An Entropy Stable Discontinuous Galerkin Finite-Element Moment Method for the Boltzmann Equation

摘要

This paper presents a numerical approximation technique for the Boltzmannequation based on a moment system approximation in velocity dependence and adiscontinuous Galerkin finite-element approximation in position dependence. Theclosure relation for the moment systems derives from minimization of a suitable{\phi}-divergence. This divergence-based closure yields a hierarchy oftractable symmetric hyperbolic moment systems that retain the fundamentalstructural properties of the Boltzmann equation. The resulting combineddiscontinuous Galerkin moment method corresponds to a Galerkin approximation ofthe Boltzmann equation in renormalized form. We present a new class ofnumerical flux functions, based on the underlying renormalized Boltzmannequation, that ensure entropy dissipation of the approximation scheme.Numerical results are presented for a one-dimensional test case.