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Low-variance direct Monte Carlo simulationsusing importance weights

Published online by Cambridge University Press:  26 August 2010

Husain A. Al-Mohssen  and
Nicolas G. Hadjiconstantinou
Husain A. Al-Mohssen
Affiliation:
Mechanical Engineering Dept., MIT, 77 Massachusetts Ave., Cambridge, MA 02139, USA. husain@mit.edu; ngh@mit.edu
Nicolas G. Hadjiconstantinou
Affiliation:
Mechanical Engineering Dept., MIT, 77 Massachusetts Ave., Cambridge, MA 02139, USA. husain@mit.edu; ngh@mit.edu
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Abstract

We present an efficient approach for reducing the statistical uncertaintyassociated with direct Monte Carlo simulations of the Boltzmann equation.As with previous variance-reduction approaches, the resulting relativestatistical uncertainty in hydrodynamic quantities (statistical uncertainty normalized by thecharacteristic value of quantity of interest) is smalland independent of the magnitude of the deviation from equilibrium,making the simulation of arbitrarily small deviations from equilibriumpossible. In contrast to previous variance-reduction methods, themethod presented here is able to substantially reduce variance withvery little modification to the standard DSMC algorithm. This is achievedby introducing an auxiliary equilibrium simulation which, via an importanceweight formulation, uses the same particle data as the non-equilibrium(DSMC) calculation; subtracting the equilibrium from the non-equilibriumhydrodynamic fields drastically reduces the statistical uncertaintyof the latter because the two fields are correlated. The resulting formulation is simple to code and provides considerable computational savings for a wide range of problems of practical interest. It is validated by comparing our results with DSMC solutions for steadyand unsteady, isothermal and non-isothermal problems; in all casesvery good agreement between the two methods is found.

Keywords

DSMCvariance reductionmicroscale gas flow

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References

Alexander, F.J., Garcia, A.L. and Cell, B.J. Alder size dependence of transport coefficients in stochastic particle algorithms. Phys. Fluids 10 (1998) 1540–1542. CrossRef
H.A. Al-Mohssen, An Excursion with the Boltzmann Equation at Low Speeds: Variance-Reduced DSMC. Ph.D. Thesis, Massachusetts Institute of Technology, Dept. of Mechanical Engineering, Cambridge (2010).
H.A. Al-Mohssen and N.G. Hadjiconstantinou, Yet Another Variance Reduction Method for Direct Monte Carlo Simulations of Low-Signal Flows, in 26th International Symposium on Rarefied Gas Dynamics, T. Abe Ed., AIP, Kyoto (2008) 257–262.
Baker, L.L. and Hadjiconstantinou, N.G., Variance reduction for Monte Carlo solutions of the Boltzmann equation. Phys. Fluids 17 (2005) 051703. CrossRef
Baker, L.L. and Hadjiconstantinou, N.G., Variance-reduced particle methods for solving the Boltzmann equation. J. Comput. Theor. Nanosci. 5 (2008) 165–174. CrossRef
Baker, L.L. and Hadjiconstantinou, N.G., Variance-reduced Monte Carlo solutions of the Boltzmann equation for low-speed gas flows: A discontinuous Galerkin formulation. Int. J. Numer. Methods Fluids 58 (2008) 381–402. CrossRef
G.A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon Press (1994).
C. Cercignani, The Boltzmann equation and its applications. Springer-Verlag (1988).
C. Cercignani, Mathematical Methods in Kinetic Theory. Plenum Press (1990).
C. Cercignani, Slow Rarefied Flows: Theory and Application to Micro-Electro-Mechanical Systems. Springer (2006).
Chun, J. and Koch, D.L., A direct simulation Monte Carlo method for rarefied gas flows in the limit of small Mach number. Phys. Fluids 17 (2005) 107107. CrossRef
Doucet, A. and Wang, X., Monte Carlo methods for signal processing: a review in the statistical signal processing context. IEEE Signal Process. Mag. 22 (2005) 152–170. CrossRef
Garcia, A.L. and Wagner, W., Time step truncation error in direct simulation Monte Carlo. Phys. Fluids 12 (2000) 2621–2633. CrossRef
P. Glasserman, Monte Carlo Methods in Financial Engineering. Springer (2004).
Hadjiconstantinou, N.G., Analysis of discretization in the direct simulation Monte Carlo. Phys. Fluids 12 (2000) 2634–2638. CrossRef
Hadjiconstantinou, N.G., The limits of Navier-Stokes theory and kinetic extensions for describing small-scale gaseous hydrodynamics. Phys. Fluids 18 (2006) 111301. CrossRef
Hadjiconstantinou, N.G., Garcia, A.L., Bazant, M.Z. and Statistical, G. He error in particle simulations of hydrodynamic phenomena. J. Comput. Phys. 187 (2003) 274–297. CrossRef
Homolle, T.M.M. and Hadjiconstantinou, N.G., Low-variance deviational simulation Monte Carlo. Phys. Fluids 19 (2007) 041701. CrossRef
Homolle, T.M.M. and Hadjiconstantinou, N.G., A low-variance deviational simulation Monte Carlo for the Boltzmann equation. J. Comput. Phys. 226 (2007) 2341–2358. CrossRef
C.D. Landon, Weighted Particle Variance Reduction of Direct Simulation Monte Carlo for the Bhatnagar-Gross-Krook Collision Operator. M.S. Thesis, Massachusetts Institute of Technology, Dept. of Mechanical Engineering, Cambridge (2010).
Ottinger, H.C., van den Brule, B.H.A.A. and Hulsen, M.A., Brownian configuration fields and variance reduced CONNFFESSIT. J. Non-Newton. Fluid Mech. 70 (1997) 255–261. CrossRef
W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes. Cambridge University Press (2007).
Radtke, G.A. and Hadjiconstantinou, N.G., Variance-reduced particle simulation of the Boltzmann transport equation in the relaxation-time approximation. Phys. Rev. E 79 (2009) 056711. CrossRef
R.Y. Rubinstein, Simulation and the Monte Carlo Method. Wiley (1981).
D.W. Scott, Multivariate Density Estimation. John Wiley & Sons (1992).
Y. Sone, Kinetic Theory and Fluid Dynamics. Birkhauser (2002).
Wagner, W., A convergence proof for Bird's direct simulation Monte Carlo method for the Boltzmann equation. J. Stat. Phys. 66 (1992) 1011–1044. CrossRef
Wagner, W., Deviational Particle Monte Carlo for the Boltzmann Equation. Monte Carlo Methods Appl. 14 (2008) 191–268. CrossRef