An efficient jump-diffusion approximation of the Boltzmann equation

TL;DR

This paper introduces a jump-diffusion particle scheme called Gamma-Boltzmann that accurately approximates the Boltzmann equation, matching all moments up to heat fluxes and achieving the correct Prandtl number, while outperforming previous models in accuracy and efficiency.

Contribution

The paper develops a novel Gamma-Boltzmann jump-diffusion model that overcomes limitations of prior Fokker-Planck-based models, providing a more accurate and computationally feasible particle solution.

Findings

Gamma-Boltzmann model accurately approximates the Boltzmann equation.

Outperforms cubic Fokker-Planck model in low particle regimes.

Remains computationally feasible in dense gas regimes.

Abstract

A jump-diffusion process along with a particle scheme is devised as an accurate and efficient particle solution to the Boltzmann equation. The proposed process (hereafter Gamma-Boltzmann model) is devised to match the evolution of all moments up to the heat fluxes while attaining the correct Prandtl number of 2/3 for monatomic gas with Maxwellian molecular potential. This approximation model is not subject to issues associated with the previously developed Fokker-Planck (FP) based models; such as having wrong Prandtl number, limited applicability, or requiring estimation of higher-order moments. An efficient particle solution to the proposed Gamma-Boltzmann model is devised and compared computationally to the direct simulation Monte Carlo and the cubic FP model [M. H. Gorji, M. Torrilhon, and P. Jenny, J. Fluid Mech. 680 (2011): 574-601] in several test cases including Couette flow and lid-driven cavity. The simulation results indicate that the Gamma-Boltzmann model yields a good approximation of the Boltzmann equation, provides a more accurate solution compared to the cubic FP in the limit of a low number of particles, and remains computationally feasible even in dense regimes.