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

TL;DR

This work addresses the angular-resolution bottleneck in $S_N$ Boltzmann neutrino transport for core-collapse supernovae by introducing a dual-resolution framework: compute advection with high angular resolution while evaluating the collision term at a lower resolution. The method employs operator splitting and inter-resolution remappings, with zenith and azimuth interpolations designed to conserve particle number and maintain periodicity, and includes an energy-subgrid to capture nucleon recoil effects. Through steady-state and time-evolution one-zone tests using source terms derived from Boltzmann data, the authors demonstrate convergence toward reference distributions and identify acceptable resolution settings, notably $(N_{\theta_\nu},N_{\phi_\nu})=(10,6)$ with $N_{\theta_\nu}^{\mathrm{poly}}=N_{\theta_\nu}+1$, and a quadratic, continuous azimuthal interpolation. The results suggest that the collision-term angular resolution used in current simulations is adequate when paired with high-resolution advection, potentially enabling improved efficiency and accuracy for multi-dimensional CCSN neutrino transport. The work also introduces an energy-subgrid approach to handle recoil in neutrino-nucleon scattering, enhancing the physical fidelity of semi-implicit collision-term treatments.

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 $φ_ν$ 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.

Figures (64)

• Figure 1:
• Figure 2:
• Figure 4:
• Figure 5:
• Figure 7: Radial profiles of background