  Any numerical method fails to provide us with acceptable results if not
equipped with appropriate boundary conditions. Catering to more realistic
applications, in the present article we have extended the work done on the one
plus one dimensional Boltzmann equation to the Boltzmann equation involving
multi-dimensions in physical and velocity space. Criteria for stable boundary
conditions, using energy estimates, have been discussed for linear symmetric
hyperbolic initial boundary value problems. Since the use of energy estimates
requires the hyperbolic system to be symmetric, the symmetric hyperbolicity of
the moment equations arising from a Hermite discretization of the Boltzmann
equation has been studied. Furthermore, an algorithm to construct a general
symmetrizer for an arbitrary order Hermite discretization has been presented. A
block structure for the multi-dimensional moment equations has been recognised
which has been used to construct stable Onsager boundary conditions. The newly
proposed Onsager boundary conditions have been used to study a Poisson heat
conduction problem using a higher order Hermite discretization; the results
have been compared to those obtained from the Maxwell's accommodation model.
