A Dual-Resolution Prescription in the $S_N$ Method for Boltzmann Neutrino Transport I: Proof of Principle and the Resolution of Collision Term

Abstract

We propose a dual-resolution prescription meant for the Boltzmann neutrino transport, in which the advection and collision terms are calculated with different angular resolutions in momentum space. The purpose is to address the issue of the low resolution that afflicts the $S_N$ method in the multi-dimensional neutrino transport simulations for core-collapse supernovae. We handle with a high resolution the advection term alone, assuming that the collision term does not require such high resolutions. To confirm this surmise as well as our new conversion scheme, from low- to high-angular resolutions and vice versa, we run a couple of experimental one-zone (in space) simulations. Neutrino scatterings on nucleons are considered with small recoils fully taken into account whereas the advection term is replaced by the angle- and energy-dependent source terms that are designed to mimic the results of a Boltzmann simulation, inducing the anisotropy in momentum space. For the conversion from a low-resolution distribution function to a high-resolution one, we employ a polynomial interpolations in the zenith and azimuth directions separately with the number conservation and continuity (and periodicity only in the azimuth direction). We find that this dual-resolution scheme works well and that the current angular resolution employed in the canonical supernova simulations with our Boltzmann solver or a bit better in the $\phi_\nu$ direction will be sufficient for the collision terms if they are coupled with the advection terms calculated with a high-angular resolution via this prescription.