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

Abstract

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.

A kinetic Fokker–Planck approach to model hard-sphere gas mixtures

“…can be found as a solution for (38). The Euler integration of the non-linear part leads to a wrong prediction of the species temperatures.…”

“…However, for steady gas flows featuring a small signal to noise ratio, this technique can dramatically reduce the number of required particles per cell. 38…”

Sensitivity characteristics of a waterproof airflow sensor based on a MEMS piezoresistive cantilever and nanohole array

“…The air permeability constant of a nanohole array can be estimated in terms of microfluidics dynamics. [21][22][23][24][25][26][27][28] If the pore diameter of a nanohole is d, the thickness is L, and the free path of a molecule is λ, then, the airflow rate through a single hole is given by the following equation from Hagen-Poiseuille law 5) applies to pore sizes of 200 nm or larger, where the Knudsen number K n is approximately 0.3. If the membrane area and aperture ratio are S and a, respectively, the number of pores N is given by the following equation If the pressure ΔP is uniformly applied to all N holes, the air permeability through the entire nanohole array can be determined as 2) and ( 6), the air transmission constant l can be expressed as…”

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

“…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. Due to the particle nature of the FP method, it can be conveniently combined with other particle methods.…”