Workshop on Numerical Methods and Analysis in CFD - Abstract

Keßler, Torsten

Entropy-based methods for rarefied gas flows

The Boltzmann equation is the fundamental equation for the description of rarefied gas flows that also remains valid in the case of increasing density, where its leading approximations yield the Euler and Navier-Stokes equations, respectively. In this talk, we develop an entropy-stable finite element-moment method for the Boltzmann equation. To that end, we employ a discontinuous-Galerkin method in space and time, and a spectral method in unbounded velocity space. We base our method on a converging sequence of approximations to the collision operator. We associate with each member of this sequence a normalisation map and an entropy. We show that each approximate collision operator inherits salient properties from Boltzmann's operator, such as the preservation of mass, momentum and energy, and that the linearisations near equilibrium agree. We prove that our method is entropy-stable for each member of the sequence of approximations to the collision operator. Finally, we apply our method to the Boltzmann equation with full collision operator and demonstrate the corresponding approximation properties, using benchmark test cases, in comparison to Direct Simulation Monte Carlo.
This is joint work with M. R. A. Abdelmalik, I. M. Gamba and S. Rjasanow.