Using the thorn AHFinder

Thorn_MinimumAHF can be used either with a minimization or a flow algorithm.

Parameters

• General parameters
  
  • To activate the thorn use:
    ahf_active = "yes"
    This parameter is set by default to "no".
    
    
  • By default the minimization algorithm is used. To switch to the flow algorithm one has to set
    ahf_flow = "yes"
    
  
  
• Parameters used in evolutions
  
  • To specify how often the finder is called the parameter:
    ahf_findevery = 1
    must be specified as the number of iterations. Here 1 is default.
    
    
  • The number of iterations after which the thorn is called the first time can be specified by:
    ahf_findafter = 0
    Here the default is 0, calling the thorn also on the initial time slice.
    
    
  • Instead specifying the number of iterations one can specify after how much coordinate time the thorn is called first. This is done by using the parameter:
    ahf_findaftertime = 0.0
    When different from zero, this parameter overrides the value of ahf_findafter. Default here is also 0.
    
  
  
• Parameters specifying the expansion of the surface in sperical harmonics.
  
  • The expansion in theta is specified by the maximal number of terms using:
    ahf_lmax = 8
    Here 8 is the default.
    
    
  • If axisymmetry is expected the surface does not need to be expanded in phi. This is set by:
    ahf_phi = "no".
    The surface is expanded in theta and phi by default.
    
    
  • The position of the center of the expansion is set by the parameters:
    ahf_xc = 0.0
    ahf_yc = 0.0
    ahf_zc = 0.0.
    Here the origin is default. The center of the expansion should be set the expected center of the apparent horizon. But it works also with a non-centered horizon as long as the center of expansion lies inside the horizon (otherwise the algorithm will fail).
    
    
  • The center of the expansion can also be allowed to move. This is controlled by:
    ahf_wander = "yes".
    However, at the moment this only works with the minimization algorithm.
    
    
  • Finally the radius of the initial sphere can also be specified. This is controlled by the parameter:
    ahf_r0 = 0.0.
    The default is 0.0, forcing the largest sphere possible in the grid.
    
    
• Parameters for the initial guess
  The initial guess can be specified by some parameters which are set to "no" by default.
  
  
  • To use on old horizon as initial guess one must set:
    ahf_guessold = "yes".
    However, if in the evolutionof the apparent horizon jumps discontinuously it might be lost by using this.
    
    
  • If no old horizon is used the inital guess can be specified further for the minimization algorithm. This algorithm is sensitive to the initial guess, so this is important. The initial guess is set up by an expansion in spherical harmonics in the first two coefficients (l=0,l=2). For both of these a number of subdivisions can be specified by:
    ahf_nn0 = 10
    ahf_nn2 = 10
    The algorithm then tests in this case 100 different combinations to find the best initial guess.
    
    
  • It is also possible to use only a sphere as initial guess. This is much faster and is done by setting:
    ahf_sloppyguess = "yes".
    In this case a number of spheres (specified by ahf_nn0) with different radii are tested for the initial guess.
    
    
  • If one want to look for an inner horizon instead of an outer one, this can be done by setting
    ahf_inner = "yes".
    This only works with the minimization algorithm.
    
    
• Parameters for surface intergrals
  The number of subdivisions in theta and phi are specified by:
  ahf_ntheta = 200
  ahf_nphi = 200.
  200 is the default for both parameters.
  
  
• Parameters indicating symmetries
  Symmetries can either be specified by reflection symmetry on single planes or by octant symmetry. Please notice that this symmetries refer to the symmetries of the surface, and not of the computational grid.
  
  
  • Reflection symmetry on the yz-plane, i.e. symmetry x -> -x, is indicated by:
    ahf_refx = "yes".
    For the symmetries y -> -y and z -> -z this is done in an analogous way by
    ahf_refy = "yes"
    ahf_refz = "yes"
    By default these parameters are set to "no".
    
    
  • Octant symmetry is specified by the parameter:
    ahf_octant = "yes".
    This is set to no by default. Possible parmeter settings are "yes" for reflection symmetries on all three coordinate planes and "high" for an additional rotational symmetry of pi/2 around the z axis.
    
    
• Parameters for the minimization algorithm
  
  • The tolerance for the minimization algorithm can be specified by:
    ahf_tol = 0.1
    with 0.1 being default.
    
    
  • The number if iterations for each step (POWELL algorithm) is specified by:
    ahf_maxiter = 10 .
    10 is the default.
    
    
  • usually the square of the expansion is minimized. To switch to minimization of the area one can use:
    ahf_minarea = "yes"
    (default is "no"). Only for time symmetric data the surface found by area minimization corresponds to an apparent horizon.
    
    
• Parameters for the flow algorithm
  
  • The maximum number of iterations for the flow algorithm is specified by:
    ahf_flowiter = 200,
    with 200 being default.
    
    
  • The tolerance can be set by
    ahf_flowtol = 0.0001.
    The default is 0.0001.
    
    
  • Further, the flow algorithm can be tuned by some additional parameters (see Carsten's paper). The default values are shown here:
    ahf_flowa = 0.01, gives the alpha parameter for the flow.
    ahf_flowb = 0.5, sets the beta parameter for the flow.
    ahf_flowh = 0.0, specifies the weight of H flow.
    ahf_flowc = 1.0, specifies the weight of C flow.
    ahf_flown = 0.0, specifies the weight of N flow (this is not yet implemented)
    
    
  The character of the differnet flows and the alpha and beta parameters are described in Carsten Gundlach's paper on his pseudo-spectral apparent horizon finder (gr-qc/9707050).
  
  
• Parameters for output
  
  • By default no logfile for MinimumAHF is written. To obtain a log file one must set:
    ahf_logfile = "yes"
    
    
  • There are two modes for verbosity:
    ahf_verbose = "yes"
    prints messages to screen at the beginning and the end of the algorithm, and
    ahf_veryverbose = "yes"
    prints messages also during the iteration process. By default the verbose mode is set to "yes", and the veryverbose mode to "no".
    
    
  • Up to now the thorn only supports 2D output of grid functions. The output is controlled be the thorn itself, not by cactus standard output. 2D output can be switched off by
    ahf_2Doutput = "no".
    In future versions of the thorn a 3D output will be possible by setting
    ahf_3Doutput = "yes".
    
    
  • Finally an area map can be found by setting
    ahf_areamap = "yes".
    This is useful for looking at the behaviour of the area for surfaces close to the horizon.
    
    
• Parameters for mask
  
  • The mask is 0 inside the horizon and 1 outside, and is used in black-hole excision (AHBC) techniques. By default the mask is off. It can be enabled by setting:
    ahf_mask = "yes".
    This sets the mask only if the finder is sure that a horizon was found. Setting also:
    ahf_weakmask = "yes".
    makes the finder set the mask also if a horizon is probably there but the resolution or lmax are to low to really resolve it.
    
    
  • There can be a buffer zone between the region inside the horizon and the region where the mask is 0. This is controlled by the shrink factor:
    ahf_maskshrink = -2.0.
    which here sets the region with where the mask is zero to be 2 gridpoints inside the horizon.
    
    

Minimal Parameter Settings

     intorder          = 2     # Second order interpolation

     ahf_active = "yes"
     ahf_flow         = "yes"

     ahf_phi          = "yes"
     ahf_sloppyguess  = "yes"

Hints for Parameter Settings

While the default settings usually work fine, they can be changed to meet special purposes:

• If the horizon is expected to be far from spherical the parameter ahf_lmax can be set to a higher value. 12 should be high enough. However values up to 20 are supported.
  
  
• If the latter parameter is set to a values higher then 8 the parameter ahf_maxiter can be raised to e.g. 14. This can be useful since more iterations can be necessary for higher coefficients of the expansion.
  
  
• Usually I suffices to use a sphere for the initial guell of the surface. So it can be set
  ahf_sloppyguess = "yes".
  
  
• The finder is not running with the convergence mode of cactus, since there are confusions with the names of the finder's output files.
  

Output to Files

• To depict the position of the horizon most important are the files ahfgrid_2d_...ieee. This files contain a 2D gridfunction. The zero level of this locates the horizon.
  
  
• The files ahf_exp_2d_...ieee show the expansion of outgoing photoons on the level set of the gridfunction ahfgrid. The horizon generally coincides with zeros of the expansion.
  
  
• The surface area of the horizon is given in ahf_area.tl
  
  
• ahf_mass.tl contains the mass of the surface.
  
  
• The coefficients of the expansion in spherical harmonics are given in ahf_coeff.alm.
  
  
• The files ahf_circ_eq.tl, ahf_meri_p1.tl and ahf_meri_p2.tl contain the equatorial circumference of the surface, the length of the meridian at phi=0, and the length of the meridian at phi=pi/2 of the surface, respectively.
  
  
• If an output of a logfile is set in the parameters, the log file for the last time the horizon was called is ahf_logfile.

Some Results with the Finder

Calculations with different grid spacings but constant grid size show convergence of the horizon area. The convergence is of the order of 4, as should be expected if the radius converges with second order.

This has been checked with different linear momenta in z direction p _z= (0M, 2M, 5M) and vanishing spin. Also for p_z=2M and a spin of 5M in x direction the horizon converges. The figure shows the case with p_z=2M and vanishing spin.

Further not only the area converges but also the shape of the horizon. For both the minimization and the flow algorithm the horizon converges to the same shape, as can be seen from the coefficients fo the expansion. The order of convergence for the coefficients is between 1.4 and 1.7.

By using the parameters ahf_xc, ahf_yc, ahf_zc it can also be shown that the finder also locates horizons which are not centered around the point given by these parameters. This works in general as long as the surface can be expanded in spherical harmonics around this point but the error increases with the off-centering.

The parameter ahf_r0 can be used e.g. when dealing with two black holes. If one searches for separate horizons one can center the finder on one of the locations of the holes and use an initial radius ahf_r0 smaller than the coordinate distance of the holes. With this parameter settings the single horizon can be found faster. But also a setup with an initial sphere of maximum radius should work at least for the flow algorithm. This has been checked with puncture data for two holes with vanishing linear and angular momentum for each hole (equivalent to Brill-Lindquist data). Here for a coordinate distance of the holes of 1.6M the separated horizons for the holes are found but no common horizon. For a coordinate distance of 1.5M a common horizon is found and also single ones, which are inner surfaces in this case. This coincides with other work where the critical coordinate distance for a single horizon is between 1.53M and 1.56M (gr-qc/9809004).

Also the Misner case was checked. Here for mu = 1.35 a common horizon is found. For mu = 1.37 separated horizons are found. From the literature we know that (e.g. gr-qc/9809004) the critical value of mu is 1.36. This is confirmed by the horizon finder.

The information of when a horizon was found can be seen in the cactus-logfile. There will be output from the thorn even if no horizon was found.