We present an efficient particle simulation method for the Boltzmann transport equation based on the low-variance deviational simulation Monte Carlo approach to the variable-hard-sphere gas. The proposed method exhibits drastically reduced statistical uncertainty for low-signal problems compared to standard particle methods such as the direct simulation Monte Carlo method. We show that by enforcing mass conservation, accurate simulations can be performed in the transition regime requiring as few as ten particles per cell, enabling efficient simulation of multidimensional problems at arbitrarily small deviation from equilibrium.
REFERENCES
1.
N. G.
Hadjiconstantinou
and O.
Simek
, “Constant-wall-temperature Nusselt number in micro and nano-channels
,” J. Heat Transfer
124
, 356
(2002
).2.
N. G.
Hadjiconstantinou
, “Dissipation in small scale gaseous flows
,” J. Heat Transfer
125
, 944
(2003
).3.
A. A.
Alexeenko
, S. F.
Gimelshein
, E. P.
Muntz
, and A. D.
Ketsdever
, “Kinetic modeling of temperature driven flows in short microchannels
,” Int. J. Therm. Sci.
45
, 1045
(2006
).4.
N. G.
Hadjiconstantinou
, “The limits of Navier-Stokes theory and kinetic extensions for describing small-scale gaseous hydrodynamics
,” Phys. Fluids
18
, 111301
(2006
).5.
Y. L.
Han
, E. P.
Muntz
, A.
Alexeenko
, and Y.
Markus
, “Experimental and computational studies of temperature gradient-driven molecular transport in gas flows through nano/microscale channels
,” Nanoscale Microscale Thermophys. Eng.
11
, 151
(2007
).6.
C.
Cercignani
, Slow Rarefied Flows: Theory and Application to Micro-Electro-Mechanical Systems
(Birkhauser-Verlag
, Basel
, 2006
).7.
G. A.
Bird
, Molecular Gas Dynamics and the Direct Simulation of Gas Flows
(Clarendon
, Oxford
, 1994
).8.
N. G.
Hadjiconstantinou
, A. L.
Garcia
, M. Z.
Bazant
, and G.
He
, “Statistical error in particle simulations of hydrodynamic phenomena
,” J. Comput. Phys.
187
, 274
(2003
).9.
L. L.
Baker
and N. G.
Hadjiconstantinou
, “Variance reduction for Monte Carlo solutions of the Boltzmann equation
,” Phys. Fluids
17
, 051703
(2005
).10.
S.
Asmussen
and P. W.
Glynn
, Stochastic Simulation: Algorithms and Analysis
(Springer
, New York
, 2007
).11.
L. L.
Baker
and N. G.
Hadjiconstantinou
, “Variance reduction in particle methods for solving the Boltzmann equation
,” Proceedings of the 4th International Conference on Nanochannels, Microchannels, and Minichannels
, Limerick, Ireland (ASME
, New York
, 2006
), pp. 377
–383
.12.
T. M. M.
Homolle
and N. G.
Hadjiconstantinou
, “Low-variance deviational simulation Monte Carlo
,” Phys. Fluids
19
, 041701
(2007
).13.
T. M. M.
Homolle
and N. G.
Hadjiconstantinou
, “A low-variance deviational simulation Monte Carlo for the Boltzmann equation
,” J. Comput. Phys.
226
, 2341
(2007
).14.
L. L.
Baker
and N. G.
Hadjiconstantinou
, “Variance-reduced particle methods for solving the Boltzmann equation
,” J. Comput. Theor. Nanosci.
5
, 165
(2008
).15.
J.
Chun
and D. L.
Koch
, “A direct simulation Monte Carlo method for rarefied gas flows in the limit of small Mach number
,” Phys. Fluids
17
, 107107
(2005
).16.
H. A.
Al-Mohssen
and N. G.
Hadjiconstantinou
, “Low-variance direct Monte Carlo using importance weights
,” Math. Modell. Numer. Anal.
44
, 1069
(2010
).17.
H. C.
Öttinger
, Stochastic Processes in Polymeric Fluids
(Springer-Verlag
, New York
, 1995
).18.
N. J.
Wagner
and H. C.
Öttinger
, “Accurate simulation of linear viscoelastic properties by variance reduction through the use of control variates
,” J. Rheol.
41
, 757
(1997
).19.
C.
Cercignani
, Mathematical Methods in Kinetic Theory
, 2nd ed. (Plenum
, New York
, 1990
).20.
C.
Cercignani
, The Boltzmann Equation and Its Applications
(Springer Verlag
, New York
, 1988
).21.
G. A.
Radtke
and N. G.
Hadjiconstantinou
, “Variance-reduced particle simulation of the Boltzmann transport equation in the relaxation-time approximation
,” Phys. Rev. E
79
, 056711
(2009
).22.
N. G.
Hadjiconstantinou
, G. A.
Radtke
, and L. L.
Baker
, “On variance reduced simulations of the Boltzmann transport equation for small-scale heat transfer applications
,” J. Heat Transfer
132
, 112401
(2010
).23.
W.
Wagner
, “Deviational particle Monte Carlo for the Boltzmann equation
,” Monte Carlo Meth. Appl.
14
, 191
(2008
).24.
G. A.
Bird
, “Monte Carlo simulation in an engineering context
,” Prog. Astronaut. Aeronaut.
74
, 239
(1981
).25.
C.
Cercignani
and A.
Daneri
, “Flow of a rarefied gas between two parallel plates
,” J. Appl. Phys.
34
, 3509
(1963
).26.
G. A.
Radtke
, N. G.
Hadjiconstantinou
, and W.
Wagner
, “Low variance particle simulations of the Boltzmann transport equation for the variable hard sphere collision model
,” in Proceedings of the 27th International Symposium on Rarefied Gas Dynamics
, Pacific Grove, CA, edited by D. A.
Levin
, I. J.
Wysong
, and A. L.
Garcia
(AIP
, Melville, NY
, 2011
).27.
J. C.
Wakefield
, A. E.
Gelfand
, and A. F. M.
Smith
, “Efficient generation of random variates via the ratio-of-uniforms method
,” Stat. Comput.
1
, 129
(1991
).28.
S. K.
Loyalka
and J. W.
Cipolla
, Jr., “Thermal creep slip with arbitrary accommodation at the surface
,” Phys. Fluids
14
, 1656
(1971
).29.
S. K.
Loyalka
, N.
Petrellis
, and T. S.
Storvick
, “Some numerical results for the BGK model: Thermal creep and viscous slip problems with arbitrary accommodation at the surface
,” Phys. Fluids
18
, 1094
(1975
).30.
M. R.
Allshouse
and N. G.
Hadjiconstantinou
, “Low-variance deviational Monte Carlo simulations of pressure driven flow in micro- and nanoscale channels
,” in Proceedings of the 26th International Symposium on Rarefied Gas Dynamics
, Kyoto, Japan, edited by T.
Abe
(AIP
, Melville, NY
, 2008
), pp. 1015
–1020
.31.
M.
Knudsen
, “Die Gesetze der Molekularströmung und der inneren Reibungsströmung der Gase durch Röhren
,” Ann. Phys.
333
, 75
(1909
).32.
D. J.
Rader
, M. A.
Gallis
, J. R.
Torczynski
, and W.
Wagner
, “DSMC convergence behavior of the hard-sphere-gas thermal conductivity for Fourier heat flow
,” Phys. Fluids
18
, 077102
(2006
).33.
T.
Ohwada
, “Higher order approximation methods
,” J. Comput. Phys.
139
, 1
(1998
).34.
T.
Doi
, “Numerical analysis of the Poiseuille flow and the thermal transpiration of a rarefied gas through a pipe with a rectangular cross section based on the linearized Boltzmann equation for a hard sphere molecular gas
,” J. Vac. Sci. Technol. A
28
, 603
(2010
).35.
For thermal creep with and , Doi (Ref. 34) reported a value of compared to 0.0473 for the LVDSMC method. Due to the small number of digits in the reported value, it was not possible to determine if 1% agreement was attained for this specific case.
36.
A.
Manela
and N. G.
Hadjiconstantinou
, “Gas-flow animation by unsteady heating in a microchannel
,” Phys. Fluids
22
, 062001
(2010
).37.
F. G.
Cheremisin
, “Solving the Boltzmann equation in the case of passing to the hydrodynamic flow regime
,” Dokl. Phys.
45
, 401
(2000
).38.
R. E.
Caflisch
and L.
Pareschi
, in Transport in Transition Regimes
, edited by N.
Ben Abdallah
, A.
Arnold
, P.
Degond
, I. M.
Gamba
, R. T.
Glassey
, C. D.
Levermore
, and C.
Ringhofer
(Springer-Verlag
, New York
, 2004
), Vol. 135
, pp. 57
–73
.39.
W.
Wagner
, “Stochastic models in kinetic theory
,” Phys. Fluids
23
, 030602
(2011
).40.
C.
Cercignani
and A.
Daneri
, “Elementary solutions of the linearized gas-dynamics Boltzmann equation and their application to the slip-flow problem
,” Ann. Phys.
20
, 219
(1962
).41.
S.
Albertoni
, C.
Cercignani
, and L.
Gotusso
, “Numerical evaluation of the slip coefficient
,” Phys. Fluids
6
, 993
(1963
).© 2011 American Institute of Physics.
2011
American Institute of Physics
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