The Gaussian moment closure for gas dynamics

The moment closure method of Levermore applied to the Boltzmann equation for rarefied gas dynamics leads to a hierarchy of symmetric hyperbolic systems of partial differential equations. The Euler system is the first member of this hierarchy of closures. In this paper we investigate the next member, the 10 moment Gaussian closure. We first reduce the collision term to an integral which may be explicitly evaluated for the special case of Maxwell molecular interaction. The resulting collision term for this case is shown to be equivalent to the term obtained by replacing the Boltzmann collision operator with the Bhatnagar, Gross, and Krook (BGK) approximation. We then analyze the Gaussian system applied to the canonical flow problem of a stationary planar shock. An analytic shock profile for the Gaussian closure is derived and compared with the numerical solutions of the Boltzmann and Navier-Stokes equations. The results show reasonable agreement for weak shocks and close agreement between the downstream Gaussian and Navier-Stokes profiles. The results also suggest what may be expected from higher moment closure systems. In particular, the presence of discontinuities in the solution are seen not to prohibit the development of significant profiles.