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The Fokkerâ€“Planck approximation to the Boltzmann equation, solved numerically by stochastic particle schemes, is used to provide estimates for rarefied gas flows. This paper presents a variance reduction technique for a stochastic particle method that is able to greatly reduce the uncertainty of the estimated flow fields when the characteristic speed of the flow is small in comparison to the thermal velocity of the gas. The method relies on importance sampling, requiring minimal changes to the basic stochastic particle scheme. 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.
[1] J.M. Reese, M.A. Gallis, D.A. Lockerby, New directions in fluid dynamics: non-equilibrium aerodynamic and microsystem flows, Philos. Trans. R. Soc. Lond. A, Math. Phys. Eng. Sci. 361 (1813) (2003) 2967-2988.
[2] G.A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon Press, 1994.
[3] E.S. Oran, C.K. Oh, B.Z. Cybyk, Direct Simulation Monte Carlo: Recent Advances and Applications, Annu. Rev. Fluid Mech. 30 (1998) 403-441.
[4] N.G. Hadjiconstantinou, A.L. Garcia, M.Z. Bazant, G. He, Statistical error in particle simulations of hydrodynamic phenomena, J. Comput. Phys. 187 (2003) 274-297, arXiv:cond-mat/0207430.
[5] T.M.M. Homolle, N.G. Hadjiconstantinou, A low-variance deviational simulation Monte Carlo for the Boltzmann equation, J. Comput. Phys. 226 (2) (2007) 2341-2358.
[6] H.A. Al-Mohssen, N.G. Hadjiconstantinou, Low-variance direct Monte Carlo simulations using importance weights, ESAIM: Math. Model. Numer. Anal. 44 (5) (2010) 1069-1083.
[7] P. Jenny, M. Torrilhon, S. Heinz, A solution algorithm for the fluid dynamic equations based on a stochastic model for molecular motion, J. Comput. Phys. 229 (4) (2010) 1077-1098.
[8] M.H. Gorji, M. Torrilhon, P. Jenny, Fokker-Planck model for computational studies of monatomic rarefied gas flows, J. Fluid Mech. 680 (2011) 574-601.
[9] M.H. Gorji, N. Andric, P. Jenny, Variance reduction for Fokker-Planck based particle Monte Carlo schemes, J. Comput. Phys. 295 (2015) 644-664.
[10] S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15 (1) (1943) 1-89.
[11] L. Ferrari, On the velocity relaxation of a Rayleigh gas: I. Assumptions and approximations in the derivation of the usual kinetic equation, Physica A 115 (1-2) (1982) 232-246.
[12] C. Cercignani, The Boltzmann Equation and Its Applications, Springer, 1988.
[13] H.G. Jenny Patrick, A kinetic model for gas mixtures based on a Fokker-Planck equation, J. Phys. Conf. Ser. 362 (1) (2012) 12042.
[14] M.H. Gorji, P. Jenny, A Fokker-Planck based kinetic model for diatomic rarefied gas flows, Phys. Fluids 25 (6) (2013) 062002.
[16] J. Chun, D.L. Koch, A direct simulation Monte Carlo method for rarefied gas flows in the limit of small Mach number, Phys. Fluids 17 (10) (2005).
[17] S. Brunner, E. Valeo, J.A. Krommes, Collisional delta-f scheme with evolving background for transport time scale simulations, Phys. Plasmas 6 (12) (1999) 4504-4521.