  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.
