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*Astron. Astrophys. 344, 533-550 (1999)*
## Neutrino transport in type II supernovae: Boltzmann solver vs. Monte Carlo method
**
Shoichi Yamada **^{*} ^{ 1,2},
Hans-Thomas Janka ^{**} ^{ 1} and
Hideyuki Suzuki ^{***} ^{ 1,3}
^{1} Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Strasse 1, Postfach 1523, D-85740 Garching, Germany
^{2} Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113, Japan
^{3} High Energy Accelerator Research Organization (KEK), Oho, Tsukuba, Ibaraki 305-0801, Japan
*Received 7 September 1998 / Accepted 1 February 1999*
**Abstract**
We have coded a Boltzmann solver based on a finite difference
scheme (S_{N} method) aiming at calculations of
neutrino transport in type II supernovae. Close comparison between the
Boltzmann solver and a Monte Carlo transport code has been made for
realistic atmospheres of post bounce core models under the assumption
of a static background. We have also investigated in detail the
dependence of the results on the numbers of radial, angular, and
energy grid points and the way to discretize the spatial advection
term which is used in the Boltzmann solver. A general relativistic
calculation has been done for one of the models. We find good overall
agreement between the two methods. This gives credibility to both
methods which are based on completely different formulations. In
particular, the number and energy fluxes and the mean energies of the
neutrinos show remarkably good agreement, because these quantities are
determined in a region where the angular distribution of the neutrinos
is nearly isotropic and they are essentially frozen in later on. On
the other hand, because of a relatively small number of angular grid
points (which is inevitable due to limitations of the computation
time) the Boltzmann solver tends to slightly underestimate the flux
factor and the Eddington factor outside the (mean) "neutrinosphere"
where the angular distribution of the neutrinos becomes highly
anisotropic. As a result, the neutrino number (and energy) density is
somewhat overestimated in this region. This fact suggests that the
Boltzmann solver should be applied to calculations of the neutrino
heating in the hot-bubble region with some caution because there might
be a tendency to overestimate the energy deposition rate in
disadvantageous situations. A comparison shows that this trend is
opposite to the results obtained with a multi-group flux-limited
diffusion approximation of neutrino transport. Employing three
different flux limiters, we find that all of them lead to a
significant underestimation of the neutrino energy density in the
semitransparent regime, and thus must yield too low values for the net
neutrino heating (heating minus cooling) in the hot-bubble region. The
accuracy of the Boltzmann solver can be improved by using a variable
angular mesh to increase the angular resolution in the region where
the neutrino distribution becomes anisotropic.
**Key words:** elementary
particles
radiative
transfer
methods:
numerical
stars: neutron
stars: supernovae: general
* shoichi@mpa-garching.mpg.de
** thj@mpa-garching.mpg.de
*** hideyuki.suzuki@kek.jp
*Send offprint requests to:* S. Yamada
This article contains no SIMBAD objects.
### Contents
© European Southern Observatory (ESO) 1999
Online publication: March 18, 1999
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