Extensions to the Navier–Stokes–Fourier equations for rarefied transport: Variational multiscale moment methods for the Boltzmann equation

In this paper, we derive a fourth-order entropy stable extension of the Navier–Stokes–Fourier equations into the transition regime of rarefied gases. We do this through a novel reformulation of the closure of conservation equations derived from the Boltzmann equation that subsumes existing methods such as the Chapman–Enskog expansion. We apply the linearized version of this extension to the stationary heat problem and the Poiseuille channel and compare our analytical solutions to asymptotic and numerical solutions of the linearized Boltzmann equation. In both model problems, our solutions compare remarkably well in the transition regime. For some macroscopic variables, this agreement even extends far beyond the transition regime.