A Dual-resolution Prescription for the Discrete Ordinates Method for Boltzmann Neutrino Transport. I. Proof of Principle and the Resolution of the Collision Term

We propose a dual-resolution prescription for the Boltzmann neutrino transport, in which the advection and collision terms are calculated at different angular resolutions in momentum space. The purpose is to address the issue of the low resolution that afflicts the discrete ordinates method in multidimensional neutrino transport simulations of core-collapse supernovae. We handle the advection term at a high resolution, assuming that the collision term does not require such high resolution. 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 scattering on nucleons is 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 anisotropy in momentum space. For the conversion from a low-resolution distribution function to a high-resolution one, we employ polynomial interpolations in the zenith ( μ ν ) and azimuth ( ϕ ν ) directions separately, with number conservation and continuity (and periodicity only in the ϕ ν direction) imposed. 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.