  Here we review a hybrid lattice Boltzmann algorithm to solve the equations of
motion of cholesteric liquid crystals. The method consists in coupling a
lattice Boltzmann solver for the Navier-Stokes equation to a finite difference
method to solve the dynamical equations governing the evolution of the liquid
crystalline order parameter. We apply this method to study the growth of
cholesteric blue phase domains, within a cholesteric phase. We focus on the
growth of blue phase II and on a thin slab geometry in which the domain wall is
flat. Our results show that, depending on the chirality, the growing blue phase
is either BPII with no or few defects, or another structure with hexagonal
ordering. We hope that our simulations will spur further experimental
investigations on quenches in micron-size blue phase samples. The computational
size that our hybrid lattice Boltzmann scheme can handle suggest that large
scale simulations of new generation of blue phase liquid crystal device are
within reach.
