Variance-reduced particle simulation of the Boltzmann transport equation in the relaxation-time approximation

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 79, Issue 5, 2009, Pages

  Radtke, Gregg A ^a   Hadjiconstantinou, Nicolas G ^a  

^a Massachusetts Institute of Technology Cambridge   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BOLTZMANN TRANSPORT EQUATION; COLLISION ALGORITHMS; EQUILIBRIUM DISTRIBUTIONS; GAS FLOWS; LOCAL EQUILIBRIUM; PARTICLE SIMULATION METHOD; PARTICLE SIMULATIONS; PHYSICAL INTERPRETATION; RADIATIVE TRANSPORT; RELAXATION TO EQUILIBRIUM; STATISTICAL UNCERTAINTY; VARIANCE REDUCTIONS;

BOLTZMANN EQUATION; PERIODIC STRUCTURES; SYSTEMS ENGINEERING;

APPROXIMATION ALGORITHMS;

EID: 67549103850     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.79.056711     Document Type: Article

References (33)

1. G. Chen, Nanoscale Energy Transfer and Conversion (Oxford University Press, New York, 2005).
2. C. Cercignani, The Boltzmann Equation and its Applications (Springer, New York, 1988).
3. C. Cercignani, Slow Rarefied Flows: Theory and Application to Micro-Electro-Mechanical Systems (Birkhauser-Verlag, Basel, 2006).
4. Y. Sone, Molecular Gas Dynamics: Theory, Techniques, and Applications (Birkhauser, Boston, 2007).
5. W. G. Vincenti and C. H. Kruger, Introduction to Physical Gas Dynamics (Wiley, New York, 1965).
6. W. A. Harrison, Solid State Theory (Dover, New York, 1980).
7. M. Lundstrom, Fundamentals of Carrier Transport, 2nd ed. (Cambridge University Press, Cambridge, 2000).
8. G. Chen, ASME J. Heat Transfer 124, 320 (2002). 10.1115/1.1447938
9. A. A. Joshi and A. Majumdar, J. Appl. Phys. 74, 31 (1993). 10.1063/1.354111
10. G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows (Clarendon, Oxford, 1994).
11. N. G. Hadjiconstantinou, A. L. Garcia, M. Z. Bazant, and G. He, J. Comput. Phys. 187, 274 (2003). 10.1016/S0021-9991(03)00099-8
12. L. L. Baker and N. G. Hadjiconstantinou, Phys. Fluids 17, 051703 (2005). 10.1063/1.1899210
13. J. Chun and D. L. Koch, Phys. Fluids 17, 107107 (2005). 10.1063/1.2107807
14. L. L. Baker and N. G. Hadjiconstantinou, in Proceedings of the 4th International Conference Nanochannels, Microchannels and Minichannels (Limerick, Ireland, 2006), pp. 377-383.
15. L. L. Baker and N. G. Hadjiconstantinou, J. Comput. Theor. Nanosci. 5, 165 (2008).
16. T. M. M. Homolle and N. G. Hadjiconstantinou, Phys. Fluids 19, 041701 (2007). 10.1063/1.2717721
17. T. M. M. Homolle and N. G. Hadjiconstantinou, J. Comput. Phys. 226, 2341 (2007). 10.1016/j.jcp.2007.07.006
18. W. Wagner, Monte Carlo Meth. Appl. 14, 191 (2008). 10.1515/MCMA.2008.010
19. N. G. Hadjiconstantinou, G. A. Radtke, and L. L. Baker, e-print arXiv:0905.2218v1 (2009).
20. L. Pareschi, in Proceedings of the ESAIM, edited by, T. Goudon, E. Sonnendrucker and, D. Talay, (EDP Sciences, Les Ulis Cedex, 2005), Vol. 15, pp. 87-120.
21. G. Dimarco and L. Pareschi, Multiscale Model. Simul. 6, 1169 (2008). 10.1137/070680916
22. This extension of the present method to solutions of the nonlinear Boltzmann equation (which is a significantly different approach to Ref.) is the subject of current work.
23. J. C. Wakefield, A. E. Gelfand, and A. F. M. Smith, Stat. Comput. 1, 129 (1991). 10.1007/BF01889987
24. In the linearized regime, all flows with τ=0 for the initial and boundary conditions are isothermal.
25. C. Cercignani and A. Daneri, J. Appl. Phys. 34, 3509 (1963). 10.1063/1.1729249
26. S. K. Loyalka, J. Chem. Phys. 63, 4054 (1975). 10.1063/1.431847
27. S. K. Loyalka and J. W. J. Cipolla, Phys. Fluids 14, 1656 (1971). 10.1063/1.1693661
28. M. R. Allshouse and N. G. Hadjiconstantinou, AIP Conf. Proc. 1084, 1015 (2009).
29. While it is tempting to generalize the technique in Ref. by shifting the velocity by δ u MB,z = 1 2 (κ+ 5 2 β) δ t and generating particles from the comparatively simpler distribution - Vcell c z c 2 β f 0 δ t, this approach results in a less efficient algorithm.
30. N. G. Hadjiconstantinou, Phys. Fluids 18, 111301 (2006). 10.1063/1.2393436
31. D. J. Rader, M. A. Gallis, J. R. Torczynski, and W. Wagner, Phys. Fluids 18, 077102 (2006). 10.1063/1.2213640
32. P. Bassanini, C. Cercignani, and C. D. Pagani, Int. J. Heat Mass Transfer 10, 447 (1967). 10.1016/0017-9310(67)90165-2
33. For this problem, the number of deviational particles in the variance-reduced method increases from zero and reaches a nearly constant value at steady state. In our variance-reduced simulation, there were approximately 105 particles at steady state, while the DSMC maintained 105 particles for the entire simulation time.