Representation Theory Based Algorithm to Compute Boltzmann’s Bilinear Collision Operator in the Irreducible Spectral Burnett Ansatz Efficiently

Abstract

Numerically solving the Boltzmann equation is computationally expensive in part due to the number of variables the distribution function depends upon. Another contributor to the complexity of the Boltzmann Equation is the quadratic collision operator describing changes in the distribution function due to colliding particle pairs. Solving it as efficiently as possible has been a topic of recent research, e.g. Cai and Torrilhon (Phys Fluids 31(12):126105, 2019. https://doi.org/10.1063/1.5127114), Wang and Cai (J Comput Phys 397:108815, 2019. https://doi.org/10.1016/j.jcp.2019.07.014), Cai et al. (Comput Fluids 200:104456, 2020. https://doi.org/10.1016/j.compfluid.2020.104456). In this paper we exploit results from representation theory to find a very efficient algorithm both in terms of memory and computational time for the evaluation of the quadratic collision operator. With this novel approach we are also able to provide a meaningful interpretation of its structure.