  Building on the framework of Zhang \& Shu
\cite{zhangShu_2010a,zhangShu_2010b}, we develop a realizability-preserving
method to simulate the transport of particles (fermions) through a background
material using a two-moment model that evolves the angular moments of a phase
space distribution function $f$. The two-moment model is closed using algebraic
moment closures; e.g., as proposed by Cernohorsky \& Bludman
\cite{cernohorskyBludman_1994} and Banach \& Larecki
\cite{banachLarecki_2017a}. Variations of this model have recently been used to
simulate neutrino transport in nuclear astrophysics applications, including
core-collapse supernovae and compact binary mergers. We employ the
discontinuous Galerkin (DG) method for spatial discretization (in part to
capture the asymptotic diffusion limit of the model) combined with
implicit-explicit (IMEX) time integration to stably bypass short timescales
induced by frequent interactions between particles and the background.
Appropriate care is taken to ensure the method preserves strict algebraic
bounds on the evolved moments (particle density and flux) as dictated by
Pauli's exclusion principle, which demands a bounded distribution function
(i.e., $f\in[0,1]$). This realizability-preserving scheme combines a suitable
CFL condition, a realizability-enforcing limiter, a closure procedure based on
Fermi-Dirac statistics, and an IMEX scheme whose stages can be written as a
convex combination of forward Euler steps combined with a backward Euler step.
Numerical results demonstrate the realizability-preserving properties of the
scheme. We also demonstrate that the use of algebraic moment closures not based
on Fermi-Dirac statistics can lead to unphysical moments in the context of
fermion transport.
