### Abstract

Original language | English |
---|---|

Pages (from-to) | 27-52 |

Number of pages | 26 |

Journal | Journal of Computational Physics |

Volume | 250 |

Early online date | 18 May 2013 |

DOIs | |

Publication status | Published - 1 Oct 2013 |

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### Keywords

- Boltzmann equation
- Lennard-Jones potential
- Sutherland's formula of shear viscosity
- Fourier spectral method
- rarefied gas dynamics

### Cite this

*Journal of Computational Physics*,

*250*, 27-52. https://doi.org/10.1016/j.jcp.2013.05.003

}

*Journal of Computational Physics*, vol. 250, pp. 27-52. https://doi.org/10.1016/j.jcp.2013.05.003

**Deterministic numerical solutions of the Boltzmann equation using the fast spectral method.** / Wu, Lei; White, Craig; Scanlon, Thomas; Reese, Jason; Zhang, Yonghao.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Deterministic numerical solutions of the Boltzmann equation using the fast spectral method

AU - Wu, Lei

AU - White, Craig

AU - Scanlon, Thomas

AU - Reese, Jason

AU - Zhang, Yonghao

PY - 2013/10/1

Y1 - 2013/10/1

N2 - The Boltzmann equation describes the dynamics of rarefied gas flows, but the multidimensional nature of its collision operator poses a real challenge for its numerical solution. In this paper, the fast spectral method [36], originally developed by Mouhot and Pareschi for the numerical approximation of the collision operator, is extended to deal with other collision kernels, such as those corresponding to the soft, Lennard–Jones, and rigid attracting potentials. The accuracy of the fast spectral method is checked by comparing our numerical solutions of the space-homogeneous Boltzmann equation with the exact Bobylev–Krook–Wu solutions for a gas of Maxwell molecules. It is found that the accuracy is improved by replacing the trapezoidal rule with Gauss–Legendre quadrature in the calculation of the kernel mode, and the conservation of momentum and energy are ensured by the Lagrangian multiplier method without loss of spectral accuracy. The relax-to-equilibrium processes of different collision kernels with the same value of shear viscosity are then compared; the numerical results indicate that different forms of the collision kernels can be used as long as the shear viscosity (not only the value, but also its temperature dependence) is recovered. An iteration scheme is employed to obtain stationary solutions of the space-inhomogeneous Boltzmann equation, where the numerical errors decay exponentially. Four classical benchmarking problems are investigated: the normal shock wave, and the planar Fourier/Couette/force-driven Poiseuille flows. For normal shock waves, our numerical results are compared with a finite difference solution of the Boltzmann equation for hard sphere molecules, experimental data, and molecular dynamics simulation of argon using the realistic Lennard–Jones potential. For planar Fourier/Couette/force-driven Poiseuille flows, our results are compared with the direct simulation Monte Carlo method. Excellent agreements are observed in all test cases, demonstrating the merit of the fast spectral method as a computationally efficient method for rarefied gas dynamics.

AB - The Boltzmann equation describes the dynamics of rarefied gas flows, but the multidimensional nature of its collision operator poses a real challenge for its numerical solution. In this paper, the fast spectral method [36], originally developed by Mouhot and Pareschi for the numerical approximation of the collision operator, is extended to deal with other collision kernels, such as those corresponding to the soft, Lennard–Jones, and rigid attracting potentials. The accuracy of the fast spectral method is checked by comparing our numerical solutions of the space-homogeneous Boltzmann equation with the exact Bobylev–Krook–Wu solutions for a gas of Maxwell molecules. It is found that the accuracy is improved by replacing the trapezoidal rule with Gauss–Legendre quadrature in the calculation of the kernel mode, and the conservation of momentum and energy are ensured by the Lagrangian multiplier method without loss of spectral accuracy. The relax-to-equilibrium processes of different collision kernels with the same value of shear viscosity are then compared; the numerical results indicate that different forms of the collision kernels can be used as long as the shear viscosity (not only the value, but also its temperature dependence) is recovered. An iteration scheme is employed to obtain stationary solutions of the space-inhomogeneous Boltzmann equation, where the numerical errors decay exponentially. Four classical benchmarking problems are investigated: the normal shock wave, and the planar Fourier/Couette/force-driven Poiseuille flows. For normal shock waves, our numerical results are compared with a finite difference solution of the Boltzmann equation for hard sphere molecules, experimental data, and molecular dynamics simulation of argon using the realistic Lennard–Jones potential. For planar Fourier/Couette/force-driven Poiseuille flows, our results are compared with the direct simulation Monte Carlo method. Excellent agreements are observed in all test cases, demonstrating the merit of the fast spectral method as a computationally efficient method for rarefied gas dynamics.

KW - Boltzmann equation

KW - Lennard-Jones potential

KW - Sutherland's formula of shear viscosity

KW - Fourier spectral method

KW - rarefied gas dynamics

UR - http://www.sciencedirect.com/S0021999113003276

U2 - 10.1016/j.jcp.2013.05.003

DO - 10.1016/j.jcp.2013.05.003

M3 - Article

VL - 250

SP - 27

EP - 52

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

ER -