  One of the longstanding issues in numerical relativity is to enable a
simulation taking account of microphysical processes (e.g., weak interactions
and neutrino cooling). We develop an approximate and explicit scheme in the
fully general relativistic framework as a first implementation of the
microphysics toward a more realistic and sophisticated modeling. In this paper,
we describe in detail a method for implementation of a realistic equation of
state, the electron capture and the neutrino cooling in a multidimensional,
fully general relativistic code. The procedure is based on the so-called
neutrino leakage scheme. To check the validity of the code, we perform a two
dimensional (2D) simulation of spherical stellar core collapse. Until the
convective activities set in, our results approximately agree, or at least are
consistent, with those in the previous so-called state-of-the-art simulations.
In particular, the radial profiles of thermodynamical quantities and the time
evolution of the neutrino luminosities agree quantitatively. The convection is
driven by negative gradients of the entropy per baryon and the electron
fraction as in the previous 2D Newtonian simulations. We clarify which gradient
is more responsible for the convection. Gravitational waves from the convection
are also calculated. We find that the characteristic frequencies of the
gravitational-wave spectra are distributed for higher frequencies than those in
Newtonian simulations due to the general relativistic effects.
