The equation of state describes the thermodynamical state of a fluid
element based on density, 
, temperature, 
, and
the composition. We use the equation of state of Lattimer_Swesty_91.
It assumes nuclear statistical equilibrium and we apply it wherever
the density is larger than 
g/cm
and the temperature
larger than
K. This region is described by a
liquid drop model for a representative nucleus with atomic number
and charge 
, surrounded by free alpha particles,
protons, and neutrons. The baryons are immersed in an electron and
positron gas in equilibrium with a photon gas. Beyond nuclear density,
where no isolated nuclei are present, the complicated population of
hadrons Glendenning_85,Pons_et_al_99 is approximated by bulk
nuclear matter comprised of protons, neutrons, and electrons. However,
the central density of the protoneutron star at bounce reaches only
about twice nuclear density and the hadron population may only develop
later, after the very dynamical postbounce phases. At the low temperature
border of nuclear statistical equilibrium, the equation of state is
connected to a Boltzmann gas of silicon atoms. In any of these cases,
once the density and temperature are given, the composition is fully
determined by the specification of the electron fraction 
.
Matter is connected to the neutrino radiation field by weak interactions.
We consider neutrinos of all three flavors and assume that they are
massless. The weak interactions enter the collision term in the Boltzmann
equation as energy- and angle-dependent emissivities, opacities, and
scattering kernels. We include the set specified by Bruenn_85:
(i) electron-type neutrino absorption on neutrons, (ii) electron antineutrino
absorption on protons, (iii) electron-type neutrino absorption on
nuclei, (iv) neutrino-nucleon scattering, (v) coherent scattering
of neutrinos on nuclei, (vi) neutrino-electron scattering, and (vii)
neutrino production from electron/positron pair annihilation. These
reactions, and their inverses, are implemented in our code as described
by Mezzacappa_Bruenn_93b,Mezzacappa_Bruenn_93c,Messer_00.
In the following code description, we will only include emissivities,
, and opacities, 
, because this is sufficient
to describe how the collision term enters the transport and hydrodynamics
equations. In the simulations, the scattering kernels are included
in the collision integral as well.
The particles treated by the equation of state are assumed to react
with each other on very short time scales such that a description
in terms of an instantaneous equilibrium is appropriate. Neutrinos
in high-density regimes can also achieve local thermal and weak equilibrium
with matter if the opacities are sufficiently high. Unlike the equilibrium
with respect to the strong interaction, however, this equilibrium
must be determined within our solution of the transport equation.
For example, in the protoneutron star at densities above 
g cm
and temperatures above
K the
neutrinos are trapped and are well-described by a Fermi-gas in thermal
equilibrium with the fluid. At lower densities, the thermalization
time scale becomes longer; then the neutrinos can propagate with a
nonequilibrium spectrum throughout these regions, to be absorbed elsewhere
or leave the star. The strong coupling of the neutrinos to the matter
at high densities and the strong coupling between different locations
mediated by neutrino transport complicates the evolution of a numerical
solution. If the problem is separated into independently updated pieces
by operator splitting, the numerical solution will only be stable
if information in the numerical implementation is shared faster between
the independent updates than in the evaluated physical processes.
The fast time scale of neutrino-matter interactions and the propagation
of neutrinos at light speed may severely restrict the time step. The
required coupling can be built directly into the numerical scheme
by an implicit finite differencing of essential parts of the transport
equation. Unfortunately, such a differencing requires knowledge of
the derivatives of the collision term with respect to all independent
state variables-i.e., density, temperature, electron fraction,
and neutrino distribution functions. Because the emissivities, opacities,
and scattering kernels strongly depend on the neutrino energies and,
in the scattering case, on the neutrino propagation directions, the
numerical evaluation of the collision term and its derivatives becomes
a nonnegligible part of the overall computational effort. Mezzacappa_Bruenn_93a
developed a storage scheme that allows the reuse of previously calculated
emissivities, opacities, and scattering kernels by linear interpolation
within a dynamical table in the independent variables of logarithmic
density,
, logarithmic temperature,
, and electron fraction, .
If one uses these same independent variables in the implicit formulation
of the Boltzmann equation, the correct partial derivatives of the
reactions directly emerge from the coefficients of the linear interpolation,
without additional computational effort. On the one hand, the reuse
of previously evaluated interactions is straightforward if the transport
equation is solved in the rest frame of the fluid, such that no transformation
of the neutrino energy or angle dependence of the interactions is
required. On the other hand, the transport equations are simpler in
the laboratory frame. In this paper we demonstrate that in the case
of spherical symmetry the complexity of the transport equation is
manageable and proceed with the analysis in a comoving frame (spacetime
coordinates and neutrino four-momentum) to take advantage of the simplifications
in the collision term. On average, we have to evaluate new collision
integrals in about two to three zones per time step (out of a hundred
zones). Although these numbers depend very much on the specific phase
of the simulation, the evaluation of nuclear physics input may still
take about half of the total execution time.