Moment realizability and the validity of the Navier–Stokes equations for rarefied gas dynamics

We present criteria for monitoring the validity of the Navier–Stokes approximation during the simulation of a rarefied gas. Our approach is based on an underlying kinetic formulation through which one can construct nondimensional non-negative definite matrices from moments of the molecular distribution. We then identify one such 3×3 matrix that can be evaluated intrinsically in the Navier–Stokes approximation. Our criteria are based on deviations of the eigenvalues of this matrix from their equilibrium value of unity. Not being tied to a particular benchmark problem, the resulting criteria are portable and may be applied to any Navier–Stokes simulation. We study its utility here by comparing stationary planar shock profiles computed using the Navier–Stokes equations with those computed using Monte Carlo simulations.