(ed.). Gravitational waves (IOP, 2001)(422s).

384 Numerical relativityfunctions, and then minimize the integral of the square of the expansion 2 overthe surface. For example, one can parameterize a surface asF(r,θ,φ)= r − h(θ, φ). (18.19)The surface under consideration will be taken to correspond to the zero level ofF. The function h(θ, φ) is then expanded in terms of spherical harmonics:h(θ, φ) =l max ∑l∑l=0 m=−la lm Y lm (θ, φ). (18.20)Similar techniques were developed by [123].At an AH the expansion integral over the surface should vanish, and we willhave a global minimum. Of course, since numerically we will never find a surfacefor which the integral vanishes exactly, one must set a given tolerance level belowwhich a horizon is assumed to have been found.Minimization algorithms for finding AHs have a few drawbacks: First, thealgorithm can easily settle down on a local minimum for which the expansion isnot zero, so a good initial guess is often required. Moreover, when more thanone marginally trapped surface is present, as is often the case, it is very difficultto predict which of these surfaces will be found by the algorithm. The algorithmcan often settle on an inner horizon instead of the true AH. Again, a good initialguess can help point the finder towards the correct horizon. Finally, minimizationalgorithms tend to be very slow when compared with ‘flow’ algorithms of the typedescribed in the next section. Typically, if N is the total number of terms in thespectral decomposition, a minimization algorithm requires of the order of a fewtimes N 2 evaluations of the surface integrals (where in our experience ‘afew’ cansometimes be as high as ten).This algorithm has been implemented in the ‘Cactus’ code for 3D numericalrelativity [34]. For more details of the application of this algorithm, see[117–119, 122].18.4.2.2 3D fast flow algorithmA second method that has been implemented in the ‘Cactus’ code is the ‘fast flow’method proposed by Gundlach [124]. Starting from an initial guess for the a lm ,itapproaches the apparent horizon through the iterationa (n+1)lm= a (n)lm − A1 + Bl(l + 1) (ρ)(n) lm(18.21)where (n) labels the iteration step, ρ is some positive definite function (‘aweight’), and (ρ) lm are the harmonic components of the function (ρ). Variouschoices for the weight ρ and the coefficients A and B parametrize a family of suchmethods. The fast flow algorithm in Cactus usesρ = 2r 2 |∇ F|[(g ij − s i s j )(ḡ ij −∇ i r∇ j r)] −1 , (18.22)