### 8.2 The shooting algorithm in axisymmetry

In an axisymmetric spacetime we can use symmetry-adapted coordinates , so (given the Strahlkörper assumption) without further loss of generality we can write the horizon shape function as . The apparent horizon equation (16) then becomes a nonlinear 2-point boundary-value ODE for the horizon shape function [146, Equation (1.1)]
where is a nonlinear 2nd order (ordinary) differential operator in as shown.

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

where is if there is “bitant” reflection symmetry across the plane, or otherwise.

As well as the more general algorithms described in the following, this may be solved by a shooting algorithm:

1. Guess the value of at one endpoint, say .
2. Use this guessed value of together with the boundary condition (19) there as initial data to integrate (“shoot”) the ODE (18) from to the other endpoint . 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 to within some tolerance, then the just-computed describes the (an) apparent horizon, and the algorithm is finished.
4. Otherwise, adjust the guessed value and try again. Because there is only a single parameter () 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 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 [159662293014534].