  It has recently been demonstrated that dynamical low-rank algorithms can
provide robust and efficient approximation to a range of kinetic equations.
This is true especially if the solution is close to some asymptotic limit where
it is known that the solution is low-rank. A particularly interesting case is
the fluid dynamic limit that is commonly obtained in the limit of small Knudsen
number. However, in this case the Maxwellian which describes the corresponding
equilibrium distribution is not necessarily low-rank; because of this, the
methods known in the literature are only applicable to the weakly compressible
case. In this paper, we propose an efficient dynamical low-rank integrator that
can capture the fluid limit -- the Navier-Stokes equations -- of the
Boltzmann-BGK model even in the compressible regime. This is accomplished by
writing the solution as $f=Mg$, where $M$ is the Maxwellian and the low-rank
approximation is only applied to $g$. To efficiently implement this
decomposition within a low-rank framework requires that certain coefficients
are evaluated using convolutions, for which fast algorithms are known. Using
the proposed decomposition also has the advantage that the rank required to
obtain accurate results is significantly reduced compared to the previous state
of the art. We demonstrate this by performing a number of numerical experiments
and also show that our method is able to capture sharp gradients/shock waves.
