Low-noise Monte Carlo simulation of the variable hard sphere gas

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.

[1]  Lorenzo Pareschi,et al.  Towards a Hybrid Monte Carlo Method for Rarefied Gas Dynamics , 2004 .

[2]  M. N. Kogan,et al.  Flow of rarefied gas between two parallel plates , 1975 .

[3]  Lowell L. Baker,et al.  Variance reduction for Monte Carlo solutions of the Boltzmann equation , 2005 .

[4]  Hans Christian Öttinger,et al.  Stochastic Processes in Polymeric Fluids , 1996 .

[5]  A. Manela,et al.  Gas-flow animation by unsteady heating in a microchannel , 2010 .

[6]  Peter W. Glynn,et al.  Stochastic Simulation: Algorithms and Analysis , 2007 .

[7]  Stochastic models in kinetic theory , 2011 .

[8]  Carlo Cercignani,et al.  Slow rarefied flows : theory and application to micro-electro-mechanical systems , 2006 .

[9]  C. Cercignani.  The Boltzmann equation and its applications , 1988 .

[10]  L. Talbot,et al.  PROCEEDINGS OF THE SECOND INTERNATIONAL SYMPOSIUM ON RAREFIED GAS DYNAMICS , 1961 .

[11]  Nicolas G. Hadjiconstantinou,et al.  Low-variance direct Monte Carlo simulations using importance weights , 2010 .

[12]  W. Steckelmacher.  Molecular gas dynamics and the direct simulation of gas flows , 1996 .

[13]  Adrian F. M. Smith,et al.  Efficient generation of random variates via the ratio-of-uniforms method , 1991 .

[14]  Lowell L. Baker,et al.  Variance Reduction in Particle Methods for Solving the Boltzmann Equation , 2006 .

[15]  Nicolas G. Hadjiconstantinou,et al.  The limits of Navier-Stokes theory and kinetic extensions for describing small-scale gaseous hydrodynamics , 2005 .

[16]  F. G. Cheremisin.  Solving the Boltzmann equation in the case of passing to the hydrodynamic flow regime , 2000 .

[17]  Alina A. Alexeenko,et al.  Experimental and Computational Studies of Temperature Gradient–Driven Molecular Transport in Gas Flows through Nano/Microscale Channels , 2007 .

[18]  N. Hadjiconstantinou,et al.  Constant-Wall-Temperature Nusselt Number in Micro and Nano-Channels , 2002 .

[19]  Alejandro L. Garcia,et al.  Statistical error in particle simulations of hydrodynamic phenomena , 2002, cond-mat/0207430.

[20]  T. Ohwada.  Higher Order Approximation Methods for the Boltzmann Equation , 1998 .

[21]  Michail A. Gallis,et al.  Direct simulation Monte Carlo convergence behavior of the hard-sphere-gas thermal conductivity for Fourier heat flow , 2006 .

[22]  Graeme A. Bird,et al.  MONTE-CARLO SIMULATION IN AN ENGINEERING CONTEXT , 1980 .

[23]  Michail A. Gallis,et al.  DSMC convergence behavior of the hard-sphere-gas thermal conductivity for Fourier heat flow. , 2005 .

[24]  Donald L. Koch,et al.  A direct simulation Monte Carlo method for rarefied gas flows in the limit of small Mach number , 2005 .

[25]  Thomas M. M. Homolle,et al.  Low-variance deviational simulation Monte Carlo , 2007 .

[26]  Lowell L. Baker,et al.  On Variance-Reduced Simulations of the Boltzmann Transport Equation for Small-Scale Heat Transfer Applications , 2010 .

[27]  Low Variance Particle Simulations of the Boltzmann Transport Equation for the Variable Hard Sphere Collision Model , 2011 .

[28]  Alina A. Alexeenko,et al.  Kinetic modeling of temperature driven flows in short microchannels , 2006, International Journal of Thermal Sciences.

[29]  Thomas M. M. Homolle,et al.  A low-variance deviational simulation Monte Carlo for the Boltzmann equation , 2007, J. Comput. Phys..

[30]  Carlo Cercignani,et al.  Elementary solutions of the linearized gas-dynamics boltzmann equation and their application to the slip-flow problem , 1962 .

[31]  Nicolas G. Hadjiconstantinou,et al.  Dissipation in Small Scale Gaseous Flows , 2003 .

[32]  M. Knudsen.  Die Gesetze der Molekularstrmung und der inneren Reibungsstrmung der Gase durch Rhren , 1909 .

[33]  Carlo Cercignani,et al.  Mathematical Methods in Kinetic Theory , 1970 .

[34]  Wolfgang Wagner,et al.  Deviational particle Monte Carlo for the Boltzmann equation , 2008, Monte Carlo Methods Appl..

[35]  Lowell L. Baker,et al.  Variance-Reduced Particle Methods for Solving the Boltzmann Equation , 2008 .

[36]  S. K. Loyalka,et al.  Some numerical results for the BGK model: Thermal creep and viscous slip problems with arbitrary accomodation at the surface , 1975 .

[37]  S. Loyalka,et al.  Thermal Creep Slip with Arbitrary Accommodation at the Surface , 1971 .

[38]  C. Cercignani,et al.  Numerical Evaluation of the Slip Coefficient , 1963 .

[40]  Gregg A Radtke,et al.  Variance-reduced particle simulation of the Boltzmann transport equation in the relaxation-time approximation. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[41]  G. Bird.  Molecular Gas Dynamics and the Direct Simulation of Gas Flows , 1994 .