  We investigate the protoneutron star (PNS) convection using our newly
developed general relativistic Boltzmann neutrino radiation-hydrodynamics code.
This is a pilot study for more comprehensive investigations later. As such, we
take a snapshot of a PNS at 2.3 seconds after bounce from a 1D PNS cooling
calculation and run our simulation for $\sim160\,\mathrm{ms}$ in 2D under
axisymmetry. The original PNS cooling calculation neglected convection entirely
and the initial condition is linearly unstable to convection. We find in our 2D
simulation that convection is instigated there indeed and expands inward after
being full-fledged. The convection is then settled to a quasi-steady state in
$\sim100\,\mathrm{ms}$, being sustained by the negative $Y_e$ gradient, which
is in turn maintained by neutrino emissions. It enhances the luminosities and
mean energies of all species of neutrinos compared to 1D. Taking advantage of
the Boltzmann solver, we analyze the possible occurrence of the neutrino fast
flavor conversion (FFC). We found that FFC is likely to occur in the regions,
where $Y_e$ is lower, and that the growth rate can be as high as $\sim
10^{-1}\,{\mathrm{cm}^{-1}}$.
