Entropy bounds for the space–time discontinuous Galerkin finite element moment method applied to the BGK–Boltzmann equation

Abstract

This paper presents a numerical analysis for the time-implicit numerical approximation of the Boltzmann equation based on a moment system approximation in velocity dependence and a discontinuous Galerkin finite-element (DGFE) approximation in time and position dependence. The implicit nature of the DGFE moment method in position and time dependence provides a robust numerical algorithm for the approximation of solutions of the Boltzmann equation. The closure relation for the moment systems derives from minimization of a suitable φ-divergence. We present sufficient conditions such that this divergence-based closure yields a hierarchy of tractable symmetric hyperbolic moment systems that retain the fundamental structural properties of the Boltzmann equation. The resulting combined space-time DGFE moment method corresponds to a Galerkin approximation of the Boltzmann equation in renormalized form. We propose a renormalization map that facilitates the approximation of multidimensional problems in an implicit manner. Moreover, upper and lower entropy bounds are derived for the proposed DGFE moment scheme. Numerical results for benchmark problems governed by the BGK-Boltzmann equation are presented to illustrate the approximation properties of the new DGFE moment method, and it is shown that the proposed velocity-space-time DGFE moment method is entropy bounded.