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

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