A fast iterative scheme for the linearized Boltzmann equation

TL;DR

This paper introduces a synthetic iterative scheme that accelerates the convergence of solutions to the linearized Boltzmann equation, especially in near-continuum flow regimes, by combining velocity distribution correction with macroscopic flow equations.

Contribution

The paper develops a novel synthetic iterative scheme that is asymptotic-preserving and significantly faster than traditional methods for solving the linearized Boltzmann equation.

Findings

The new scheme converges faster than conventional methods in transition and near-continuum regimes.

It requires less spatial resolution due to its asymptotic-preserving property.

The scheme successfully models gas flows, including mixtures with Lennard-Jones potential.

Abstract

An iterative scheme can be used to find a steady-state solution to the Boltzmann equation, however, it is very slow to converge in the near-continuum flow regime. In this paper, a synthetic iterative scheme is developed to speed up the solution of the linearized Boltzmann equation. The velocity distribution function is first solved by the conventional iterative scheme, then it is corrected such that the macroscopic flow velocity is governed by a diffusion equation which is asymptotic-preserving in the Navier-Stokes limit. The efficiency of the new scheme is verified by calculating the eigenvalue of the iteration, as well as solving for Poiseuille and thermal transpiration flows. The synthetic iterative scheme is significantly faster than the conventional iterative scheme in both the transition and the near-continuum flow regimes. Moreover, due to the asymptotic-preserving properties, the SIS needs less spatial resolution in the near-continuum flow regimes, which makes it even faster than the conventional iterative scheme. Using this synthetic iterative scheme, and the fast spectral approximation of the linearized Boltzmann collision operator, Poiseuille and thermal transpiration flows between two parallel plates, through channels of circular/rectangular cross sections, and various porous media are calculated over the whole range of gas rarefaction. Finally, the flow of a Ne-Ar gas mixture is solved based on the linearized Boltzmann equation with the Lennard-Jones potential for the first time, and the difference between these results and those using hard-sphere intermolecular potential is discussed.