An efficient jump-diffusion approximation of the Boltzmann equation

• Published in: Journal of computational physics Vol. 490; p. 112308
• Main Authors: Mies, Fabian, Sadr, Mohsen, Torrilhon, Manuel
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.10.2023
• Subjects: Couette flow; Fokker-Planck equation; Jump process; Lid-driven cavity; Particle scheme; Couette flow; Jump process; Particle scheme; Fokker-Planck equation; Lid-driven cavity
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2023.112308

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 (Gorji et al. (2011) [10]) 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. •Approximate Boltzmann equation with correct Prandtl number via a jump-diffusion.•Reduced statistical noise by utilizing only third-order moments.•Positive-semidefinite diffusion tensor in all regimes.•Validation against DSMC and cubic Fokker-Planck for Couette flow and lid-driven cavity.