An efficient particle Fokker–Planck algorithm for rarefied gas flows

Gorji, M. Hossein; Jenny, Patrick

doi:10.1016/j.jcp.2013.12.046

Cited by 64 publications

(31 citation statements)

References 27 publications

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“…This Prandtl number problem could be fixed with the introduction of a cubic drift model, 2,19 which was demonstrated for various challenging test cases. 20 This issue is also discussed in the recent work by Singh et al, 21 in which they proposed a way to fix the Prandtl number resulting from the Fokker-Planck equation. They achieved that by introducing an extra streaming in the particle position to fix the transport properties including the Prandtl number.…”

Section: Introductionmentioning

confidence: 94%

“…The advantage of this approach has been demonstrated for a large variety of relevant flows and up to Kn ≈ 2 the FP method proved to deliver accurate velocity fields. 2,20 It has to be mentioned, however, that opposed to DSMC the scheme (12) and (13) honors momentum and energy conservation only statistically. Thus, a disadvantage of models based on Langevin equations is that additional noise gets introduced, which also affects the bias error.…”

Section: Physics Of Fluidsmentioning

confidence: 99%

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Controlling the bias error of Fokker-Planck methods for rarefied gas dynamics simulations

2019

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Direct simulation Monte-Carlo (DSMC) is the most established method for rarefied gas flow simulations. It is valid from continuum to near vacuum, but in cases involving small Knudsen numbers (Kn), it suffers from high computational cost. The Fokker-Planck (FP) method, on the other hand, is almost as accurate as DSMC for small to moderate Kn, but it does not have the computational drawback of DSMC, if Kn is small [P. Jenny, M. Torrilhon, and S. Heinz, “A solution algorithm for the fluid dynamic equations based on a stochastic model for molecular motion,” J. Comput. Phys. 229, 1077–1098 (2010) and H. Gorji, M. Torrilhon, and P. Jenny, “Fokker–Planck model for computational studies of monatomic rarefied gas flows,” J. Fluid Mech. 680, 574–601 (2011)]. Especially attractive is the combination of the two approaches leading to the FP-DSMC method. Opposed to other hybrid methods, e.g., coupled DSMC/Navier-Stokes solvers, it is relatively straightforward to couple DSMC with the FP method since both are based on particle solution algorithms sharing the same data structure and having similar components. Regarding the numerical accuracy of such particle methods, one has to distinguish between spatial truncation errors, time stepping errors, statistical errors and bias errors. In this paper, the bias error of the FP method is analyzed in detail, and it is shown how it can be reduced without increasing the particle number to an exorbitant level. The effectiveness of the discussed bias error reduction scheme is demonstrated for uniform shear flow, for which an analytical reference solution was derived.

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Section: Introductionmentioning

confidence: 94%

Section: Physics Of Fluidsmentioning

confidence: 99%

Controlling the bias error of Fokker-Planck methods for rarefied gas dynamics simulations

2019

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“…It has been used to model electrons, dense liquids and more recently has received attention for its ability to model rarefied gas flows [12]. More recently it has been extended to describe flows of monatomic gas mixtures [13], diatomic molecules [14] and has been coupled to DSMC [15]. The equation for the one-particle distribution function f (x, v, t), over a state-space comprised of the position x ∈ R 3 , velocity v ∈ R 3 and time t ∈ R + takes the form:…”

Section: The Fokker-planck Collision Operatormentioning

confidence: 99%

Importance sampling variance reduction for the Fokker–Planck rarefied gas particle method

Collyer

Connaughton

Lockerby

2016

Journal of Computational Physics

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Models and methods that are able to accurately and efficiently predict the flows of low-speed rarefied gases are in high demand, due to the increasing ability to manufacture devices at micro and nano scales. One such model and method is a Fokker-Planck approximation to the Boltzmann equation, which can be solved numerically by a stochastic particle method. The stochastic nature of this method leads to noisy estimates of the thermodynamic quantities one wishes to sample when the signal is small in comparison to the thermal velocity of the gas. Recently, Gorji et al have proposed a method which is able to greatly reduce the variance of the estimators, by creating a correlated stochastic process which acts as a control variate for the noisy estimates. However, there are potential difficulties involved when the geometry of the problem is complex, as the method requires the density to be solved for independently. Importance sampling is a variance reduction technique that has already been shown to successfully reduce the noise in direct simulation Monte Carlo calculations. In this paper we propose an importance sampling method for the Fokker-Planck stochastic particle scheme. The method requires minimal change to the original algorithm, and dramatically reduces the variance of the estimates. We test the importance sampling scheme on a homogeneous relaxation, planar Couette flow and a lid-driven-cavity flow, and find that our method is able to greatly reduce the noise of estimated quantities. Significantly, we find that as the characteristic speed of the flow decreases, the variance of the noisy estimators becomes independent of the characteristic speed.

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“…The issue of the Prandtl number was then fixed in the cubic model derived in [2]. Further validations of the cubic FP model are presented in [3]. For simplicity, in this study the focus remains on the FP model with the linear drift term (4).…”

Section: The Fokker-planck Model With Linear Driftmentioning

confidence: 99%

“…

Recently, Fokker-Planck based particle Monte Carlo schemes have been proposed and evaluated for simulations of rarefied gas flows [1][2][3]. In this paper, the variance reduction for particle Monte Carlo simulations based on the Fokker-Planck model is considered.

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mentioning

confidence: 99%