(ed.). Gravitational waves (IOP, 2001)(422s).

Recent applications and progress 395Figure 18.3. (a) The full (dotted) curve is the AH for the full 3D data set (6, 0.2, 1)((6, 0, 0)) att = 9inthex–z plane. (b) The {l = 2, m = 0} waveform for the 3D(6, 0.2, 1) case at r = 4 (full curve) is compared to the axisymmetric (6, 0, 0) case (dottedcurve). The chain curve shows the fit of the 3D case to the two lowest lying QNMs fora BH of mass 0.99. (c) Same comparison for the {l = 4, m = 0} waveform. (d) Samecomparison for the non-axisymmetric {l = 2, m = 2} waveform.axisymmetry and with 3D nonlinear numerical codes, but with standard 3 + 1formulations, as well as with perturbative methods. In summary, these newformulations are able to extend the evolutions by a considerable amount, sinceinstabilities that develop during grid stretching caused by singularity avoidingslicings are highly suppressed. 3D black hole evolutions that crashed previouslyby t = 20–30M now routinely extend several times longer. However, by suchlate times the pathological peaks in metric functions associated with singularityavoiding slicings cannot be resolved, and then of course evolutions become veryinaccurate. This is actually a major improvement! Previously, with traditionalformulations, instabilities would develop as large gradients were created near theblack hole, causing the codes to crash prematurely. As discussed above and shownin many papers [121,131,132,146], the evolutions can actually be very accurate—allowing the extraction of very delicate waveforms from a large spectrum ofmodes—and remain so until the code crashes. With the new formulations, wefind that we can break far through the former crash barrier, but as features becomeunder-resolved the results naturally become less accurate. For a fuller discussion