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In this paper we give an introduction to the Boltzmann equation for neutrino transport used in core collapse supernova models as well as a detailed mathematical description of the \emph{Isotropic Diffusion Source Approximation} (IDSA). Furthermore, we present a numerical treatment of a reduced Boltzmann model problem based on time splitting and finite volumes and revise the discretization of the IDSA for this problem. Discretization error studies carried out on the reduced Boltzmann model problem and on the IDSA show that the errors are of order one in both cases. By a numerical example, a detailed comparison of the reduced model and the IDSA is carried out and interpreted. For this example the IDSA modeling error with respect to the reduced Boltzmann model is numerically determined and localized.

We examine magnetorotationally driven supernovae as sources of $r$-process elements in the early Galaxy. On the basis of thermodynamic histories of tracer particles from a three-dimensional magnetohydrodynamical core-collapse supernova model with approximated neutrino transport, we perform nucleosynthesis calculations with and without considering the effects of neutrino absorption reactions on the electron fraction ($Y_{e}$) during post-processing. We find that the peak distribution of $Y_{e}$ in the ejecta is shifted from $\sim0.15$ to $\sim0.17$ and broadened toward higher $Y_{e}$ due to neutrino absorption. Nevertheless, in both cases the second and third peaks of the solar $r$-process element distribution can be well reproduced. The rare progenitor configuration that was used here, characterized by a high rotation rate and a large magnetic field necessary for the formation of bipolar jets, could naturally provide a site for the strong $r$-process in agreement with observations of the early galactic chemical evolution., Comment: 5 pages, 4 figures

We present a numerical method for handling the resolution of a general transport equation for radiative particles, aimed at physical problems with a general spherical geometry. Having in mind the computational time difficulties encountered in problems such as neutrino transport in astrophysical supernovae, we present a scheme based on full spectral methods in 6d spherical coordinates. This approach, known to be suited when the characteristic length of the dynamics is much smaller than the domain size, has the potential advantage of a global speedup with respect to usual finite difference schemes. An analysis of the properties of the Liouville operator expressed in our coordinates is necessary in order to handle correctly the numerical behaviour of the solution. This reflects on a specific (spherical) geometry of the computational domain. The numerical tests, performed under several different regimes for the equation, prove the robustness of the scheme: their performances also point out to the suitability of such an approach to large scale computations involving transport physics for mass less radiative particles., Comment: 18 pages