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

Abstract

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

A kinetic model of the Boltzmann equation for non-vibrating polyatomic gases

“…For more information about the deflection angle θ, collision kernel B, and postcollision velocity v , see Wu et al (2013).…”

“…Usually we choose γ = 0, but to solve the linearized BCO we choose γ = (2ω − 1)/2 to double the computational efficiency (Wu et al 2013(Wu et al , 2014. Finally, the macroscopic quantities are calculated as: The BCO (3.2) can be solved by the fast spectral method (Wu et al 2013) with a computational cost of O(M 2 N 3 v log(N v )), while the other collision operators in (3.1) can be solved by the discrete velocity method (Huang & Giddens 1967) with a cost of O(N 3 v ). Since we have two VDFs, the computational memory required is twice that for monatomic gases; however, the computational cost only increases slightly.…”

“…However, its intricate collision operator makes a solution difficult to obtain by deterministic numerical methods. For monatomic gases, the computational cost of the Boltzmann collision operator (BCO) is usually of the order of N 7 v , although this can be reduced to O(M 2 N 3 v log N v ) using the fast spectral method for some special collision kernels (Mouhot & Pareschi 2006;Wu et al 2013;Wu, Reese & Zhang 2014), where M 2 and N v are the number of discrete solid angles and velocity grid points in each velocity direction, respectively.…”

Preface: 29th International Symposium on Rarefied Gas Dynamics

“…Here we consider the inverse power-law intermolecular interaction, where the shear viscosity µ is proportional to T ω , with ω being the viscosity index whose values for typical gases can be found in [14]; specifically, the hard-sphere and Maxwell gas has a viscosity index of 0.5 and 1, respectively. We choose the collision kernel to be [12,13] …”

“…In this paper, we investigate the influence of the intermolecular potential on the CRBS line shape and evaluate the accuracy of the s6 kinetic model using a deterministic numerical method, where we solve the Boltzmann collision operator using the fast spectral method [12,13].…”