A fast spectral method for the Boltzmann equation for monatomic gas mixtures

Lei Wu, Jun Zhang, Jason M. Reese, Yonghao Zhang

Research output: Contribution to journalArticle

20 Citations (Scopus)

Abstract

Although the fast spectral method has been established for solving the Boltzmann equation for single-species monatomic gases, its extension to gas mixtures is not easy because of the non-unitary mass ratio between the di↵erent molecular species. The conventional spectral method can solve the Boltzmann collision operator for binary gas mixtures but with a computational cost of the order m3rN6, where mr is the mass ratio of the heavier to the lighter species, and N is the number of frequency nodes in each frequency direction. In this paper, we propose a fast spectral method for binary mixtures of monatomic gases that has a computational cost O(pmrM2N4 logN), where M2 is the number of discrete solid angles. The algorithm is validated by comparing numerical results with analytical Bobylev-
Krook-Wu solutions for the spatially-homogeneous relaxation problem, for mr up to 36. In spatially-inhomogeneous problems, such as normal shock waves and planar Fourier/Couette flows, our results compare well with those of both the numerical kernel and the direct simulation Monte Carlo methods. As an application, a two-dimensional temperature-driven flow is investigated, for which other numerical methods find it difficult to resolve the flow field at large Knudsen numbers. The fast spectral method is accurate and elective in simulating highly rarefied gas flows, i.e. it captures the discontinuities and fine structures in the velocity distribution functions.
LanguageEnglish
Pages602-621
Number of pages10
JournalJournal of Computational Physics
Volume298
Early online date30 Jun 2015
DOIs
Publication statusPublished - 1 Oct 2015

Fingerprint

Boltzmann equation
monatomic gases
spectral methods
Binary mixtures
Gas mixtures
gas mixtures
Velocity distribution
Gases
Shock waves
mass ratios
binary mixtures
Distribution functions
Flow of gases
Costs
Numerical methods
Flow fields
Monte Carlo methods
normal shock waves
costs
Knudsen flow

Keywords

  • Boltzmann equation
  • gas mixtures
  • Fourier spectral method
  • rarefied gas dynamics

Cite this

@article{f666e01e3f8542459b9f7cb6a66f3fc8,
title = "A fast spectral method for the Boltzmann equation for monatomic gas mixtures",
abstract = "Although the fast spectral method has been established for solving the Boltzmann equation for single-species monatomic gases, its extension to gas mixtures is not easy because of the non-unitary mass ratio between the di↵erent molecular species. The conventional spectral method can solve the Boltzmann collision operator for binary gas mixtures but with a computational cost of the order m3rN6, where mr is the mass ratio of the heavier to the lighter species, and N is the number of frequency nodes in each frequency direction. In this paper, we propose a fast spectral method for binary mixtures of monatomic gases that has a computational cost O(pmrM2N4 logN), where M2 is the number of discrete solid angles. The algorithm is validated by comparing numerical results with analytical Bobylev-Krook-Wu solutions for the spatially-homogeneous relaxation problem, for mr up to 36. In spatially-inhomogeneous problems, such as normal shock waves and planar Fourier/Couette flows, our results compare well with those of both the numerical kernel and the direct simulation Monte Carlo methods. As an application, a two-dimensional temperature-driven flow is investigated, for which other numerical methods find it difficult to resolve the flow field at large Knudsen numbers. The fast spectral method is accurate and elective in simulating highly rarefied gas flows, i.e. it captures the discontinuities and fine structures in the velocity distribution functions.",
keywords = "Boltzmann equation, gas mixtures, Fourier spectral method, rarefied gas dynamics",
author = "Lei Wu and Jun Zhang and Reese, {Jason M.} and Yonghao Zhang",
note = "OA for RCUK",
year = "2015",
month = "10",
day = "1",
doi = "10.1016/j.jcp.2015.06.019",
language = "English",
volume = "298",
pages = "602--621",
journal = "Journal of Computational Physics",
issn = "0021-9991",

}

A fast spectral method for the Boltzmann equation for monatomic gas mixtures. / Wu, Lei; Zhang, Jun; Reese, Jason M.; Zhang, Yonghao.

In: Journal of Computational Physics, Vol. 298, 01.10.2015, p. 602-621.

Research output: Contribution to journalArticle

TY - JOUR

T1 - A fast spectral method for the Boltzmann equation for monatomic gas mixtures

AU - Wu, Lei

AU - Zhang, Jun

AU - Reese, Jason M.

AU - Zhang, Yonghao

N1 - OA for RCUK

PY - 2015/10/1

Y1 - 2015/10/1

N2 - Although the fast spectral method has been established for solving the Boltzmann equation for single-species monatomic gases, its extension to gas mixtures is not easy because of the non-unitary mass ratio between the di↵erent molecular species. The conventional spectral method can solve the Boltzmann collision operator for binary gas mixtures but with a computational cost of the order m3rN6, where mr is the mass ratio of the heavier to the lighter species, and N is the number of frequency nodes in each frequency direction. In this paper, we propose a fast spectral method for binary mixtures of monatomic gases that has a computational cost O(pmrM2N4 logN), where M2 is the number of discrete solid angles. The algorithm is validated by comparing numerical results with analytical Bobylev-Krook-Wu solutions for the spatially-homogeneous relaxation problem, for mr up to 36. In spatially-inhomogeneous problems, such as normal shock waves and planar Fourier/Couette flows, our results compare well with those of both the numerical kernel and the direct simulation Monte Carlo methods. As an application, a two-dimensional temperature-driven flow is investigated, for which other numerical methods find it difficult to resolve the flow field at large Knudsen numbers. The fast spectral method is accurate and elective in simulating highly rarefied gas flows, i.e. it captures the discontinuities and fine structures in the velocity distribution functions.

AB - Although the fast spectral method has been established for solving the Boltzmann equation for single-species monatomic gases, its extension to gas mixtures is not easy because of the non-unitary mass ratio between the di↵erent molecular species. The conventional spectral method can solve the Boltzmann collision operator for binary gas mixtures but with a computational cost of the order m3rN6, where mr is the mass ratio of the heavier to the lighter species, and N is the number of frequency nodes in each frequency direction. In this paper, we propose a fast spectral method for binary mixtures of monatomic gases that has a computational cost O(pmrM2N4 logN), where M2 is the number of discrete solid angles. The algorithm is validated by comparing numerical results with analytical Bobylev-Krook-Wu solutions for the spatially-homogeneous relaxation problem, for mr up to 36. In spatially-inhomogeneous problems, such as normal shock waves and planar Fourier/Couette flows, our results compare well with those of both the numerical kernel and the direct simulation Monte Carlo methods. As an application, a two-dimensional temperature-driven flow is investigated, for which other numerical methods find it difficult to resolve the flow field at large Knudsen numbers. The fast spectral method is accurate and elective in simulating highly rarefied gas flows, i.e. it captures the discontinuities and fine structures in the velocity distribution functions.

KW - Boltzmann equation

KW - gas mixtures

KW - Fourier spectral method

KW - rarefied gas dynamics

UR - http://www.sciencedirect.com/science/journal/00219991

U2 - 10.1016/j.jcp.2015.06.019

DO - 10.1016/j.jcp.2015.06.019

M3 - Article

VL - 298

SP - 602

EP - 621

JO - Journal of Computational Physics

T2 - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

ER -