Accurate and efficient computation of the Boltzmann equation for Kramer's problem

TL;DR

This paper introduces a synthetic iteration scheme (SIS) for solving the linear Boltzmann equation efficiently and accurately in Kramer's problem, analyzing gas-surface interactions across various potentials and boundary conditions.

Contribution

The novel SIS method improves computational efficiency and accuracy for the linear Boltzmann equation in Kramer's problem, incorporating asymptotic-preserving flow velocity guidance.

Findings

SIS achieves high accuracy and efficiency in solving the LBE.

Knudsen layer characteristics depend on intermolecular potential and boundary conditions.

Knudsen layer functions can be fitted by power series using asymptotic theory.

Abstract

In this work, a novel synthetic iteration scheme (SIS) is developed for the LBE to find solutions to Kramer's problem accurately and efficiently: the velocity distribution function is first solved by the conventional iterative scheme, then it is modified such that in each iteration i) the flow velocity is guided by an ordinary differential equation that is asymptotic-preserving at the Navier-Stokes limit and ii) the shear stress is equal to the average shear stress. Based on the Bhatnagar-Gross-Krook model, the SIS is assessed to be efficient and accurate. Then we investigate the Kramer's problem for gases interacting through the inverse power-law, shielded Coulomb, and Lennard-Jones potentials, subject to diffuse-specular and Cercignani-Lampis gas-surface boundary conditions. When the tangential momentum accommodation coefficient (TMAC) is not larger than one, the Knudsen layer function is strongly affected by the potential, where its value and width increase with the effective viscosity index of gas molecules. Moreover, the Knudsen layer function exhibits similarities among different values of TMAC when the intermolecular potential is fixed. For Cercignani-Lampis boundary condition with TMAC larger than one, both the viscous slip coefficient and Knudsen layer function are affected by the intermolecular potential, especially when the "backward" scattering limit is approached. With the asymptotic theory by Jiang and Luo for the singular behavior of the velocity gradient in the vicinity of the solid surface, we find that the whole Knudsen layer function can be well fitted by power series.