An entropy stable discontinuous Galerkin finite-element moment method for the Boltzmann equation

This paper presents a numerical approximation technique for the Boltzmann equation based on a moment-system approximation in velocity dependence and a discontinuous Galerkin finite-element approximation in position dependence. The closure relation for the moment systems derives from minimization of a suitable φ-divergence. This divergence-based closure yields a hierarchy of tractable symmetric hyperbolic moment systems that retain the fundamental structural properties of the Boltzmann equation. The approximation in position dependence is based on the discontinuous Galerkin finite-element method. The resulting combined discontinuous Galerkin moment method corresponds to a Galerkin approximation of the Boltzmann equation in renormalized form. The new moment–closure formulation engenders a new upwind numerical flux function, based on half-space integrals of the approximate distribution. We establish that the proposed upwind flux ensures entropy dissipation of the discontinuous Galerkin finite-element approximation. Numerical results are presented for a one-dimensional test case.