Building on the framework of Zhang & Shu ( ) , 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 ( ) and Banach & Larecki ( ) .
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 .
The IMEX scheme is formally only first-order accurate , but works well in the diffusion limit , and — without interactions with the background — reduces to the optimal second-order strong stability-preserving explicit Runge-Kutta scheme of Shu & Osher ( ) .
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 .