Scientific background

In 1963, Graeme Bird published a short article in Physics of Fluids [1] in which he described a stochastic algorithm for simulating gas dynamics at the molecular scale. This numerical method, which today is known as Direct Simulation Monte Carlo [2], had a tremendous impact on the development of theoretical and applied kinetic theory where it became the dominant computational technique.

From the time of its introduction, during the early years of space exploration, DSMC has been widely used by the aerospace community, from applied mathematicians to design engineers [3]. Increasingly one finds DSMC used in mechanical engineering to simulate nano-scale flows, where the mean free path is comparable to the characteristic length scale of the flow [4,5,6].

Since DSMC is a stochastic algorithm for solving the full, nonlinear Boltzmann equation [7] the method and its variants also has many applications in physics and chemistry, such as plasmas [8,9], gas-phase reactions [10,11], and planetary atmospheres [12]. One of the more recent applications is in the simulation of granular gases [13,14,15]. DSMC is also used extensively in the applied mathematics community, particularly in the field of stochastic processes [16,17] and for the simulation of a variety of kinetic equations (e.g., traffic flow [18]).

Many improvements and extensions to DSMC have been introduced, especially in the past few years. For example, there are several variants that generalize the method to the simulation of dense gases [19,20,21] and liquids [22]. A highly efficient DSMC variant proposed recently by Malevanets and Kapral [23] is a promising alternative for model the dynamics of dilute suspensions. By combining a particle scheme, such as DSMC, with a conventional hydrodynamic computation (e.g., numerically solving the Navier-Stokes equations) allows one to span from microscopic to macroscopic scales; the development of such hybrids is an active branch of study in computational fluid mechanics [24,25,26].


[1] G.A. Bird, Approach to Translational Equilibrium in a Rigid Sphere Gas, Phys. Fluids, 6, 1518 (1963).
[2] G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, (Clarendon, Oxford, 1994).
[3] M.S. Ivanov and S.F. Gimelshein, Computational Hypersonic Rarefied Flows, Annu. Rev. Fluid Mech. 30 469 (1998).
[4] F. Alexander, A. Garcia and B. Alder, Direct Simulation Monte Carlo for Thin Film Bearings, Phys. Fluids, 6, 3854 (1994).
[5] N.G. Hadjiconstantinou and O. Simek, Constant-Wall-Temperature Nusselt Number in Micro and Nano Channels, to appear in J. Heat Transfer (2002).
[6] Q. Sun and I.D. Boyd, A Direct Simulation Method for Subsonic, Micro-Scale Gas Flows, J. Comp. Phys., 179 400 (2002).
[7] Cercignani, C., Rarefied Gas Dynamics, (Cambridge University Press, Cambridge, UK, 2000)
[8] A. Vasenkov and B.D. Shizgal, Numerical study of a direct current plasma-sheath based on kinetic theory, Phys. Plasmas 9 691 (2002).
[9] K. Nanbu, Particle-Model-Based Simulation of Plasmas for Etching, J. Plasma Fusion Res. 77 1137 (2001).
[10] F. Baras and M. Malek Mansour, Particle Simulations of Chemical Systems, Adv. Chem Phys., 100, 393-474 (1997).
[11] A. Lemarchand and B. Nowakowski, Different description levels of chemical wave front and propagation speed selection J. Chem. Phys. 111 6190 (1999).
[12] J. Zhang, D. B. Goldstein, P.L. Varghese, N.E. Gimelshein, S.F. Gimelshein, D.A. Levin, and L. Trafton, DSMC Modeling of Gasdynamics, Radiation and Fine Particulates in ionian Volcanic Jets, Proceedings of 23rd International Symposium on Rarefied Gas Dynamics (2002).
[13] H.J. Herrmann and S. Luding, Modeling granular media on the computer, Contin. Mech. Thermodyn., 104, 189--231, 1998.
[14] D. Risso and P. Cordero, Dynamics of rarefied granular gases, Phys. Rev. E, {\bf 65} 021304 (2002).
[15] A. Frezzotti, DSMC Simulation of the vertical structure of planetary rings, Astronomy & Astrophysics 380, 761--775 (2001).
[16] W. Wagner. A convergence proof for Bird's direct simulation Monte Carlo method for the Boltzmann equation. J. Statist. Phys., 66 1011 (1992).
[17] S. Rjasanow and W. Wagner, On time counting procedure in the DSMC method for rarified gases, Math. Comput. Simulation, 48 153 (1998).
[18] A. Klar, B. Aw, T. Materne, and M. Rascle, Derivation of continuum flow traffic models from microscopic Follow the leader models, to appear in SIAM J. Appl. Math.
[19] F. Alexander, A. Garcia and B. Alder, A Consistent Boltzmann Algorithm, Phys. Rev. Lett. 74, 5212 (1995).
[20] A. Frezzotti. A particle scheme for the numerical solution of the Enskog equation Phys. Fluids, 9, 1329 (1997).
[21] J.M. Montanero and A. Santos, Simulation of the Enskog equation a-la-Bird, Phys. Fluids, 9, 2057--2060, (1997).
[22] N. Hadjiconstantinou, A. Garcia, and B. Alder, The Surface Properties of a van der Waals Fluid, Physica A 281, 337-347 (2000).
[23] A. Malevanets and R. Kapral, Mesoscopic Model for Solvent Dynamics J. Chem. Phys., 110, 8605 (1999).
[24] J. Eggers and A. Beylich, New Algorithms for Application in the Direct Simulation Monte Carlo Method, Prog. Astro. Aero. 159, 166 (1994).
[25] S. Tiwari and A. Klar, Coupling of the Boltzmann and Euler equations with adaptive domain decomposition procedure, J. Comp. Phys. 144, 710 (1998).
[26] A. Garcia, J. Bell, Wm. Y. Crutchfield and B. Alder, Adaptive Mesh and Algorithm Refinement using Direct Simulation Monte Carlo, J. Comp. Phys. 154, 134 (1999).

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