Fast algorithms for computing the Boltzmann collision operator

Mouhot, Clément; Pareschi, Lorenzo

doi:10.1090/s0025-5718-06-01874-6

Cited by 181 publications

(240 citation statements)

References 42 publications

(72 reference statements)

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“…(7)) with direct solutions of the full Boltzmann equation using the numerical method reported in [27,28], where the molecular velocity distribution function can be obtained accurately. The results for u a and u c at the left wall boundary and the middle channel point are shown in Figs.…”

Section: Simulation Resultsmentioning

confidence: 99%

Assessment of the ellipsoidal-statistical Bhatnagar–Gross–Krook model for force-driven Poiseuille flows

Meng

Reese

et al. 2013

Journal of Computational Physics

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We investigate the accuracy of the ellipsoidal-statistical Bhatnagar-Gross-Krook (ES-BGK) kinetic model for planar force-driven Poiseuille flows. Our numerical simulations are conducted using the deterministic discrete velocity method, for Knudsen numbers (Kn) ranging from 0.05 to 10. While we provide numerically accurate data, our aim is to assess the accuracy of the ES-BGK model for these flows. By comparing with data from the direct simulation Monte Carlo (DSMC) method and the Boltzmann equation, the ES-BGK model is found to be able to predict accurate velocity and temperature profiles in the slip flow regime (0:01 < Kn 6 0:1), for both low-speed and high-speed flows. In the transition flow regime (0:1 < Kn 6 10), however, the model does not quantitatively capture the viscous heating effect.

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Section: Simulation Resultsmentioning

confidence: 99%

Assessment of the ellipsoidal-statistical Bhatnagar–Gross–Krook model for force-driven Poiseuille flows

Meng

Reese

et al. 2013

Journal of Computational Physics

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“…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.…”

mentioning

confidence: 99%

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

White

Scanlon

et al. 2014

J. Fluid Mech.

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A kinetic model of the Boltzmann equation for non-vibrating polyatomic gases is proposed, based on the Rykov model for diatomic gases. We adopt two velocity distribution functions (VDFs) to describe the system state; inelastic collisions are the same as in the Rykov model, but elastic collisions are modelled by the Boltzmann collision operator (BCO) for monatomic gases, so that the overall kinetic model equation reduces to the Boltzmann equation for monatomic gases in the limit of no translational-rotational energy exchange. The free parameters in the model are determined by comparing the transport coefficients, obtained by a Chapman-Enskog expansion, to values from experiment and kinetic theory. The kinetic model equations are solved numerically using the fast spectral method for elastic collision operators and the discrete velocity method for inelastic ones. The numerical results for normal shock waves and planar Fourier/Couette flows are in good agreement with both conventional direct simulation Monte Carlo (DSMC) results and experimental data. Poiseuille and thermal creep flows of polyatomic gases between two parallel plates are also investigated. Finally, we find that the spectra of both spontaneous and coherent Rayleigh-Brillouin scattering (RBS) compare well with DSMC results, and the computational speed of our model is approximately 300 times faster. Compared to the Rykov model, our model greatly improves prediction accuracy, and reveals the significant influence of molecular models. For coherent RBS, we find that the Rykov model could overpredict the bulk viscosity by a factor of two.

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“…The method we use is an extension of the spectral method introduced in [7,16] for the classical collision operator.…”

Section: Computing the Quantum Collision Operator Q Qmentioning

confidence: 99%

A numerical scheme for the quantum Boltzmann equation with stiff collision terms

Filbet

Jin

2011

ESAIM: M2AN

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Abstract.Numerically solving the Boltzmann kinetic equations with the small Knudsen number is challenging due to the stiff nonlinear collision terms. A class of asymptotic-preserving schemes was introduced in [F. Filbet and S. Jin, J. Comput. Phys. 229 (2010) 7625-7648] to handle this kind of problems. The idea is to penalize the stiff collision term by a BGK type operator. This method, however, encounters its own difficulty when applied to the quantum Boltzmann equation. To define the quantum Maxwellian (Bose-Einstein or Fermi-Dirac distribution) at each time step and every mesh point, one has to invert a nonlinear equation that connects the macroscopic quantity fugacity with density and internal energy. Setting a good initial guess for the iterative method is troublesome in most cases because of the complexity of the quantum functions (Bose-Einstein or Fermi-Dirac function). In this paper, we propose to penalize the quantum collision term by a 'classical' BGK operator instead of the quantum one. This is based on the observation that the classical Maxwellian, with the temperature replaced by the internal energy, has the same first five moments as the quantum Maxwellian. The scheme so designed avoids the aforementioned difficulty, and one can show that the density distribution is still driven toward the quantum equilibrium. Numerical results are presented to illustrate the efficiency of the new scheme in both the hydrodynamic and kinetic regimes. We also develop a spectral method for the quantum collision operator.

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Fast algorithms for computing the Boltzmann collision operator

Cited by 181 publications

References 42 publications

Assessment of the ellipsoidal-statistical Bhatnagar–Gross–Krook model for force-driven Poiseuille flows

Assessment of the ellipsoidal-statistical Bhatnagar–Gross–Krook model for force-driven Poiseuille flows

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

A numerical scheme for the quantum Boltzmann equation with stiff collision terms

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