When considering moment methods for the resolution of the free-transport term of the 1-D kinetic equation, the hyperbolic quadrature method of moments (HyQMOM) closure introduced in [12] leads to a globally hyperbolic system of conservation equations. Here, the HLL scheme used for its resolution is first proved to be realizable, i.e., allows computed moments to remain moments of a positive measure, exploiting a new property shown for the closure. Then, the accuracy of HyQMOM is evaluated for two test cases from the literature. The first is related to rarefied gas dynamics, where the internal structure of stationary shock waves is simulated for non-equilibrium cases. The second is related to the description of a population of inertial particles, where two populations cross each other without interacting.