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
As well as the more general algorithms described in the following, this may be solved by a shooting
- Guess the value of at one endpoint, say .
- 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.
- 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.
- 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 [159, 66, 2, 29, 30, 145, 3, 4].