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8.2 The shooting algorithm in axisymmetry

In an axisymmetric spacetime we can use symmetry-adapted 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 (16View Equation) then becomes a nonlinear 2-point boundary-value ODE for the horizon shape function h [146Jump To The Next Citation Point, Equation (1.1)]
Θ ≡ Θ (h,∂θh,∂θθh;gij,∂kgij,Kij) = 0, (18 )
where Θ(h ) is a nonlinear 2nd order (ordinary) differential operator in h as shown.

Taking the angular coordinate θ to have the usual polar-spherical topology, local smoothness of the apparent horizon gives the boundary conditions

∂θh = 0 at θ = 0 and θ = θmax, (19 )
where θmax 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 algorithm31:

  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 (19View Equation) there as initial data to integrate (“shoot”) the ODE (18View Equation) from θ=0 to the other endpoint θ= θmax. 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 (19View Equation) at θ=θ max to within some tolerance, then the just-computed h(θ) describes the (an) apparent horizon, and the algorithm is finished.
  4. Otherwise, adjust the guessed value h(θ=0 ) ≡ h∗ and try again. Because there is only a single parameter (h ∗) to be adjusted, this can be done using one of the 1-dimensional zero-finding 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 [159662293014534Jump To The Next Citation Point].


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