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7.5 Geometry interpolation

Θ depends on the slice geometry variables gij, ∂kgij, and Kij at the horizon position25. In practice these variables are usually only known on the (3-dimensional) numerical grid of the underlying numerical-relativity simulation26, so they must be interpolated to the horizon position and, more generally, to the position of each intermediate-iterate trial shape the apparent horizon finding algorithm tries in the process of (hopefully) converging to the horizon position.

Moreover, usually the underlying simulation gives only gij and Kij, so gij must be numerically differentiated to obtain ∂kgij. As discussed by Thornburg [156Jump To The Next Citation Point, Section 6.1], it is somewhat more efficient to combine the numerical differentiation and interpolation operations, essentially doing the differentiation inside the interpolator27.

Thornburg [156Jump To The Next Citation Point, Section 6.1] argues that for an elliptic-PDE algorithm (Section 8.5), for best convergence of the nonlinear elliptic solver the interpolated geometry variables should be smooth (differentiable) functions of the trial horizon surface position. He argues that that the usual Lagrange polynomial interpolation does not suffice here (in some cases his Newton’s-method iteration failed to converge) because this interpolation gives results which are only piecewise differentiable28. He uses Hermite polynomial interpolation to avoid this problem. Cook and Abrahams [51Jump To The Next Citation Point], and Pfeiffer et al. [124Jump To The Next Citation Point] use bicubic spline interpolation; most other researchers either do not describe their interpolation scheme or use Lagrange polynomial interpolation.


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