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

Abstract

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.

On the conservative property of particle-based Fokker–Planck method for rarefied gas flows

“…52 Numerical errors in the effective transport properties caused by temporal discretization were analyzed by Zhang et al 53 and Fei et al 54,55 using the Green-Kubo relations. Gorji et al 56 and Collyer et al 57 applied variance reduction techniques to reduce the noise in the FP simulation of low speed flows. Recently, Jenny et al 58 analyzed the bias error of the FP method.…”

Particle number control for direct simulation Monte-Carlo methodology using kernel estimates

“…Kernel based density estimators have been employed for reconstructing the probability density of particles in low-variance DSMC or Fokker-Planck Monte-Carlo schemes. 20,21 On the reduction side, elaborate schemes have been introduced in numerous works. 18,22,23 For example, in Ref.…”