  We derive a second-order realizability-preserving scheme for moment models
for linear kinetic equations. We apply this scheme to the first-order
continuous and discontinuous models in slab and three-dimensional geometry
derived in a previous paper as well as the classical full-moment $M_N$ models.
We provide extensive numerical analysis as well as our code to show that the
new class of models can compete or even outperform the full-moment models in
reasonable test cases.
