Direct Simulation and the Boltzmann Equation

Abstract

The direct simulation Monte Carlo method for the numerical solution of problems in rarefied gas dynamics is described and discussed. It is shown that the procedures adopted in this method can be directly related to the Boltzmann equation and that the two are entirely consistent. It is concluded that the results obtained from the method constitute a solution of the Boltzmann equation.

A discontinuous Galerkin finite-element method for a 1D prototype of the Boltzmann equation

“…according to: 6) with m the number of samples in the Monte-Carlo process. The functionf m (t) can be conceived of as the Riesz representation (in L 2 (Σ)) of the functional w → i,j w(x i,j (t), ξ i,j (t)).…”

“…Notwithstanding successful applications to simplified problems (see, e.g., [1,17]), this computational complexity is prohibitive for practical applications. Current simulation methods for 2 and 3-dimensional problems are generally based on Bird's Direct-Simulation-Monte-Carlo method [6,7] or its Nanbu extension [2,33]. The computational complexity of DSMC methods depends sensitively on the Knudsen number, however, and the computational cost of the methods becomes prohibitive in the continuum limit.…”

Atomistic phenomena in dense fluid shock waves

“…100, 000 nitrogen molecules are randomly distributed within the fluid portion of the domain and given random initial velocities and rotation rates drawn from a Maxwell-Boltzmann distribution. The rigid nitrogen molecule is modeled by a two-center Lennard-Jones (6,12) potential with the parameters as given in Ref. [22], but without the five point-charges.…”

“…Bird, who first introduced DSMC [4], was also the first to apply this technique to the shock wave problem [5]. It was not clear initially whether the procedure was actually solving the Boltzmann equation until Bird himself [6] proved this. DSMC was applied numerous times to the shock structure problem in the following years and today is the work horse in rarefied gas dynamics.…”

“…The data analysis showing some unanticipated effects (in particular, Secs. [4][5][6] were primarily meant to verify the proper behavior of the code and convergence of the 2 simulations. The objective of this paper is then less to document in detail each one of the findings (yet they are described with some background for their understanding), but to show that even classical problems might not be as fully explored as they appear.…”