  We present a newly developed moving-mesh technique for the multi-dimensional
Boltzmann-Hydro code for the simulation of core-collapse supernovae (CCSNe).
What makes this technique different from others is the fact that it treats not
only hydrodynamics but also neutrino transfer in the language of the 3+1
formalism of general relativity (GR), making use of the shift vector to specify
the time evolution of the coordinate system. This means that the transport part
of our code is essentially general relativistic although in this paper it is
applied only to the moving curvilinear coordinates in the flat Minknowski
spacetime, since the gravity part is still Newtonian. The numerical aspect of
the implementation is also described in detail. Employing the axisymmetric
two-dimensional version of the code, we conduct two test computations:
oscillations and runaways of proto-neutron star (PNS). We show that our new
method works fine, tracking the motions of PNS correctly. We believe that this
is a major advancement toward the realistic simulation of CCSNe.
