Direct Simulation Monte Carlo Algorithms for ...

Direct Simulation Monte Carlo Algorithms for Simulation of Non-equilibrium Gas Flows S. K. Stefanov Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bontchev str., bl. 4 1113 Sofia, Bulgaria Abstract. The paper deals with some basic problems of the DSMC method concerning the generation of new collision algorithms with improved stochastic properties, allowing simulation of non-equilibrium gas flows with a smaller number of particles in cells compared to the traditional collision schemes. The modified algorithms avoid the events of repeated collisions, which are ones of the basic sources of bias from the theoretical Boltzmann collision frequency. The considered modifications are validated on the one-dimensional unsteady-state problem of strong shock wave formation and the macroscopic characteristics of the shock wave obtained by traditional and new collision algorithms are compared. The efficiency of the modified DSMC method is illustrated by numerical results obtained from the simulation of three-dimensional Rayleigh-Bénard convection of a rarefied gas. Keywords: DSMC method, kinetic theory, rarefied gas flows, micro gas flows. PACS: 47.11.-j, 47.45.-n, 47.70.Nd, 05.10.Ln

INTRODUCTION The emergence of Micro-Electro-Mechanical-Systems (MEMS) as a key enabling technology has led to the development of an increasing number of gas-phase microfluidic systems [1]. Potential applications are numerous and include miniaturized heat exchangers, portable gas chromatography systems, miniaturized gas sensors and novel highthroughput gas flow cytometers [2]. However, one of the most important issues influencing MEMS research is the growing realization that microflows are dominated by non-continuum or rarefaction effects [3]. In this view, it becomes obvious that most of the classical fluid dynamics problems must be reconsidered when applied to flows through micro-channels or more complex micro-systems [3, 4]. Since the characteristic length of MEMS/NEMS is comparable with the mean free path of the gas molecules, the traditional computational fluid dynamics (CFD) methods, based on the Euler or NavierStokes-Fourier equations, fail to predict the flows related with this devices. One of the most successful methods, developed for solving problems in rarefied gas dynamics and microfluidics, is the Direct Simulation Monte Carlo (DSMC) method, originally proposed by the Australian scientist G. Bird in 1963 [5]. Since then, DSMC has been applied to an impressive array of different problems ranging from hypersonic to subsonic flows. The paper deals with some basic problems of the DSMC method concerning the generation of new collision algorithms with improved stochastic properties, allowing simulation of non-equilibrium gas flows with a smaller number of particles in cells compared to the traditional collision schemes. The modified algorithms avoid the events of repeated collisions, which areTO oneBEofINSERTED basic sources of bias fromPAGE the theoretical Boltzmann CREDIT LINE (BELOW) ON THE FIRST OF EACH PAPER CP1301, Applications of Mathematics in Technical and Natural Sciences (AMiTaNS ’10) edited by M. D. Todorov and C. I. Christov © 2010 American Institute of Physics 978-0-7354-0856-2/10/$30.00

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collision frequency. The considered modifications are validated on the one-dimensional unsteady-state problem of strong shock wave formation and the macroscopic characteristics of the shock wave obtained by traditional and new collision algorithms are compared. The efficiency of the modified DSMC method is illustrated by numerical results, obtained from the simulation of three-dimensional Rayleigh-Bénard convection of a rarefied gas.

THE GENERAL DSMC SCHEME The DSMC technique uses a finite set of model particles, denoted by their positions and velocities {xi , ξi }, i = 1, . . . , N, that move and collide in a computational domain to perform a stochastic simulation of the real molecular gas dynamics. The basic concept of the method is built on a discretization in time and space of the real gas dynamics process and splitting in each time step the motion into two successive stages of free molecular motion and binary intermolecular collisions within the grid cells . A detailed mathematical description of the motion of a rarefied gas system can be given by an evolutionary kinetic equation in the following non-closed form with respect to the velocity distribution function f (t, x, ξ ): h i ∂ f (t, x, ξ ) = −D[ f (t, x, ξ )] + Q f (2) (t, x, ξ , x∗ , ξ∗ ) , ∂t

(1)

where f (t, x, ξ ) = f (1) (t, x, ξ ) and f (2) (t, x, ξ , x∗ , ξ∗ ) are one-particle and two-particle distribution functions of the particle velocities ξ and ξ∗ at time t and spacial coordinate x, D denotes a linear differential operator describing the free particle motion and Q is a non-linear integral operator describing the particle binary interactions. For more details concerning equation (1) and its relation to the Boltzmann equation we refer τ ,h τ the numerical to Cercignani’s monograph [6]. We denote with operators SQ and SD algorithms approximating the action of the collision and convective terms in Eq. (1), τ ,h respectively. If we denote with SQ+D the operator evaluating the solution of equation (1) at tk + 1 from the state at tk then the splitting method is expressed with the approximation τ ,h τ τ ,h SQ+D ≈ SD SQ .

(2)

Using the result, obtained by Bobylev and Ohwada [7], one can show that the splitting method approximates the Boltzmann equation with accuracy O(τ + h). The accuracy with respect to time step can be improved by using the Strang splitting symmetric scheme [8]: n h io τ /2 τ /2 τ τ SQ+D ( f0 ) = SQ SD SQ ( f0 ) + O(τ 3 ). (3) τ ,h Further, the considered collision algorithms are presented by operator SQ . A detailed description of the general standard two-stage DSMC algorithm can be found in the Bird’s monograph [5]. The problems related to the algorithm convergence to the Boltzmann equation solution are considered by Wagner [9].

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During a particle simulation the following two stages are performed over each time step (tk ,tk+1 ), k = 1, ..., K : τ ,h Stage 1. Operator SQ (a binary collision procedure applied to each cell of the computational grid). τ (Free particle motion). Each particle x , ξ , i = 1, ..., N is moved Stage 2. Operator SD i i over the time step τ to its new position:

x0 i = xi + ξi τ .

(4)

The boundary conditions are also simulated within Stage 2. The first stage of modeling the binary collisions in cells is more complicated and over the years serious efforts have been made to improve the “Time Counter” collision scheme originally proposed by Graham Bird [10]. Later, as a result of subsequent theoretical investigations, several collision schemes with better characteristics have been proposed: “Null-Collision” [11], “Ballot-Box” [12],“Modified-Nanbu” [13], “Majorant Collision Frequency” [14], and “No Time Counter (NTC)”[5]. The most frequently used scheme has become the Bird’s NTC scheme and further in the text we will refer to it as the “standard scheme.” In the main, all these schemes require a large number of particles per cell (N ∼ 10 − 20) in order to obtain reliable results. The same is true for the recently proposed Monte Carlo schemes for near-continuum [15] and low speed [16, 17] gas flows which take into account the corresponding asymptotic properties of the Boltzmann equation. In particular, the second approach [16, 17], suggests that a considerable variance reduction can be achieved in the numerical solution by simulating only the deviation from the local equilibrium state. However, accurate simulation seems to need a large number of particles per cell to avoid the stochastic errors arising in the procedure of particle generation and cancelation that is an intrinsic part of the proposed low-variance algorithm. The source of these stochastic errors is closely related to the generation within each time step of random values from a small sample-size set of particles representing the local velocity distribution function within a cell. This leads to a systematic bias estimation of important parameters in the simulation. Similar considerations highlight the reasons for the limitations of the standard DSMC method when run with small numbers of model particles per cell. Consider in more details the standard NTC scheme: Stage 1. (No Time Counter binary collision procedure) Three steps are included in the “No Time Counter” collision procedure performed in each cell l, l = 1, ..., M: • computing the number of particle pairs Nc to be checked for a collision from the formula ¯ ¯ D E ± ¯ ¯ (l) Nc = 1 2∆N (l) N (l) (σ g)max τ / ¯dx(l) ¯ , (5) (l)

where (σ g)max is a superior parameter, which is updated continuously during the simulation, N (l) is the number of particles in cell l, ∆ is scaling factor;

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• (acceptance-rejection method) each pair (i, j), 1 ≤ i < j ≤ N (l) , chosen at ran-

dom from the particle subset N (l) , is checked for a collision with probability pi j =

σi j gi j , (σ g)max

(6)

where σ ¯ ¯ i j is the effective collision cross-section of pair (i, j) and gi j = ¯ξi − ξ j ¯. • if the collision is accepted then the particle velocities are changed to their post-collision values ¯ ¤ £¡ ¢ ¯ ξ 0 i = 12 ξi + ξ j + ¯ξi − ξ j ¯ ω (7) ¯ ¤ £¡ ¢ ¯ ξ 0 j = 21 ξi + ξ j − ¯ξi − ξ j ¯ ω . In the system of equations (7) ³ ω is a ´unity vector pointing out from new direction of relative velocity ξ 0 i − ξ 0 j . The end of vector ω is uniformly distributed on a sphere of unit radius. During stage 1, the particle positions are not changed. The required number of computations of the NTC algorithm in a cell l is O(N (l) ). A rigorous mathematical proof of the algorithm convergence for large number of particles in cells to the Boltzmann equation was given by Wagner [9] in 1992. A recent discussion on the theoretical background of the collision algorithms can be found in Sone’s monograph [18]. The standard NTC collision algorithm allows multiple repeated collisions of one and the same particle pair. Let us estimate the effect of n repeated collisions of the particle pair (i, j) belonging to a given cell. Let particles have initial velocities (ξi (0) , ξ j (0) ). Applying equation (7) n times we obtain the post-collision velocities after n repeated collisions: ·µ ¶ ¯ ¸ ¯ (n−1) ¯ ¯ ξ 0 i (n) = 12 ξi (n−1) + ξ j + ¯ξi (n−1) − ξ j (n−1) ¯ ω (n)

ξ 0 j (n)

¯ ¶ ¯ ¸ ·µ ¯ (n−1) (n−1) (n−1) ¯ (n−1) ¯ ¯ ω (n) . = +ξj − ¯ξi −ξj ξi ¯

(8)

1 2

(n−1)

(n−1) +¯ ξ j ) = (ξi (0) + ξ j ¯Taking into account ¯ that ¯ (ξi ¯ (n−1) ¯ ¯ (n−1) ¯ ¯ξi ¯ = ¯ξi (0) − ξ j (0) ¯ we obtain −ξj ¯ ¯ ¯ ¯

ξ 0 i (n) = ξ 0 j (n)

1 2

(0)

) and

¶ ¯ ¸ ·µ ¯ (0) ¯ ¯ + ¯ξi (0) − ξ j (0) ¯ ω (n) ξi (0) + ξ j

¯ ·µ ¶ ¯ ¸ ¯ (0) (0) (0) ¯ (0) ¯ ξi + ξ j = − ¯ξi − ξ j ¯¯ ω (n) .

(9)

1 2

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Thus the post-collision velocities obtained as a result of using n successively repeated elastic collisions are statistically distributed in the same way as if they are obtained by the realization of only one collision of the chosen pair. In the case of non-elastic repeated collisions, a reduction of the effect of collisions could also be made evident. As a consequence, the major effect on a standard DSMC simulation with small number of particles in cells is a reduction in the local collision frequency, which only converges to the Boltzmann collision frequency for large enough number of particles per cell.

COLLISION ALGORITHMS WITHOUT REPEATED COLLISIONS In order to improve the stochastic properties of the binary collision process in cells of the computational grid containing small number of particles per cell one have to modify or replace the standard binary collision algorithm by other one that avoid the repeated collisions. Such a collision algorithm known under the name “Bernoulli trials” (BT) scheme, was proposed by Yanitskiy [12]. Here in addition to the BT scheme a simplified algorithm based on BT scheme is presented. For brevity hereafter, we will call the BT scheme - algorithm A, and the simplification 1 - algorithm B. Algorithm A: For each cell l (l = 1, ..., M) (with volume V (l) = |dx(l) | = hα and reference point x(l) ) of the grid mapped exactly on the considered domain the following steps are carried out: • all particles are given local indices with regard to their cell disposition; • in each cell the following Bernoulli trials algorithm [12] (see also [18]) is realized: - each pair of particles in cell l with velocities (ξi , ξ j ), i < j = 1, . . . , N (l) is checked for collision with probability W =4

σi j g i j τ . V (l)

(10)

It is worth noting that, for an appropriate non-dimensional form of the variables, the scaling factor 4 = 1. In dimensional form 4 is equal to the number of real gas molecules represented by each simulated particle; - if the collision is accepted the velocities (ξi , ξ j ) are changed to their postcollision velocities (ξ 0 i , ξ 0 j ) according to Eq. (7), otherwise the velocities are left unchanged. The Bernoulli trials procedure avoids the appearance of repeated collisions within step τ . The following stochastic condition is important to 1

the derivation of the algorithm will be published soon

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hold Prob{W ≥ 1} → 0.

(11)

As one can see the required number of computations of algorithm A in a cell l is O((N (l) )2 ). The efficiency of algorithm A is less than the NTC efficiency and becomes comparable with it when N (l) → 1. In order to improve the efficiency of the algorithm A the following simplification is proposed by the author 2 . Algorithm B: • a sequence of pairs i = 1, . . . , (N (l) − 1) is chosen from N (l) particles in cell l as follows: - the first particle i is the particle with index i in the particle list created for cell l; h i - the second particle j ∈ i + 1, N (l) is chosen with probability 1/k from

k = (N (l) − i) particles taking place in the list after particle i. • particle pair (i, j) is checked for collision with probability ˆ i j = k σi j gi j τ , W V (l)

(12)

ˆ i j must satisfy the condition where probability W ˆ i j ≥ 1} → 0; Prob{W

(13)

• if collision is accepted then velocities (ξi , ξ j ) are changed to the post-collision

values (ξ 0 i , ξ 0 j ), otherwise they remain unchanged.

The required number of computations of algorithm B in a cell l is proportional to N (l) − 1, which makes the efficiency of algorithm B comparable to the standard NTC collision procedure. At the same time repeated collisions are avoided. This fact allows the use of a rather smaller number of particles per cell than the required by the standard algorithm without lost of computational accuracy (see the next section). It is worth noting that the better efficiency of algorithm B with respect to the number of particles in cell requires a stronger limitation of the time step, i.e., a smaller time step (see condition (13)). At the same time both algorithms, A and B, avoid the realization of repeated collisions within a time step τ . In the framework of the general Strang splitting scheme algorithms A or B are applied twice with step τ /2 realizing the action of operator (τ /2,h) SQ .

2

to be published soon.

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COMPUTATIONAL APPLICATIONS OF THE MODIFIED DSMC METHOD Strong shock wave formation In order to validate numerically the proposed modification in the DSMC method the one-dimensional unsteady-state problem of the formation of a strong shock wave is considered. The wave is creating by a piston that impulsively starts to move with a constant velocity of 2285.5 m/s in the x-direction. The gas in front of the piston uses the variable hard sphere model of argon at a temperature of T0 = 273o K and a number density of n0 = 1020 m−3 . The piston can travel within an interval of 1 m, i.e., approximately 100 mfps in the undisturbed gas. This is exactly the same case as described in Bird’s book [5](§13.2). Bird’s original program DSMC1U has been used to obtain the results by the standard DSMC method. We have used the same source code to obtain the modified scheme, substituting the “No Time Counter” collision procedure by the Bernoulli trials simplified one (algorithm B). The results are normalized√ by the molecular mean free path for length, the most probable molecular speed Vth = 2RT0 for velocity and T0 for temperature. Figure 1 presents the profiles of x-velocity (left) and temperature (right) along the x-direction, computed by both the standard NTC algorithm and simplified BT algorithm B on a grid with 400 uniform cells with different number of particles per cell. The reference profiles (shown as thick solid lines) are computed by the standard “No Time Counter” NTC collision scheme with hNi = 100.0 particles per cell. The profiles computed by the NTC scheme with hNi = 10.0 and hNi = 2.0 (shown in stars) increasingly deviate from the reference profile in the lower part of the shock wave front, where the average number of particles per cell is smaller. The characteristic swelling in the lower part of the shock front is an effect of the insufficient number of effective collisions owing to an increase of repeated collisions of particle pairs with high relative velocity. Unlike these results, all profiles, obtained by algorithm B (shown in circles), are in a good agreement with the reference profiles.

Three-dimensional Rayleigh-Bénard flow In this subsection the computational simulation of the formation and development of the three-dimensional Rayleigh-Bénard (RB)convection flows is demonstrated by using the modified DSMC method, which includes the Strang splitting procedure and the Bernoulli trials collision algorithm A ( for details see [19]. The calculations have been performed within a range of Knudsen numbers Kn (= `0 /Lz ) = 0.002 − 0.03, Froude numbers Fr (= Vth2 /gLz ) = 0.8 − 1500, and a fixed temperature ratio r (= Tc /Th ) = 0.1, where `0 is mean free path, Lzp is the distance between the hot and cold walls with temperatures Tc and Th , Vth = (2RTh ) is the most probable molecular velocity in an equilibrium gas with temperature Th , g is a constant acceleration acting on the gas molecules. We have observed a variety of possible final states (attractors), namely: pure heat conduction, stable vortex roll convection, periodic, wavy, and chaotic regimes. The DSMC computations of the problem are performed in a three-dimensional box with

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7

35 Standard, N=2 Standard, N=10 Standard, N=100 Algorithm B, N=1 Algorithm B, N=2 Algorithm B, N=10

6 5

Standard, N=2 Standard, N=10 Standard, N=100 Algorithm B, N=1 Algorithm B, N=2 Algorithm B, N=10

30 25 T/T0

U/Vth

4 3

20 15

2 10

1

5

0 −1 10

20

30

40 50 x/mfp

60

70

0 10

80

20

30

40 50 x/mfp

60

70

80

FIGURE 1. Formation of a strong shock wave in front of a piston moving with a constant velocity of 2285.5 m/s in the x-direction; (left) comparison of velocity profiles at times t = 1.0 × 10−4 s and t = 2.0 × 10−4 s; (right) comparison of temperature profiles at the same times

imposed periodicity at the lateral sides. The gas flow is simulated for a set of different Knudsen and Froude numbers at a fixed temperature ratio r = 0.1. The dependence of the pattern formation on the aspect ratio of three-dimensional computational domain A = Lx /Lz : Ly /Lz : Lz /Lz is also studied, where Lx , Ly and Lz are the box dimensions in the coordinate directions. As we have used everywhere a computational domain with Lx = Ly hereafter we shall use A = Lx /Lz instead of the longer definition above. The size of the rectangular uniform computational grid depends on the Knudsen number and the aspect ratio A. For the larger value Kn = 0.02 and the aspect ratio A = 2.0, calculations have been performed twice on grids 100 × 100 × 50 and 200 × 200 × 100, respectively, in order to validate the accuracy of the method. For Kn = 0.02 and A = 6.0 the grid is 300 × 300 × 50. At Kn = 0.002 and A = 2.0 the basic grid contains of 200 × 200 × 100 basic cells, which are subdivided into subcells in order to meet the computational requirements of the method. The instantaneous fields are sampled by time averaging over 500 time steps 4t (4t is less than the inverse value of mean collision frequency). Finally, the obtained three-dimensional instantaneous macroscopic fields are processed by a simple average filter in order to remove the high-frequency statistical fluctuations from the simulation results. Depending on the Knudsen number and the aspect ratio A, the total number of model particles varies from 2 × 106 for Kn = 0.02 and A = 1.5 to 64.0 × 106 for Kn = 0.002 and A = 2.0. Figure 2 presents the vertical velocity fields of the rising convective complex pattern at z = 0.8 for the case with A = 5. Since the shape and orientation of the convective cell is difficult to analyze from figures showing (x, y)flow fields at a section with small aspect ratio, in Figure 2 we present the contours of the vertical z-velocity component in a 2A × 2A-section, which represents the same patterns in the x- and y-directions. This way of presentation of the periodic solution illustrates the shape and orientation of the convective cells more clearly The darkest spots (blue in color) show the areas where the gas is moving downward; the areas of rising gas (red in color) are separated from the flux going downward by lighter stripes (green and yellow in color)

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FIGURE 2. Contours of the vertical z-component of velocity for the odd aspect ratio A = 5 . The symmetry lines of the periodic configuration are given by straight solid lines; regular complex convective patterns in form of squares (in the centers) surrounded by eight pentagons are clearly observed

From the contour image for A = 5, which presents the periodic configuration in the doubled area, a complex pattern design consisting of a central square and surrounding eight pentagonal cells is formed. There are four sets of lines in Figure 2: horizontal, vertical, and two inclined at angle ±π /4. If we extend the picture horizontally by adding infinitely many periods it would turn out that each of these lines would be a symmetry line. Thus these four sets of lines define a group of four symmetries taking place in the considered case of a stable convective flow with an A = 5 imposed periodicity. We have found the three-dimensional RB flow at a Knudsen number Kn = 0.002 and Fr = 1.0 to be chaotic. The z-velocity level contours obtained from the DSMC calculation are plotted in Figure 3 for a typical instantaneous configuration of convective cells. We remind that the darkest stripes (blue in color) display the area where the gas flow goes downward. Again, for a better presentation the considered section is doubled in both the x- and y-directions as done in the previous figures displaying contour plots of the zvelocity. Unlike the cases at Kn = 0.02, where the convective cells are described as zones of non-mixing convection motions, the chaotic flow obtained at Kn = 0.002 is separated into unstable moving cells delineated by the stripes of gas flows going downward. Inside the cell the convective stream is upward.

CONCLUDING REMARKS The comparison analysis on the shock wave formation problem demonstrates that, under certain conditions, the modified DSMC scheme can be used successfully for simulation of gas flows with very small numbers of particles per cell compared with the standard DSMC method. The simplified BT algorithm B has an efficiency of the same order such

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0.06 3.5 0.04 3

0.02 0

2.5

y

−0.02 2 −0.04 1.5

−0.06 −0.08

1 −0.1 0.5

−0.12 −0.14 0.5

1

1.5

2 x

2.5

3

3.5

FIGURE 3. Typical instantaneous contour field of the vertical z-component of velocity (view from above at (x, y)-plane z = 0.8) realized within the established irregular regime for Kn = 0.002, Fr = 1.0 and aspect ratios A = 2

as the standard NTC collision algorithm. As shown in the last section an important advantage of the new algorithm is the reduced amount of memory required during a simulation. This is particularly useful in DSMC calculations of complex vortex and unstable gas flows on two- or threedimensional fine meshes when the use of the standard scheme requires a huge total number of particles. The modified DSMC method can be used successfully for simulation of micro gas flows in a systems with a complex geometry requiring very fine grids.

ACKNOWLEDGMENTS The research leading to these results has received funding from the European Community’s Seventh Framework Programme FP7/2007-2013 under grant agreement ITN GASMEMS No 215504. The author acknowledges the financial support provided by the NSF of Bulgaria under Grant No DID 02/20-2009.

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