# Evaluation of the bilinear collision operator of the Boltzmann equation with the irreducible Burnett Ansatz

• Evaluation des Bilinearen Kollisionsoperators der Boltzmanngleichung mit dem Irreduziblen Burnett Ansatz

Aachen : RWTH Aachen University (2023)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2023

Abstract

Solving the Boltzmann equation numerically is an area of ongoing research. There are already existing results based on the Spectral-Fourier Approach by Pareschi & Russo (2000) and Gamba & Tharkabhushanam (2009), and based on the spectral Hermite ansatz introduced by Grad (1949) and linearizing the collision operator by Cai, Torrilhon (2015).Wang and Cai were recently (2019) able to calculate the bilinear collision operator based on the spectral Hermite ansatz. However, their method is computationally expensive. In response, Cai & Fan & Wang (2020) developed and implemented a more efficient algorithm based on the spectral Burnett ansatz. This ansatz has also been looked into analytically (Kumar 1966) and numerically (Gamba 2018). Inspired by Struchtrup's (2005) tensorial approach to the collision operator in Equation 6.35 in "Macroscopic Transport Equations for Rarefied Gas Flows: Approximation Methods in Kinetic Theory", we chose a different approach for calculating the Spectral-Burnett approximation of the Boltzmann collision operator. The main difference lies in the exploitation of features of the basis set, which is adapted to the irreducible subspaces with respect to the orthogonal group of the polynomial space, as it consists of real solid spherical harmonics multiplied with Laguerre polynomials. While this thesis discusses the mostly analytical calculation of collision coefficients for a variety of potentials, the focus is on the understanding of these coefficients provided by representation theory. This understanding is used as inspiration for an algorithm, which has been implemented as c++ code to calculate the numerical solution to the space-homogeneous Boltzmann equation for any distribution. Utilizing the properties of the irreducible subspaces, compared to the previous works we are able to significantly reduce the memory and computation time required for calculating the numerical solution. We could achieve this due to the uniqueness up to a constant of linear maps between the irreducible subspaces. Representation theory allowed us to decompose the collision tensor into two and identify all bilinear maps, that evaluate to zero due to the mathematical properties of the irreducible subspaces in question.

Institutions

• Department of Mathematics [110000]
• Chair of Applied and Computational Mathematics [115010]