Scientists implemented a full Boltzmann solver to follow neutrinos in space, momentum and time in a way that includes general relativity. The Boltzmann equation describes how particles like neutrinos move and collide. Solving it directly in the full six‑dimensional phase space is usually too expensive, so most groups use simplified models. This work extends an existing general‑relativistic magnetohydrodynamics code called Gmunu to handle the full problem in multiple spatial coordinate systems.

The team discretizes, or breaks up, the problem in two ways. They split position space (the usual 3D space) using Cartesian, cylindrical or spherical grids in the lab frame. They split momentum space (the neutrino energies and directions) in spherical coordinates in the comoving frame — the frame that moves with the fluid. They use the finite volume method, a standard numerical technique that keeps track of conserved quantities inside small cells. For particle interactions they expand the interaction kernels with a first‑order Legendre series — a simple angular approximation similar to what M1 moment methods do.

A major focus was making the method practical. A direct implicit solver for all coupled interactions would need huge memory and time, so the authors developed optimizations. They exploit the Legendre expansion to lower the effective dimensionality of the linear systems to solve. In a test based on a one‑dimensional snapshot of a core‑collapse supernova, this approach gave a speed‑up of about 300 times compared to a dense matrix solver when using 14 angular bins and including coupling between energy groups and neutrino species.

The solver was validated on standard test problems. It shows quadratic convergence as the grid is refined in space, energy and angles. In one‑dimensional tests the implementation conserved energy to about 1% when using 20 energy bins. The authors also compared their full Boltzmann results to an M1 moment scheme in simplified 1D setups. They found roughly 10% differences in the neutrino luminosity in free‑streaming (low‑interaction) regions, a discrepancy they mainly attribute to the approximate closure used by M1. Average neutrino energies agreed between the two methods. In a core‑collapse supernova test at the time of core bounce, fluid profiles matched well between the methods for 20 energy bins.