The Boltzmann transport equation models the evolution of the distribution function of a rarefied gas through phase space. Taking the first five moments of this equation leads to conservation equations for mass, momentum and energy. However, closure relations for the deviatoric stress and heat flux are needed if one wishes to use these conservation equations in place of the Boltzmann equation. In [1], we introduced the Variational Multiscale (VMS) Moment method as a framework for systematically deriving such closure relations. Furthermore, we used this framework to derive a novel entropy stable set of conservation equations that extends the Navier-Stokes-Fourier equations to rarefied gas flows. The VMS method can be interpreted as replacing the distribution function of the Boltzmann equation with a function that is finite-dimensional in phase space velocity, i.e. a discretization. While the objective of deriving closures is to enable the use of the conservation equations in place of the Boltzmann equation, it is natural to ask how well the underlying hydrodynamic discretization approximates the original distribution function. In other words, if a hydrodynamic discretization effectively describes the conservation moments in a given flow regime, does that imply that it is a good approximation of the distribution function of the Boltzmann equation? Focusing on the hard spheres linearized Boltzmann equation applied to the one-dimensional stationary heat transfer and Couette flow problems, we use a generalization of the method in [2] to compute hydrodynamic discretizations due to the Navier-Stokes-Fourier equations and our entropy stable extension. We then compare these discretizations to a numerical solution of the linearized Boltzmann equation. Among other observations, we find that the hydrodynamic discretization due the entropy stable extension is a closer match to the linearized Boltzmann solution than that due to the Navier-Stokes-Fourier equations. This is consistent with the former being an improvement upon the Navier-Stokes-Fourier equations. References [1] Baidoo, F. A., Gamba, I. M., Hughes, T. J. R., & Abdelmalik, M. R. A. (2024). Extensions to the Navier-Stokes-Fourier Equations for Rarefied Transport: Variational Multiscale Moment Methods for the Boltzmann Equation. arXiv:2407.17334. [2] Gust, E. D., & Reichl, L. E. (2009). Molecular dynamics simulation of collision operator eigenvalues. Physical Review E, 79(3), 031202.