Physical problems governed by nonlinear equations can develop small scale structures, which are rarely
easy to predict. The canonical example in hydrodynamics is turbulence, which results in short wavelength
features up to the viscous scale. An example in general relativity of arbitrarily small scales is
the one of critical phenomena [125, 223], where the solution develops a self-similar behavior
revealing a universal approach to a singular one describing a naked singularity. Uncovering this
phenomena is crucially required to dynamically adjust the grid structure to respond to the
(exponentially) ever-shrinking features of the solution. Recently, such need was also demonstrated quite
clearly by the resolution of the final fate of unstable black strings [279, 280]. This work followed
the dynamics, in five-dimensional spacetimes, of an unstable black string: a black hole with
topology , with the asymptotic length of over the black hole mass per unit length
above the critical value for linearized stability [216, 217]. As the evolution unfolds, pieces of
the string shrink and elongate (so that the area increases), yielding another unstable stage;
see Figure 10. This behavior repeats in a self-similar manner, the black string developing a
fractal structure of thin black strings joining spherical black holes. This behavior was followed
through four generations and the numerical grid refined in some regions up to a factor of
compared to the initial one. This allowed the authors to extrapolate the observed behavior
and conclude that the spacetime will develop naked singularities in finite time from generic
conditions, thereby providing a counterexample to the cosmic censorship conjecture in five
dimensions.

Figure 10: Early (left) and late (right) stages of the apparent horizon describing the evolution of an
unstable black string. Courtesy: Luis Lehner and Frans Pretorius.