (an) apparent horizon, and the algorithm is finished.

4.
Otherwise, adjust the guessed value h(θ=θ) ≡ h_{*} and try again. Because there is only a single parameter (h_{*}) to be adjusted, this can be done using one of the 1dimensional zerofinding algorithms discussed in Appendix A.
This algorithm is fairly efficient and easy to program. By trying a sufficiently wide range of initial guesses h_{*} this algorithm can give a high degree of confidence that all apparent horizons have been located, although this, of course, increases the cost.
Shooting algorithms of this type have been used by many researchers, for example [159, 66, 2, 29, 30, 145, 3, 4].
11.3 Minimization algorithms
This family of algorithms defines a scalar norm ‖ · ‖ on the expansion Θ over the space of possible trial surfaces, typically the meansquared norm
where the integral is over all solid angles on a trial surface.
Assuming the horizon surface to be a Strahlkörper and adopting the spectral representation (5) for the horizon surface, we can view the norm (20) as being defined on the space of spectral coefficients {a_{ℓm}}.
This norm clearly has a global minimum ‖Θ‖ = 0 for each solution of the apparent horizon equation (16). To find the apparent horizon we numerically search the spectralcoefficient space for this (a) minimum, using a generalpurpose “functionminimization” algorithm (code) such as Powell’s algorithm^{Footnote 32}.
Evaluating the norm (20) requires a numerical integration over the horizon surface: We choose some grid of N_{ang} points on the surface, interpolate the slice geometry fields (g_{ij}, ∂_{k}g_{ij}, and K_{ij}) to this grid (see Section 7.5), and use numerical quadrature to approximate the integral. In practice this must be done for many different trial surface shapes (see Section 8.3.2), so it is important that it be as efficient as possible. Anninos et al. [8] and Baumgarte et al. [26] discuss various ways to optimize and/or parallelize this calculation.
Unfortunately, minimization algorithms have two serious disadvantages for apparent horizon finding: They can be susceptible to spurious local minima, and they’re very slow at high angular resolutions. However, for the (fairly common) case where we want to find a common apparent horizon as soon as it appears in a binary blackhole (or neutronstar) simulation, minimization algorithms do have a useful ability to “anticipate” the formation of the common apparent horizon, in a manner similar to the pretracking algorithms discussed in Section 8.6. I discuss the properties of minimization algorithms further in the following.
11.3.1 Spurious local minima
While the norm (20) clearly has a single global minimum ‖Θ‖ = 0 for each MOTS Θ = 0, it typically also has a large number of other local minima with Θ ≠ 0, which are “spurious” in the sense that they do not correspond (even approximately) to MOTSs^{Footnote 33}. Unfortunately, generalpurpose “functionminimization” routines only locate local minima and, thus, may easily converge to one of the spurious Θ ≠ 0 minima.
What this problem means in practice is that a minimization algorithm needs quite a good (accurate) initial guess for the horizon shape in order to ensure that the algorithm converges to the true global minimum Θ = 0 rather than to one of the spurious Θ ≠ 0 local minima.
To view this problem from a different perspective, once the functionminimization algorithm does converge, we must somehow determine whether the “solution” found is the true one, Θ = 0, or a spurious one, Θ ≠ 0. Due to numerical errors in the geometry interpolation and the evaluation of the integral (20), ‖Θ‖ will almost never evaluate to exactly zero; rather, we must set a tolerance level for how large ‖Θ‖ may be. Unfortunately, in practice it is hard to choose this tolerance: If it is too small, the genuine solution may be falsely rejected, while if it is too large, we may accept a spurious solution (which may be very different from any of the true solutions).
Anninos et al. [8] and Baumgarte et al. [26] suggest screening out spurious solutions by repeating the algorithm with varying resolutions of the horizonsurface grid and checking that O shows the proper convergence towards zero. This seems like a good strategy, but it is tricky to automate and, again, it may be difficult to choose the necessary error tolerances in advance.
When the underlying simulation is a spectral one, Pfeiffer et al. [124, 121] report that in practice, spurious solutions can be avoided by a combination of two factors:

The underlying spectral solution can inherently be “interpolated” (evaluated at arbitrary positions) to very high accuracy.

Pfeiffer et al. use a large number of quadrature points (typically an order of magnitude larger than the number of coefficients in the expansion (5)) in numerically evaluating the integral (20).
11.3.2 Performance
For convenience of exposition, suppose the spectral representation (5) of the horizonshape function h uses spherical harmonics Y_{ℓm}. (Symmetric tracefree tensors or other basis sets do not change the argument in any important way.) If we keep harmonics up to some maximum degree ℓ_{max}, the number of coefficients is then N_{coeff} = (ℓ_{max}+1)^{2}. ℓ_{max} is set by the desired accuracy (angular resolution) of the algorithm and is typically on the order of 6 to 12.
To find a minimum in an N_{coeff}dimensional space (here the space of surfaceshape coefficients {a_{ℓm}}), a generalpurpose functionminimization algorithm typically needs on the order of \(5N_{{\rm{coeff}}}^2\) to \(10N_{{\rm{coeff}}}^2\) iterations^{Footnote 34}. Thus the number of iterations grows as \(\ell _{\max}^4\).
Each iteration requires an evaluation of the norm (20) for some trial set of surfaceshape coefficients {a_{ℓm}}, which requires \({\mathcal O}({N_{{\rm{coeff}}}}) = {\mathcal O}(\ell _{\max}^2)\) work to compute the surface positions, together with \({\mathcal O}({N_{{\rm{ang}}}})\) work to interpolate the geometry fields to the surface points and compute the numerical quadrature of the integral (20).
Thus the total work for a single horizon finding is \({\mathcal O}(\ell _{\max}^6 + {N_{{\rm{ang}}}}\ell _{max}^4)\). Fortunately, the accuracy with which the horizon is found generally improves rapidly with ℓ_{max}, sometimes even exponentially^{Footnote 35}. Thus, relatively modest values of ℓ_{max} (typically in the range 8–12) generally suffice for adequate accuracy. Even so, minimization horizon finders tend to be slower than other methods, particularly if high accuracy is required (large ℓ_{max} and N_{ang}). The one exception is in axisymmetry, where only spherical harmonics Y_{ℓm} with m=0 need be considered. In this case minimization algorithms are much faster, though probably still slower than shooting or ellipticPDE algorithms.
11.3.3 Anticipating the formation of a common apparent horizon
Consider the case where we want to find a common apparent horizon as soon as it appears in a binary blackhole (or neutronstar) simulation. In Section 8.6 I discuss “horizon pretracking” algorithms which can determine — before it appears — approximately where (in space) and when (in time) the common apparent horizon will appear.
Minimization algorithms can provide a similar functionality: Before the common apparent horizon forms, trying to find it via a minimization algorithm will (hopefully) find the (a) surface which minimizes the error norm ∥Θ∥ (defined by Equation (20)). This surface can be viewed as the current slice’s closest approximation to a common apparent horizon, and as the evolution proceeds, it should converge to the actual common apparent horizon.
However, it is not clear whether minimization algorithms used in this way suffer from the problems discussed in Section 8.6.2. In particular, it is not clear whether, in a realistic binarycoalescence simulation, the minimum∥Θ∥ surfaces would remain smooth enough to be represented accurately with a reasonable ℓ_{max}.
11.3.4 Summary of minimization algorithms/codes
Minimization algorithms are fairly easy to program and have been used by many researchers, for example [43, 69, 102, 8, 26, 4]. However, at least when the underlying simulation uses finite differencing, minimization algorithms are susceptible to spurious local minima, have relatively poor accuracy, and tend to be quite slow. I believe that the other algorithms discussed in the following sections are generally preferable. If the underlying simulation uses spectral methods, then minimization algorithms may be (relatively) somewhat more efficient and robust.
Alcubierre’s apparent horizon finder AHFinder [4] includes a minimization algorithm based on the work of Anninos et al. [8]^{Footnote 36}. It is implemented as a freely available module (“thorn”) in the Cactus computational toolkit (see Table 2).
11.4 Spectral integraliteration algorithms
Nakamura, Kojima, and Oohara [113] developed a functionaliteration spectral algorithm for solving the apparent horizon equation (16).
This algorithm begins by choosing the usual polarspherical topology for the angular coordinates (θ,ϕ), and rewriting the apparent horizon equation (16) in the form
where Δ is the flatspace angular Laplacian operator, and G is a complicated nonlinear algebraic function of the arguments shown, which remains regular even at θ=0 and θ=π.
Next we expand h in spherical harmonics (5). Because the left hand side of Equation (21) is just the flatspace angular Laplacian of h, which has the Y_{ℓm} as orthogonal eigenfunctions, multiplying both sides of Equation (21) by Y*_{ℓm} (the complex conjugate of Y_{ℓm}) and integrating over all solid angles gives
for each ℓ and m except ℓ = m = 0.
Based on this, Nakamura, Kojima, and Oohara [113] proposed the following functionaliteration algorithm for solving Equation (21):

1.
Start with some (initialguess) set of horizonshape coefficients {a_{ℓm}}. These determine a surface shape via Equation (5).

2.
Interpolate the geometry variables to this surface shape (see Section 7.5).

3.
For each ℓ and m except ℓ = m = 0, evaluate the integral (22) by numerical quadrature to obtain a nextiteration coefficient a_{ℓm}.

4.
Determine a nextiteration coefficient a_{00} by numerically solving (finding a root of) the equation
$$\int {Y_{00}^{\ast}G\;d\Omega = 0.}$$(23)This can be done using any of the 1dimensional zerofinding algorithms discussed in Appendix A.

5.
Iterate until all the coefficients {a_{ℓm}} converge.
Gundlach [80] observed that the subtraction and inversion of the flatspace 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 ℓ (high angular resolution) because the Y_{ℓm} 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.
11.4.1 Kemball and Bishop’s modifications of the NakamuraKojimaOohara algorithm
Kemball and Bishop [93] investigated the behavior of the NakamuraKojimaOohara algorithm and found that its (only) major weakness seems to be that the a_{00}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 ℓ_{max} of sphericalharmonic expansion used. They also give an analysis that the algorithm’s cost is \({\mathcal O}(\ell _{\max}^4)\), and its accuracy \(\varepsilon = {\mathcal O}(1/{\ell _{\max}})\), i.e. the cost is \({\mathcal O}(1/{\varepsilon ^4})\). This accuracy is surprisingly low: Exponential convergence with ℓ_{max} is typical of spectral algorithms and would be expected here. I do not know of any published work which addresses this discrepancy.
11.4.2 Lin and Novak’s variant of the NakamuraKojimaOohara algorithm
Lin and Novak [104] have developed a variant of the NakamuraKojimaOohara algorithm which avoids the need for a separate search for a_{00} at each iteration: Write the apparent horizon equation (16) in the form
where Δ is again the flatspace angular Laplacian operator and where λ is a nonzero scalar function on the horizon surface. Then choose λ as
where f_{ij} the flat metric of polar spherical coordinates, ∇ is the associated 3covariant derivative operator, and F is the level set function (7).
Lin and Novak [104] showed that all the sphericalharmonic coefficients a_{ℓm} (including a_{00}) 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 3.1, 5.8, 17, 88, and 313 seconds to find the apparent horizon in a test slice to a relative error (measured in the horizon area) of 9 × 10^{−4}, 5 × 10^{−5}, 6 × 10^{−7}, 9 × 10^{−10}, and 3 × 10^{−12} respectively^{Footnote 37}. This implementation is freely available as part of the Lorene toolkit for spectral computations in numerical relativity (see Table 2).
11.4.3 Summary of spectral integraliteration algorithms
Despite what appears to be fairly good numerical behavior and reasonable ease of implementation, the original NakamuraKojimaOohara 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 NakamuraKojimaOohara algorithm. Lin and Novak [104] have develped a variant of the NakamuraKojimaOohara algorithm which avoids the need for a separate search for the a_{00} 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).
11.5 EllipticPDE algorithms
The basic concept of ellipticPDE algorithms is simple: We view the apparent horizon equation (16) as a nonlinear elliptic PDE for the horizon shape function h on the angularcoordinate space and solve this equation by standard finitedifferencing techniques^{Footnote 38}, generally using Newton’s method to solve the resulting set of nonlinear algebraic (finitedifference) equations. Algorithms of this type have been widely used both in axisymmetry and in fully generic slices.
11.5.1 Angular coordinates, grid, and boundary conditions
In more detail, ellipticPDE algorithms assume that the horizon is a Strahlkörper about some local coordinate origin, and choose an angular coordinate system and a finitedifference grid of N_{ang} points on S^{2} in the manner discussed in Section 2.2.
The most common choices are the usual polarspherical coordinates (θ, ϕ) and a uniform “latitude/longitude” grid in these coordinates. Since these coordinates are “unwrapped” relative to the actual S^{2} trialhorizonsurface topology, the horizon shape function h satisfies periodic boundary conditions across the artificial grid boundary at ϕ = 0 and ϕ = 2π. The north and south poles θ = 0 and θ = π are trickier, but Huq et al. [89, 90], Shibata and Uryū [147], and Schnetter [132, 133]^{Footnote 39} “reaching across the pole” boundary conditions for these artificial grid boundaries.
Alternatively, Thornburg [156] avoids the z axis coordinate singularity of polarspherical coordinates by using an “inflatedcube” system of six angular patches to cover S^{2}. Here each patch’s nominal grid is surrounded by a “ghost zone” of additional grid points where h is determined by interpolation from the neighboring patches. The interpatch interpolation thus serves to tie the patches together, enforcing the continuity and differentiability of h 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.
11.5.2 Evaluating the expansion Θ
The next step in the algorithm is to evaluate the expansion Θ given by Equation (16) on the angular grid given a trial horizon surface shape function h on this same grid (6).
Most researchers compute Θ via 2dimensional 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 (θ, ϕ) latitude/longitude grid the Θ(h, ∂_{u}h, ∂_{uv}h) function in Equation (16) is singular on the z axis (at the north and south poles θ = 0 and θ = π) 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 Θ, based on 3dimensional Cartesian finite differencing:

1.
They observe that the scalar field F defined by Equation (7) can be evaluated at any (3dimensional) position in the slice by computing the corresponding (r, θ, ϕ) using the usual flatspace formulas, then interpolating h in the 2dimensional (θ, ϕ) surface grid.

2.
Rewrite the apparent horizon condition (15) in terms of F and its (3dimensional) Cartesian derivatives,
$$\Theta \equiv \Theta (F,{\partial _i}F,{\partial _{ij}}F;{g_{ij}},{\partial _k}{g_{ij}},{K_{ij}}) = 0.$$(27) 
3.
For each (latitude/longitude) grid point on the trial horizon surface, define a 3×3×3point local Cartesian grid centered at that point. The spacing of this grid should be such as to allow accurate finite differencing, i.e. in practice it should probably be roughly comparable to that of the underlying numericalrelativity simulation’s grid.

4.
Evaluate F on the local Cartesian grid as described in Step 1 above.

5.
Evaluate the Cartesian derivatives in Equation (27) by centered 2nd order Cartesian finite differencing of the F values on the local Cartesian grid.
Comparing the different ways of evaluating Θ, 2dimensional angular finite differencing of Equation (16) seems to me to be both simpler (easier to program) and likely more efficient than 3dimensional Cartesian finite differencing of Equation (27).
11.5.3 Solving the nonlinear elliptic PDE
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 N_{ang} nonlinear algebraic equations for the horizon shape function h at the N_{ang} angular grid points; these can be solved by Newton’s method in N_{ang} dimensions. (As explained by Thornburg [153, Section VIII.C], this is usually equivalent to applying the NewtonKantorovich 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 \({1 \over 4} \ {\rm to} \ {1 \over 3}\) of the horizon radius is often a reasonable estimate^{Footnote 40}.
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 generalpurpose ellipticsolver library [22, 23, 24] to solve the equations. This offers a wide variety of Newtonlike 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 functionaliteration 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.
11.5.4 The Jacobian matrix
Newton’s method, and all its variants, require an explicit computation of the Jacobian matrix
where the indices _{I} and _{J} label angular grid points on the horizon surface (or equivalently on S^{2}).
Notice that J includes contributions both from the direct dependence of Θ on h, ∂_{u}h, and ∂_{uv}h, and also from the indirect dependence of Θ on h through the positiondependence of the geometry variables g_{ij}, ∂_{k}g_{ij}, and K_{ij} (since Θ depends on the geometry variables at the horizon surface position, and this position is determined by h). Thornburg [153] discusses this indirect dependence in detail.
There are two basic ways to compute the Jacobian matrix.

Numerical Perturbation:
The simplest way to determine the Jacobian matrix is by “numerical perturbation”, where for each horizonsurface grid point j, h is perturbed by some (small) amount ε at the _{J} th grid point (that is, h_{I} → h_{I} + εδ_{IJ}), and the expansion Θ is recomputed^{Footnote 41}. The _{J} th column of the Jacobian matrix (28) is then estimated as
$${{\bf{J}}_{{\rm{IJ}}}} \approx {{{\Theta _{\rm{I}}}(h + \varepsilon {\delta _{{\rm{IJ}}}}){\Theta _{\rm{I}}}(h)} \over \varepsilon}.$$(29)Curtis and Reid [53], and Stoer and Bulirsch [150, Section 5.4.3] discuss the optimum choice of ε in this algorithm^{Footnote 42}.
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].

Symbolic Differentiation:
A more efficient, although somewhat more complicated, way to determine the Jacobian matrix is the “symbolic differentiation” algorithm described by Thornburg [153] and also used by Pasch [118], Shibata et al. [146, 147], and Thornburg [156]. Here the internal structure of the finite differenced Θ(h) function is used to directly determine the Jacobian matrix elements.
This algorithm is best illustrated by an example which is simpler than the full apparent horizon equation: Consider the flatspace Laplacian in standard (θ, ϕ) polarspherical coordinates,
$$\Delta h \equiv {\partial _{\theta \theta}}h + {{{\partial _\theta}h} \over {\tan \theta}} + {{{\partial _{\phi \phi}}h} \over {{{\sin}^2}\theta}}.$$(30)Suppose we discretize this with centered 2nd order finite differences in θ and ϕ. Then neglecting finitedifferencing truncation errors, and temporarily adopting the usual notation for 2dimensional grid functions, h_{i, j} = h(θ=θ_{i}, ϕ=ϕ_{j}), our discrete approximation to Δh is given by
$${(\Delta h)_{i,j}} = {{{h_{i  1,j}}  2{h_{i,j}} + {h_{i + 1,j}}} \over {{{(\Delta \theta)}^2}}} + {1 \over {\tan \theta}}{{{h_{i + 1,j}}  {h_{i + 1,j}}  {h_{i  1,j}}} \over {2\Delta \theta}} + {1 \over {{{\sin}^2}\theta}}{{{h_{i,j  1}}  2{h_{i,j}} + {h_{i,j + 1}}} \over {{{(\Delta \phi)}^2}}}.$$(31)The Jacobian of Δh is thus given by
$${{\partial {{(\Delta h)}_{(i,j)}}} \over {\partial {h_{(k,\ell)}}}} = \left\{{\begin{array}{*{20}c} {{1 \over {{{(\Delta \theta)}^2}}} \pm {1 \over {2\tan \theta \Delta \theta}}} & {{\rm{if}}\;(k,\ell) = (i \pm 1,j),} \\ {{1 \over {{{\sin}^2}\theta {{(\Delta \phi)}^2}}}} & {{\rm{if}}\;(k,\ell) = (i \pm 1,j),} \\ { {2 \over {{{(\Delta \theta)}^2}}}  {2 \over {{{\sin}^2}\theta {{(\Delta \phi)}^2}}}} & {{\rm{if}}\;(k,\ell) = (i,j),} \\ 0 & {{\rm{otherwise}}.} \\ \end{array}} \right.$$(32)Thornburg [153] describes how to generalize this to nonlinear differential operators without having to explicitly manipulate the nonlinear finite difference equations.
11.5.5 Solving the linear equations
All the algorithms described in Section 8.5.3 for treating nonlinear elliptic PDEs require solving a sequence of linear systems of N_{ang} equations in N_{ang} unknowns. N_{ang} 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 positivedefinite matrices while, due to the angular boundary conditions^{Footnote 43} (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 discussions^{Footnote 44}. 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 program^{Footnote 45}. Schnetter [132, 133] reports good results with an ILUpreconditioned GMRES solver from the PETSc library. Thornburg [156] experimented with both an ILUCG solver and a direct sparse LU 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.
11.5.6 Sample results
As an example of the results obtained with this type of apparent horizon finder, Figure 10 shows the numericallycomputed apparent horizons (actually, MOTSs) at two times in a headon 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.)
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, AHFinder^{Footnote 46}. 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 numericallycomputed 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.)
11.5.7 Summary of ellipticPDE algorithms/codes
EllipticPDE apparent horizon finders have been developed by many researchers, including Eardley [67]^{Footnote 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.
EllipticPDE 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 “fastflow” apparent horizon finder AHFinder (discussed in more detail in Section 8.7)^{Footnote 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).
EllipticPDE 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 implemented^{Footnote 49}. Their typical radius of convergence is on the order of \({1 \over 3}\) 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 ellipticPDE 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 meshrefinement driver for Cactus. TGRapparentHorizon2D is no longer maintained, but AHFinderDirect is actively supported and is now used by many different research groups^{Footnote 50}.
11.6 Horizon pretracking
Schnetter et al. [133, 135] introduced the important concept of “horizon pretracking”. They focus on the case where we want to find a common apparent horizon as soon as it appears in a binary blackhole (or neutronstar) simulation. While a global (flow) algorithm (Section 8.7) could be used to find this common apparent horizon, these algorithms tend to be very slow. They observe that the use of a local (ellipticPDE) algorithm for this purpose is somewhat problematic:
The common [apparent] horizon […] appears instantaneously at some late time and without a previous good guess for its location. In practice, an estimate of the surface location and shape can be put in by hand. The quality of this guess will determine the rate of convergence of the finder and, more seriously, also determines whether a horizon is found at all. Gauge effects in the strong field region can induce distortions that have a large influence on the shape of the common horizon, making them difficult to predict, particularly after a long evolution using dynamical coordinate conditions. As such, it can be a matter of some expensive trial and error to find the common apparent horizon at the earliest possible time. Further, if a common apparent horizon is not found, it is not clear whether this is because there is none, or whether there exists one which has only been missed due to unsuitable initial guesses — for a fast apparent horizon finder, a good initial guess is crucial.
Pretracking tries (usually successfully) to eliminate these difficulties by determining — before it appears — approximately where (in space) and when (in time) the common apparent horizon will appear.
11.6.1 Constantexpansion surfaces
The basic idea of horizon pretracking is to consider surfaces of constant expansion (“CE surfaces”), i.e. smooth closed orientable 2surfaces in a slice satisfying the condition
where the expansion E is a specified real number. Each marginally outer trapped surface (including the apparent horizon) is thus a CE surface with expansion E = 0; more generally Equation (33) defines a 1parameter family of 2surfaces in the slice. As discussed by Schnetter et al. [133, 135], for asymptotically flat slices containing a compact strongfield region, some of the E < 0 members of this family typically foliate the weakfield region.
In the binarycoalescence context, for each t = constant slice we define E_{*} to be the smallest E ≥ 0 for which a CE surface (containing both strongfield regions) exists with expansion E. If E_{*} = 0 this “minimumexpansion CE surface” is the common apparent horizon, while if E_{*} > 0 this surface is an approximation to where the common apparent horizon will appear. We expect the minimumexpansion CE surface to change continuously during the evolution and its expansion E_{*} to decrease towards 0. Essentially, horizon pretracking follows the time evolution of the minimumexpansion CE surface and uses it as an initial guess for (searching for) the common apparent horizon.
11.6.2 Generalized constantexpansion surfaces
Schnetter [133] implemented an early form of horizon pretracking, which followed the evolution of the minimumexpansion constantexpansion surface, and found that it worked well for simple test problems. However, Schnetter et al. [135] found that for more realistic binaryblackhole coalescence systems the algorithm needs to be extended:

While the expansion is zero for a common apparent horizon, it is also zero for a 2sphere at spatial infinity. Figure 12 illustrates this for Schwarzschild spacetime. Notice that for small positive E_{*} there will generally be two distinct CE surfaces with E = E_{*}, an inner surface just outside the horizon and an outer one far out in the weakfield region. The inner CE surface converges to the common apparent horizon as E_{*} decreases towards 0; this surface is the one we would like the pretracking algorithm to follow. Unfortunately, without measures such as those described below, there is nothing to prevent the algorithm from following the outer surface, which does not converge to the common apparent horizon as E_{*} decreases towards 0.

In a realistic binarycoalescence simulation, the actual minimumexpansion CE surface may be highly distorted and, thus, hard to represent accurately with a finiteresolution angular grid.
Schnetter et al. [135] discuss these problems in more detail, arguing that to solve them, the expansion Θ should be generalized to a “shape function” H given by one of
CE surfaces are then generalized to surfaces satisfying
for some specified E ≥ 0.
Note that unlike H_{1}, both H_{r} and H_{r}2 are typically monotonic with radius. Neither H_{r} nor H_{r}2 are 3covariantly defined, but they both still have the property that E = 0 in Equation (35) implies the surface is a MOTS, and in practice they work better for horizon pretracking.
11.6.3 Goal functions
To define the single “smallest” surface at each time, Schnetter et al. [135] introduce a second generalization, that of a “goal function” G, which maps surfaces to real numbers. The pretracking search then attempts, on each time slice, to find the surface (shape) satisfying H = E with the minimum value of G. They experimented with several different goal functions,
where in each case the overbar (̅) denotes an average over the surface^{Footnote 51}.
11.6.4 The pretracking search
Schnetter’s [133] original implementation of horizon pretracking (which followed the evolution of the minimumexpansion CE surface) used a binary search on the desired expansion E. Because E appears only on the right hand side of the generalized CE condition (35), it is trivial to modify any apparent horizon finder to search for a surface of specified expansion E. (Schnetter used his TGRapparentHorizon2D ellipticPDE apparent horizon finder described in Section 8.5.7 for this.) A binary search on E can then be used to find the minimum value E_{*}.^{Footnote 52}
Implementing a horizon pretracking search on any of the generalized goal functions (36) is conceptually similar but somewhat more involved: As described by Schnetter et al. [135] for the case of an ellipticPDE apparent horizon finder^{Footnote 53}, we first write the equation defining a desired pretracking surface as
where p is the desired value of the goal function G. Since H is the only term in Equation (37) which varies over the surface, it must be constant for the equation to be satisfied. In this case \(H  \bar H\) vanishes, so the equation just gives G = p, as desired.
Because \({\bar H}\) depends on H at all surface points, directly finite differencing Equation (37) would give a nonsparse Jacobian matrix, which would greatly slow the linearsolver phase of the ellipticPDE apparent horizon finder (Section 8.5.5). Schnetter et al. [135, Section III.B] show how this problem can be solved by introducing a single extra unknown into the discrete system. This gives a Jacobian which has a single nonsparse row and column, but is otherwise sparse, so the linear equations can still be solved efficiently.
When doing the pretracking search, the cost of a single binarysearch iteration is approximately the same as that of finding an apparent horizon. Schnetter et al. [135, Figure 5] report that their pretracking implementation (a modified version of Thornburg’s AHFinderDirect [156] ellipticPDE apparent horizon finder described in Section 8.5.7) typically takes on the order of 5 to 10 binarysearch iterations^{Footnote 54}. The cost of pretracking is thus on the order of 5 to 10 times that of finding a single apparent horizon. This is substantial, but not prohibitive, particularly if the pretracking algorithm is not run at every time step.
11.6.5 Sample results
As an example of the results obtained from horizon pretracking, Figure 13 shows the expansion Θ for various pretracking surfaces (i.e. various choices for the shape function H in a headon binary black hole collision). Notice how all three of the shape functions (34) result in pretracking surfaces whose expansions converge smoothly to zero just when the apparent horizon appears (at about t = 1.1).
As a further example, Figure 14 shows the pretracking surfaces (more precisely, their cross sections projected into the black holes’ orbital plane) at various times in a spiraling binary black hole collision.
11.6.6 Summary of horizon pretracking
Pretracking is a very valuable addition to the horizon finding repertoire: It essentially gives a local algorithm (in this case, an ellipticPDE algorithm) most of the robustness of a global algorithm in terms of finding a common apparent horizon as soon as it appears. It is implemented as a higherlevel algorithm which uses a slightlymodified ellipticPDE apparent horizon finding algorithm as a “subroutine”.
The one significant disadvantage of pretracking is its cost: Each pretracking search typically takes 5 to 10 times as long as finding an apparent horizon. Further research to reduce the cost of pretracking would be useful.
Schnetter et al.’s pretracking implementation [135] is implemented as a set of modifications to Thornburg’s AHFinderDirect [156] apparent horizon finder. Like the original AHFinderDirect, the modified version is a freely available “thorn” in the Cactus toolkit (see Table 2).
11.7 Flow algorithms
Flow algorithms define a “flow” on 2surfaces, i.e. they define an evolution of 2surfaces in some pseudotime λ, such that the apparent horizon is the λ → ∞ limit of a (any) suitable starting surface. Flow algorithms are different from other apparent horizon finding algorithms (except for zerofinding in spherical symmetry) in that their convergence does not depend on having a good initial guess. In other words, flow algorithms are global algorithms (Section 7.7).
To find the (an) apparent horizon, i.e. an outermost MOTS, the starting surface should be outside the largest possible MOTS in the slice. In practice, it generally suffices to start with a 2sphere of areal radius substantially greater than 2 m_{adm}.
The global convergence property requires that a flow algorithm always flow from a large starting surface into the apparent horizon. This means that the algorithm gains no particular benefit from already knowing the approximate position of the apparent horizon. In particular, flow algorithms are no faster when “tracking” the apparent horizon (repeatedly finding it at frequent intervals) in a numerical time evolution. (In contrast, in this situation a local apparent horizon finding algorithm can use the most recent previouslyfound apparent horizon as an initial guess, greatly speeding the algorithm’s convergence^{Footnote 55}).
Flow algorithms were first proposed for apparent horizon finding by Tod [157]. He initially considered the case of a timesymmetric slice (one where K_{ij} = 0). In this case, a MOTS (and thus an apparent horizon) is a surface of minimal area and may be found by a “mean curvature flow”
where x^{i} are the spatial coordinates of a horizonsurface point, s^{i} is as before the outwardpointing unit 3vector normal to the surface, and k ≡ ∇_{k} s^{k} is the mean curvature of the surface as embedded in the slice. This is a gradient flow for the surface area, and Grayson [79] has proven that if the slice contains a minimumarea surface, this will in fact be the stable λ → ∞ limit of this flow. Unfortunately, this proof is valid only for the timesymmetric case.
For nontimesymmetric slices, Tod [157] proposed generalizing the mean curvature flow to the “expansion flow”
There is no theoretical proof that this flow will converge to the (an) apparent horizon, but several lines of argument make this convergence plausible:

The expansion flow is identical to the mean curvature flow (38) in the principal part.

The expansion flow’s velocity is clearly zero on an apparent horizon.

More generally, a simple argument due to Bartnik [25]^{Footnote 56} shows that the expansion flow can never move a (smooth) test surface through an apparent horizon. Suppose, to the contrary, that the test surface \({\mathcal T}\) is about to move through an apparent horizon \({\mathcal H}\), i.e. since both surfaces are by assumption smooth, that \({\mathcal T}\) and \({\mathcal H}\) touch at single (isolated) point P. At that point, \({\mathcal T}\) and \({\mathcal H}\) obviously have the same gij and K_{ij}, and they also have the same s^{i} (because P is isolated). Hence the only term in Θ (as defined by Equation (15)) which differs between \({\mathcal T}\) and \({\mathcal H}\) is ∇_{i}s^{i}. Clearly, if \({\mathcal T}\) is outside \({\mathcal H}\) and they touch at the single isolated point P, then relative to \({\mathcal H},{\mathcal T}\) must be concave outwards at P, so that \({\nabla _i}{s^i}({\mathcal T}) < {\nabla _i}{s^i}({\mathcal H})\). Thus the expansion flow (39) will move \({\mathcal T}\) outwards, away from the apparent horizon. (If \({\mathcal T}\) lies inside \({\mathcal H}\) the same argument holds with signs reversed appropriately.)
Numerical experiments by Bernstein [28], Shoemaker et al. [148, 149], and Pasch [118] show that in practice the expansion flow (39) does in fact converge robustly to the apparent horizon.
In the following I discuss a number of important implementation details for, and refinements of, this basic algorithm.
11.7.1 Implicit pseudotime stepping
Assuming the Strahlkörper surface parameterization (4), the expansion flow (39) is a parabolic equation for the horizon shape function h.^{Footnote 57} This means that any fully explicit scheme to integrate it (in the pseudotime λ) must severely restrict its pseudotime step Δλ for stability, and this restriction grows (quadratically) worse at higher spatial resolutions^{Footnote 58}. This makes the horizon finding process very slow.
To avoid this restriction, practical implementations of flow algorithms use implicit pseudotime integration schemes; these can have large pseudotime steps and still be stable. Because we only care about the λ → ∞ limit, a highly accurate pseudotime integration is not important; only the accuracy of approximating the spatial derivatives matters. Bernstein [28] used a modified Du FortFrankel scheme [64]^{Footnote 59} but found some problems with the surface shape gradually developing highspatialfrequency noise. Pasch [118] reports that an “exponential” integrator (Hochbrucket al. [85]) works well, provided the flow’s Jacobian matrix is computed accurately^{Footnote 60}. The most common choice is probably that of Shoemaker et al. [148, 149], who use the iterated CrankNicholson (“ICN”) scheme^{Footnote 61}. They report that this works very well; in particular, they do not report any noise problems.
By refining his finiteelement grid (Section 2.3) in a hierarchical manner, Metzger [109] is able to use standard conjugategradient elliptic solvers in a multigridlike fashion^{Footnote 62}, using each refinement level’s solution as an initial guess for the next higher refinement level’s iterative solution. This greatly speeds the flow integration: Metzger reports that the performance of the overall surfacefinding algorithm is “of the same order of magnitude” as that of Thornburg’s AHFinderDirect [156] ellipticPDE apparent horizon finder (described in Section 8.5.7).
In a more general context than numerical relativity, Osher and Sethian [116] have discussed a general class of numerical algorithms for integrating “fronts propagating with curvaturedependent speed”. These flow a levelset function (Section 2.1) which implicitly locates the actual “front”.
11.7.2 Varying the flow speed
Another important performance optimization of the standard expansion flow (39) is to replace Θ in the righthand side by a suitable nonlinear function of Θ, chosen so the surface shrinks faster when it is far from the apparent horizon. For example, Shoemaker et al. [148, 149] use the flow
for this purpose, where Θ_{0} is the value of Θ on the initialguess surface, and c (which is gradually decreased towards 0 as the iteration proceeds) is a “goal” value for Θ.
11.7.3 Surface representation and the handling of bifurcations
Since a flow algorithm starts with (topologically) a single large 2sphere, if there are multiple apparent horizons present the surface must change topology (bifurcate) at some point in the flow. Depending on how the surface is represented, this may be easy or difficult.
Pasch [118] and Shoemaker et al. [148, 149] use a levelset function approach (Section 2.1). This automatically handles any topology or topology change. However, it has the drawback of requiring the flow to be integrated throughout the entire volume of the slice (or at least in some neighborhood of each surface). This is likely to be much more expensive than only integrating the flow on the surface itself. Shoemaker et al. also generate an explicit Strahlkörper surface representation (Section 2.2), monitoring the surface shape to detect an imminent bifurcation and reparameterizing the shape into 2 separate surfaces if a bifurcation happens.
Metzger [109] uses a finiteelement surface representation (Section 2.3), which can represent any topology. However, if the flow bifurcates, then to explicitly represent each apparent horizon the code must detect that the surface selfintersects, which may be expensive.
11.7.4 Gundlach’s “fast flow”
Gundlach [80] introduced the important concept of a “fast flow”. He observed that the subtraction and inversion of the flatspace Laplacian in the NakamuraKojimaOohara spectral integraliteration algorithm (Section 8.4) is an example of “a standard way of solving nonlinear elliptic problems numerically, namely subtracting a simple linear elliptic operator from the nonlinear one, inverting it by pseudospectral algorithms and iterating”. Gundlach then interpreted the NakamuraKojimaOohara algorithm as a type of flow algorithm where each pseudotime step of the flow corresponds to a single functionaliteration step of the NakamuraKojimaOohara algorithm.
In this framework, Gundlach defines a 2parameter family of flows interpolating between the NakamuraKojimaOohara algorithm and Tod’s [157] expansion flow (39),
where A ≥ 0 and B ≥ 0 are parameters, ρ > 0 is a weight functional which depends on h through at most 1st derivatives, Δ is the flatspace Laplacian operator, and (1 — BΔ)−^{1} denotes inverting the operator (1 — BΔ). Gundlach then argues that intermediate “fast flow” members of this family should be useful compromises between the speed of the NakamuraKojimaOohara algorithm and the robustness of Tod’s expansion flow. Based on numerical experiments, Gundlach suggests a particular choice for the weight functional ρ and the parameters A and B. The resulting algorithm updates highspatialfrequency components of h essentially the same as the NakamuraKojimaOohara algorithm but should reduce lowspatialfrequency error components faster. Gundlach’s algorithm also completely avoids the need for numerically solving Equation (23) for the a_{00} coefficient in the NakamuraKojimaOohara algorithm.
Alcubierre’s AHFinder [4] horizon finder includes an implementation of Gundlach’s fast flow algorithm^{63}. AHFinder is implemented as a freely available module (“thorn”) in the Cactus computational toolkit (see Table 2) and has been used by many research groups.
11.7.5 Summary of flow algorithms/codes
Flow algorithms are the only truly global apparent horizon finding algorithms and, as such, can be much more robust than local algorithms. In particular, flow algorithms can guarantee convergence to the outermost MOTS in a slice. Unfortunately, these convergence guarantees hold only for timesymmetric slices.
In the forms which have strong convergence guarantees, flow algorithms tend to be very slow. (Metzger’s algorithm [109] is a notable exception: It is very fast.) There are modifications which can make flow algorithms much faster, but then their convergence is no longer guaranteed. In particular, practical experience has shown that in some binary black hole coalescence simulations (Alcubierre et al. [5], Diener et al. [62]), “fast flow” algorithms (Section 8.7.4) can miss common apparent horizons which are found by other (local) algorithms.
Alcubierre’s apparent horizon finder AHFinder [4] includes a “fast flow” algorithm based on the work of Gundlach [80]^{Footnote 63}. It is implemented as a freely available module (“thorn”) in the Cactus computational toolkit (see Table 2) and has been used by a number of research groups.
12 Summary of Algorithms/Codes for Finding Apparent Horizons
There are many apparent horizon finding algorithms, with differing tradeoffs between speed, robustness of convergence, accuracy, and ease of programming.
In spherical symmetry, zerofinding (Section 8.1) is fast, robust, and easy to program. In axisymmetry, shooting algorithms (Section 8.2) work well and are fairly easy to program. Alternatively, any of the algorithms for generic slices (summarized below) can be used with implementations tailored to the axisymmetry.
Minimization algorithms (Section 8.3) are fairly easy to program, but when the underlying simulation uses finite differencing these algorithms are susceptible to spurious local minima, have relatively poor accuracy, and tend to be very slow unless axisymmetry is assumed. When the underlying simulation uses spectral methods, then minimization algorithms can be somewhat faster and more robust.
Spectral integraliteration algorithms (Section 8.4) and ellipticPDE algorithms (Section 8.5) are fast and accurate, but are moderately difficult to program. Their main disadvantage is the need for a fairly good initial guess for the horizon position/shape.
In many cases Schnetter’s “pretracking” algorithm (Section 8.6) can greatly improve an ellipticPDE algorithm’s robustness, by determining — before it appears — approximately where (in space) and when (in time) a new outermost apparent horizon will appear. Pretracking is implemented as a modification of an existing ellipticPDE algorithm and is moderately slow: It typically has a cost 5 to 10 times that of finding a single horizon with the ellipticPDE algorithm.
Finally, flow algorithms (Section 8.7) are generally quite slow (Metzger’s algorithm [109] is a notable exception) but can be very robust in their convergence. They are moderately easy to program. Flow algorithms are global algorithms, in that their convergence does not depend on having a good initial guess.
Table 2 lists freelyavailable apparent horizon finding codes.
Notes
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.
For convenience of exposition I use spherical harmonics here, but there are no essential differences if other basis sets are used.
I discuss the choice of this angular grid in more detail in Section 8.5.1.
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.
Chruściel and Galloway [49] showed that if a “cloud of sand” falls into a large black hole, each “sand grain” generates a nondifferentiable caustic in the event horizon.
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.
I briefly review ODE integration algorithms and codes in Appendix B.
In practice the differentiation can usefully be combined with the interpolation; I outline how this can be done in Section 7.5.
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.
Diener [60] describes how the algorithm can be enhanced to also determine (integrate) individual null generators of the event horizon. This requires interpolating the 4metric to the generator positions but (still) not taking any derivatives of the 4metric.
Walker [162] mentions an implementation for fully generic slices but only presents results for the axisymmetric case.
Equivalently, Diener [60] observed that the locus of any given nonzero value of the levelset function F is itself a null surface and tends to move (exponentially) closer and closer to the event horizon as the backwards evolution proceeds.
They describe how Richardson extrapolation can be used to improve the accuracy of the solutions from \({\mathcal O}(\varepsilon)\) to \({\mathcal O}({\varepsilon ^2})\), but it appears that this has not been done for their published results.
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 t = constant slice of Minkowski spacetime contains many nonsmooth “MOTSs”: The surface of any closed polyhedron in such a slice satisfies all the other conditions to be an MOTS.
As an indication of the importance of the “closed” requirement, Hawking [81] observed that if we consider two spacelikeseparated 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.
Wald and Iyer [161] proved this by explicitly constructing a family of angularly anisotropic slices in Schwarzschild spacetime which approach arbitrarily close to r = 0 yet contain no apparent horizons. However, Schnetter and Krishnan [136] have recently studied the behavior of apparent horizons in various anisotropic slices in Schwarzschild and Vaidya spacetimes, finding that the Wald and Iyer behavior seems to be rare.
This worldtube 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 2surface contained in a (single) slice.
Ashtekar and Galloway [17] have recently proved “a number of physically interesting constraints” on this slicingdependence.
Notice that in order for the 3divergence in Equation (15) to be meaningful, s^{i} (defined only as a field on the MOTS) must be smoothly continued off the surface and extended to a field in some 3dimensional neighborhood of the surface. The offsurface continuation is nonunique, but it is easy to see that this does not affect the value of Θ on the surface.
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], Gottlieb and Orszag [75], and Boyd [37] for general discussions of spectral methods, and (for example) Ansorg et al. [12, 11, 10, 13], Bonazzola et al. [35, 33, 34], Grandclement et al. [77], Kidder et al. [95, 96, 97], and Pfeiffer et al. [120, 124, 123, 122] for applications to numerical relativity.
Conceptually, an interpolator generally works by locally fitting a fitting function (usually a lowdegree 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 ∂_{k}g_{ij} values can be obtained very cheaply at the same time as the g_{ij} values.
Thornburg [154, Appendix F] gives a more detailed discussion of the nonsmoothness of Lagrangepolynomial interpolation errors.
Note that ∂_{r}g_{θθ} is a known coefficient field here, not an unknown; if necessary, it can be obtained by numerically differentiating g_{θθ}. Therefore, despite the appearance of the derivative, Equation (17) is still an algebraic equation for the horizon radius h, not a differential equation.
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 nonvacuum spherically symmetric spacetimes.
Ascher, Mattheij, and Russel [15, Chapter 4] give a more detailed discussion of shooting methods.
There is a simple heuristic argument (see, for example, Press et al. [125, Section 9.6]) that at least some spurious local minima should be expected. We are trying to solve a system of N_{ang} nonlinear equations, Θ_{I} = 0 (one equation for each horizonsurface grid point). Equivalently, we are trying to find the intersection of the N_{ang} codimensionone hypersurfaces Θ_{I} = 0 in surfaceshape space. The problem is that anywhere two or more of these hypersurfaces approach close to each other, but do not actually intersect, there is a spurious local minimum in ‖Θ‖.
A simple counting argument suffices to show that any generalpurpose functionminimization algorithm in n dimensions must involve at least \({\mathcal O}({n^2})\) function evaluations (see, for example, Press et al. [125, Section 10.6]): Suppose the function to be minimized is f: ℜ^{n} → ℜ, and suppose f has a local minimum near some point x_{0} ∈ ℜ^{n}. Taylorexpanding f in a neighborhood of x_{0} gives \(f(\times) = f({\times _0}) + {{\rm{a}}^T}(\times  {\times _0}) + {(\times  {\times _0})^T}{\rm{B(}} \times  {\times _0}) + {\mathcal O}(\Vert \times  {\times _0}{\Vert^3})\), where a ∈ ℜ^{n}, B ∈ ℜ^{n×n} is symmetric, and v^{T} denotes the transpose of the column vector v ∈ ℜ^{n}.
Neglecting the higher order terms (i.e. approximating f as a quadratic form in x in a neighborhood of x_{0}), and ignoring f(x_{0}) (which does not affect the position of the minimum), there are a total of \(N = n + {1 \over 2}n(n + 1)\) coefficients in this expression. Changing any of these coefficients may change the position of the minimum, and at each function evaluation the algorithm “learns” only a single number (the value of f at the selected evaluation point), so the algorithm must make at least \(N = {\mathcal O}({n^2})\) function evaluations to (implicitly) determine all the coefficients.
Actual functions are not exact quadratic forms, so in practice there are additional \({\mathcal O}(1)\) multiplicative factors in the number of function evaluations. Minimization algorithms may also make additional performance and/or spaceversustime tradeoffs to improve numerical robustness or to avoid explicitly manipulating n × n Jacobian matrices.
AHFinder also includes a “fast flow” algorithm (Section 8.7).
For comparison, the ellipticPDE AHFinderDirect horizon finder (discussed in Section 8.5.6), running on a roughly similar processor, takes about 1.8 seconds to find the apparent horizon in a similar test slice to a relative error of 4 × 10^{−4}.
In theory this equation could also be solved by a spectral method on using spectral differentiation to evaluate the angular derivatives. (See [70, 75, 37, 12, 11, 10, 13, 35, 33, 34, 77, 95, 96, 97, 120, 124, 123, 122] for further discussion of spectral methods.) This should yield a highly efficient apparent horizon finder. However, I know of no published work taking this approach.
Thornburg [153] used a MonteCarlo survey of horizonshape perturbations to quantify the radius of convergence of Newton’s method for apparent horizon finding. He found that if strong highspatialfrequency perturbations are present in the slice’s geometry then the radius of convergence may be very small. Fortunately, this problem rarely occurs in practice.
A very important optimization here is that Θ only needs to be recomputed within the finitedifference domain of dependence of the _{J} th grid point.
Because of the onesided finite differencing, the approximation (29) is only \({\mathcal O}(\varepsilon)\) accurate. However, in practice this does not seriously impair the convergence of a horizon finder, and the extra cost of a centeredfinitedifferencing \({\mathcal O}({\varepsilon ^2})\) approximation is not warranted.
Or the interpatch interpolation conditions in Thornburg’s multiplegridpatch scheme [156].
This paper does not say how the author finds apparent horizons, but [68, page 135] cites a preprint of this as treating the apparenthorizon equation as a 2point (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.
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.
The convergence problems, which Thornburg [153] noted when highspatialfrequency perturbations are present in the slice’s geometry, seem to be rare in practice.
In addition, at least two different research groups have now ported, or are in the process of porting, AHFinderDirect to their own (nonCactus) numerical relativity codes.
Schnetter et al. [135] use a simple arithmetic mean over all surface grid points. In theory this average could be defined 3covariantly by taking the induced metric on the surface into account, but in practice they found that this was not worth the added complexity.
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 constantexpansion surface for a slightly larger expansion E as an initial guess, gradually “walking down” the value of E to find the minimum value E_{*}. Thornburg [156, Appendix C] describes such a “continuationalgorithm binary search” algorithm in detail.
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.
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 1dimensional 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.
Alternatively, a flow algorithm could use the most recent previouslyfound 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 previouslyfound MOTS). However, the algorithm would only need to flow the surface a small distance, so the algorithm should be fairly fast.
Linearizing the Θ(h) function (16) gives a negative Laplacian in h as the principal part.
For a spatial resolution Δx, an explicit scheme is generally limited to a pseudotime step Δλ ≲ (Δx)^{2}.
Richtmyer and Morton [129, Section 7.5] give a very clear presentation and analysis of the Du FortFrankel scheme.
More precisely, Pasch [118] found that that an exponential integrator worked well when the flow’s Jacobian matrix was computed exactly (using the symbolicdifferentiation technique described in Section 8.5.4). However, when the Jacobian matrix was approximated using the numericalperturbation technique described in Section 8.5.4, Pasch found that the pseudotime integration became unstable at high numerical resolutions. Pasch [118] also notes that the exponential integrator uses a very large amount of memory.
AHFinder also includes a minimization algorithm (Section 8.3).
The parabola generically has two roots, but normally only one of them lies between x_{−} and x_{+}.
The numericalanalysis 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 “timeintegration problem for ODEs” to (try to) avoid this confusion. In this appendix, sansserif lowercase letters abc… z denote variables and functions in ℜ^{n} (for some fixed dimension n), and sansserif uppercase letters ABC … Z denote n × n realvalued matrices.
LSODA can also automatically detect stiff systems of ODEs and adjust its integration scheme so as to handle them efficiently.
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Acknowledgements
I thank the many researchers who answered my email queries on various aspects of their work. I thank the anonymous referees for their careful reading of the manuscript and their many helpful comments. I thank Badri Krishnan for many useful conversations on the properties of apparent, isolated, and dynamical horizons. I thank Scott Caveny and Peter Diener for useful conversations on eventhorizon finders. I thank Peter Diener, Luciano Rezzolla, and Virginia J. Vitzthum for helpful comments on various drafts of this paper. I thank Peter Diener and Edward Seidel for providing unpublished figures.
I thank the many authors named in this review for granting permission to reprint figures from their published work. I thank the American Astronomical Society, the American Physical Society, and IOP Publishing for granting permission to reprint figures published in their journals. The American Physical Society requires the following disclaimer regarding such reprinted material:
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I thank the Alexander von Humboldt Foundation, the AEI visitors program, and the AEI postdoctoral fellowship program for financial support.
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Appendices
Solving a Single Nonlinear Algebraic Equation
In this appendix I briefly outline numerical algorithms and codes for solving a single 1dimensional nonlinear algebraic equation f (x) = 0, where the continuous function f: ℜ → ℜ is given.
The process generally begins by evaluating f on a suitable grid of points and looking for sign changes. By the intermediate value theorem, each sign change must bracket at least one root. Given a pair of such ordinates x_{−} and x_{+}, there are a variety of algorithms available to accurately and efficiently find the (a) root:
If ∣x_{+} − x_{−}∣ is small, say on the order of a finitedifference grid spacing, then closedform approximations are probably accurate enough:

The simplest approximation is a simple linear interpolation of f between x− and x_{+}.

A slightly more sophisticated algorithm, “inverse quadratic interpolation”, is to use three ordinates, two of which bracket a root, and estimate the root as the root of the (unique) parabola which passes through the three given (x, f (x)) points^{Footnote 64}.
For larger ∣x+ − x−∣, iterative algorithms are necessary to obtain an accurate root:

Bisection (binary search on the sign of f) is a wellknown iterative scheme which is very robust. However, it is rather slow if high accuracy is desired.

Newton’s method can be used, but it requires that the derivative f′ be available. Alternatively, the secant algorithm (similar to Newton’s method but estimating f′ from the most recent pair of function evaluations) gives similarly fast convergence without requiring f′ to be available. Unfortunately, if ∣f′∣ is small enough at any iteration point, both these algorithms can fail to converge, or more generally they can generate “wild” trial ordinates.

Probably the most sophisticated algorithm is that of van Wijngaarden, Dekker, and Brent. This is a carefully engineered hybrid of the bisection, secant, and inverse quadratic interpolation algorithms, and generally combines the rapid convergence of the secant algorithm with the robustness of bisection. The van WijngaardenDekkerBrent algorithm is described by Forsythe, Malcolm, and Moler [71, Chapter 7], Kahaner, Moler, and Nash [92, Chapter 7], and Press et al. [125, Section 9.3]. An excellent implementation of this, the Fortran subroutine ZEROIN, is freely available from http://www.netlib.org/fmm/.
The Numerical Integration of Ordinary Differential Equations
The timeintegration problem^{Footnote 65} for ordinary differential equations (ODEs) is traditionally written as follows: We are given an integer n > 0 (the number of ODEs to integrate), a “righthandside” function f: ℜ^{n} ×ℜ → ℜ^{n}, and the value y(0) of a function y: ℜ → ℜ^{n} satisfying the ODEs
We wish to know (or approximate) y(t) for some finite interval t ∈ [0, t_{max}].
This is a wellstudied problem in numerical analysis. See, for example, Forsythe, Malcolm, and Moler [71, Chapter 6] or Kahaner, Moler, and Nash [92, Chapter 8] for a general overview of ODE integration algorithms and codes, or Shampine and Gordon [140], Hindmarsh [84], or Brankin, Gladwell, and Shampine [38] for detailed technical accounts.
For our purposes, it suffices to note that highly accurate, efficient, and robust ODEintegration codes are widely available. In fact, there is a strong tradition in numerical analysis of free availability of such codes. Notably, Table 3 lists several freelyavailable ODE codes. As well as being of excellent numerical quality, these codes are also very easy to use, employing sophisticated adaptive algorithms to automatically adjust the step size and/or the precise integration scheme used^{Footnote 66}. These codes can generally be relied upon to produce accurate results both more efficiently and more easily than a handcrafted integrator. I have used the LSODE solver in several research projects with excellent results.
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Thornburg, J. Event and Apparent Horizon Finders for 3 + 1 Numerical Relativity. Living Rev. Relativ. 10, 3 (2007). https://doi.org/10.12942/lrr20073
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DOI: https://doi.org/10.12942/lrr20073