Entropy stable Hermite approximation of the linearised Boltzmann equation for inflow and outflow boundaries

To obtain a symmetric hyperbolic moment system from the linearised Boltzmann equation, we approximate the entropy variable (derivative of the entropy functional) with the help of multi-variate polynomials in the velocity space. Choosing the entropy functional to be quadratic, we retrieve the Grad's approximation for the linearised Boltzmann equation. We develop a necessary and sufficient condition for the entropy stability of the Grad's approximation on bounded position domain with inflow and outflow boundaries. These conditions show the importance of using the Onsager Boundary Conditions (OBCs) Rana and Struchtrup (2016) [28] for obtaining entropy stability and we use them to prove that a broad class of Grad's approximations, equipped with boundary conditions obtained through continuity of odd fluxes Grad (1949) [17], are entropy unstable. Entropy stability, for the Grad's approximation, is obtained through entropy stabilization of the boundary conditions obtained through the continuity of odd fluxes. Since many practical implementations require the prescription of an inflow velocity, the entropy bounds for two possible methods to achieve the same is discussed in detail, both for the linearised Boltzmann equation and its Hermite approximation. We use the Discontinuous Galerkin (DG) discretization in the physical space, to study several benchmark problems to ascertain the physical accuracy of the proposed entropy stable Grad's approximation.