We present and discuss a variance-reduced stochastic particle simulation method for solving the relaxation-time model of the Boltzmann transport equation. The variance reduction, achieved by simulating only the deviation from equilibrium, results in a significant computational efficiency advantage compared with traditional stochastic particle methods in the limit of small deviation from equilibrium. More specifically, the proposed method can efficiently simulate arbitrarily small deviations from equilibrium at a computational cost that is independent of the deviation from equilibrium, which is in sharp contrast to traditional particle methods. The proposed method is developed and validated in the context of dilute gases; despite this, it is expected to directly extend to all fields (carriers) for which the relaxation-time approximation is applicable.
1.
Chen
, G.
, 2005, Nanoscale Energy Transport and Conversion
, Oxford University Press
, New York
.2.
Majumdar
, A.
, 1993, “Microscale Heat Conduction in Dielectric Thin Films
,” ASME J. Heat Transfer
0022-1481, 115
, pp. 7
–16
.3.
Chen
, G.
, 2002, “Ballistic-Diffusive Equations for Transient Heat Conduction From Nano to Macroscales
,” ASME J. Heat Transfer
0022-1481, 124
, pp. 320
–328
.4.
Haji-Sheikh
, A.
, Minkowycz
, W. J.
, and Sparrow
, E. M.
, 2002, “Certain Anomalies in the Analysis of Hyperbolic Heat Conduction
,” ASME J. Heat Transfer
0022-1481, 124
, pp. 307
–319
.5.
Manela
, A.
, and Hadjiconstantinou
, N. G.
, 2007, “On the Motion Induced in a Gas Confined in a Small-Scale Gap Due to Instantaneous Heating
,” J. Fluid Mech.
0022-1120, 593
, pp. 453
–462
.6.
Hadjiconstantinou
, N. G.
, 2002, “Constant-Wall-Temperature Nusselt Number in Micro and Nano Channels
,” ASME J. Heat Transfer
0022-1481, 124
, pp. 356
–364
.7.
Hadjiconstantinou
, N. G.
, 2003, “Dissipation in Small Scale Gaseous Flows
,” ASME J. Heat Transfer
0022-1481, 125
, pp. 944
–947
.8.
Hadjiconstantinou
, N. G.
, 2006, “The Limits of Navier-Stokes Theory and Kinetic Extensions for Describing Small-Scale Gaseous Hydrodynamics
,” Phys. Fluids
0031-9171, 18
, p. 111301
.9.
Hadjiconstantinou
, N. G.
, 2000, “Analysis of Discretization in the Direct Simulation Monte Carlo
,” Phys. Fluids
0031-9171, 12
, pp. 2634
–2638
.10.
Frangi
, A.
, Frezzotti
, A.
, and Lorenzani
, S.
, 2007, “On the Application of the BGK Kinetic Model to the Analysis of Gas-Structure Interactions in MEMS
,” Comput. Struct.
0045-7949, 85
, pp. 810
–817
.11.
Baker
, L. L.
, and Hadjiconstantinou
, N. G.
, 2005, “Variance Reduction for Monte Carlo Solutions of the Boltzmann Equation
,” Phys. Fluids
0031-9171, 17
, p. 051703
.12.
Bird
, G. A.
, 1994, Molecular Gas Dynamics and the Direct Simulation of Gas Flows
, Clarendon
, Oxford
.13.
Peterson
, R. B.
, 1994, “Direct Simulation of Phonon-Mediated Heat Transfer in a Debye Crystal
,” ASME J. Heat Transfer
0022-1481, 116
, pp. 815
–822
.14.
Mazumder
, S.
, and Majumdar
, A.
, 2001, “Monte Carlo Study of Phonon Transport in Solid Thin Films Including Dispersion and Polarization
,” ASME J. Heat Transfer
0022-1481, 123
, pp. 749
–759
.15.
Hadjiconstantinou
, N. G.
, Garcia
, A. L.
, Bazant
, M. Z.
, and He
, G.
, 2003, “Statistical Error in Particle Simulations of Hydrodynamic Phenomena
,” J. Comput. Phys.
0021-9991, 187
, pp. 274
–297
.16.
Baker
, L. L.
, and Hadjiconstantinou
, N. G.
, 2008, “Variance-Reduced Particle Methods for Solving the Boltzmann Equation
,” J. Comput. Theor. Nanosci.
1546-1955, 5
, pp. 165
–174
.17.
Homolle
, T. M. M.
, and Hadjiconstantinou
, N. G.
, 2007, “Low-Variance Deviational Simulation Monte Carlo
,” Phys. Fluids
0031-9171, 19
, p. 041701
.18.
Homolle
, T. M. M.
, and Hadjiconstantinou
, N. G.
, 2007, “A Low-Variance Deviational Simulation Monte Carlo for the Boltzmann Equation
,” J. Comput. Phys.
0021-9991, 226
, pp. 2341
–2358
.19.
Cercignani
, C.
, 1990, Mathematical Methods in Kinetic Theory
, Plenum
, New York
.20.
Cercignani
, C.
, 1988, The Boltzmann Equation and Its Applications
, Springer-Verlag
, New York
.21.
Gurevich
, V. L.
, 1986, Transport in Phonon Systems
, North-Holland
, New York
.22.
Lundstrom
, M.
, 2000, Fundamentals of Carrier Transport
, 2nd ed., Cambridge University Press
, Cambridge
.23.
Davidson
, B.
, and Sykes
, J. B.
, 1957, Neutron Transport Theory
, Clarendon
, Oxford
.24.
Modest
, M. F.
, 2003, Radiative Heat Transfer
, 2nd ed., Academic
, New York
.25.
Baker
, L. L.
, and Hadjiconstantinou
, N. G.
, 2008, “Variance Reduced Monte Carlo Solutions of the Boltzmann Equation for Low-Speed Gas Flows: A Discontinuous Galerkin Formulation
,” Int. J. Numer. Methods Fluids
0271-2091, 58
, pp. 381
–402
.26.
Sone
, Y.
, 2002, Kinetic Theory and Fluid Dynamics
, Birkhauser
, Boston
.27.
Bassanini
, P.
, Cercignani
, C.
, and Pagani
, C. D.
, 1967, “Comparison of Kinetic Theory Analyses of Linearized Heat Transfer Between Parallel Plates
,” Int. J. Heat Mass Transfer
0017-9310, 10
, pp. 447
–460
.28.
Bassanini
, P.
, Cercignani
, C.
, and Pagani
, C. D.
, 1968, “Influence of the Accommodation Coefficient on the Heat Transfer in a Rarefied Gas
,” Int. J. Heat Mass Transfer
0017-9310, 11
, pp. 1359
–1369
.29.
Manela
, A.
, and Hadjiconstantinou
, N. G.
, 2008, “Gas Motion Induced by Unsteady Boundary Heating in a Small-Scale Slab
,” Phys. Fluids
0031-9171, 20
, p. 117104
.30.
Radtke
, G. A.
, and Hadjiconstantinou
, N. G.
, 2009, “Variance-Reduced Particle Simulation of the Boltzmann Transport Equation in the Relaxation-Time Approximation
,” Phys. Rev. E
1063-651X, 79
, p. 056711
.31.
Wakefield
, J. C.
, Gelfand
, A. E.
, and Smith
, A. F. M.
, 1991, “Efficient Generation of Random Variables via the Ratio-of-Uniforms Method
,” Stat. Comput.
0960-3174, 1
, pp. 129
–133
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