Timestep Dependence Of Transport Coefficients In The
Direct Simulation Monte Carlo
Nicolas G. Hadjiconstantinou
Department of Mechanical Engineering
Massachusetts Institute of Technology
77 Massachusetts Avenue, Cambridge, MA 02139
Abstract. Using Green-Kubo theory we calculate the timestep dependence of transport coefficients (viscosity, thermal conductivity, coefficient of diffusion) in the direct simulation Monte Carlo (DSMC). Because
DSMC is discrete in time, we formulate a continuous-time model of DSMC that allows application of the
Green-Kubo formulation (N.G. Hadjiconstantinou, "Analysis of discretization in the direct simulation Monte
Carlo", to appear in Physics of Fluids). The transport coefficients in equilibrium are found to be quadratic
functions of the timestep, resulting in errors (with respect to the dilute gas Enskog result) of the order of
5% for timesteps of the order of one mean free time. The results are in very good agreement with numerical
simulations performed by Garcia and Wagner (A.L. Garcia and W. Wagner, "Time step truncation error in
direct simulation Monte Carlo", to appear in Physics of Fluids).
I
INTRODUCTION
In a recent paper, Alexander et al. [1] used the Green-Kubo theory as applied to hard spheres by Wainright
[2] to calculate the dependence of transport coefficients on the cell size in the Direct Simulation Monte Carlo
(DSMC) [3,4]. This work has shown that the error (defined as the deviation from the dilute gas Enskog theory)
is a quadratic function of the cell size, and is of the order of 10% for cell sizes of the order of one mean free
path. Unfortunately this work was limited to the limit of zero timestep because the Green-Kubo theory cannot
be readily applied to DSMC in the limit of finite non-zero timestep because DSMC is discrete in time.
We have developed [5] a continuous-time analogue of DSMC that allows the use of Green-Kubo theory for
the calculation of the effect of finite timestep on the transport coefficients. In this analogue, or model, the
collisions that occur instantaneously during the "collide" part of the DSMC algorithm are distributed uniformly
throughout the timestep. It is shown that this redistribution of collision times generates molecular trajectories
different from the trajectories generated by the DSMC algorithm. However, the change in trajectories and its
effect can be formulated analytically in the limit of a small cell size. Subsequently, the change in trajectories
is compensated for by supplying spatial shifts during the collision process, such that the particle dynamics are
equivalent: no changes to the collision pairs are made, while particles under the model dynamics have the same
final positions as molecules following the DSMC algorithm starting at the same initial position.
Application of the Green-Kubo theory implies the consideration of problems that are steady. However, we
expect these results to hold for transient problems on hydrodynamic timescales that are typically much longer
than the molecular relaxation timescale. We quantify the deviation between the DSMC solution and the correct
non-equilibrium steady state by obtaining measures of the deviation of the transport coefficients from their
theoretical values. We will refer to the deviation of the transport coefficients from the exact Enskog values for
dilute gases (which DSMC reproduces in the limit of infinite number of particles and vanishing timestep and
cell size [61) as the truncation error.
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis
© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00
381
(s?,v?)
FIGURE 1. Schematic of a representative collision between particles in DSMC for cell size L —>• 0. The particles
collide at T = At/2.
II
EQUIVALENT CONTINUOUS-TIME MODEL
In what follows we will limit our illustration to two dimensions; generalization to three dimensions follows
directly. Consider two particles, i and j, that collide at timestep n (Fig. 1); their initial locations at the start
of the timestep are (x®,y®) and (o^yP). Let r be the time variable over which DSMC coarse grains, that
is T = mod(t. At), where t = nAt -f r is time. The above particles travel with velocities Vi and iTj, and at
r = At/2 (when collisions take place) will be in the same cell so that they are chosen as collision partners.
After colliding, the particles acquire new velocities v\ and v^ and travel ballistically for the remainder of the
timestep r = At/2 to their final positions (xt,yt) and (x|,y|). Note that in our notation a timestep starts
l/2At before the instantaneous collision process and ends l/2At after the instantaneous collision process. This
being purely a matter of convention has no effect on the DSMC dynamics: particles travel ballistically between
collisions for a time At (we exclude the presence of external fields that may introduce particle accelerations)
and then instantaneous collisions accounting for the whole time interval At take place. Our convention merely
introduces a shift in the global time counter with collisions taking place at t = At/2, 3At/2, 5At/2,... with
ballistic motion in between, instead of t = 0, At, 2At,... with ballistic motion in between. We now proceed
to examine the case in which At is finite and thus the DSMC algorithm becomes discrete in time. For the
Green-Kubo formalism to be applied we propose a continuous-time model that is dynamically equivalent to
DSMC.
Consider a model where the particles that according to DSMC would have collided at r = At/2 (that is, at
r = At/2 were in the same cell) instead collide (each colliding pair independently) at tc that lies uniformly
between r = 0 and T = At. The distribution function for the particle location (at collision r = tc) relative to
the DSMC collision point depends on the difference between the collision time tc and the DSMC collision time
T = At/2: we can write the single particle distribution (in one dimension) in the case that the collision cell is
centered on x = 0 as
rL/2
f(x;tc) = I
p(x] q(x x ] t c ) dx,
(1)
J-L/2
where p(x) is the probability (uniform in this approximation) for a particle to be in the cell at x, and q(x\x] tc)
is the probability for a particle that is at x at r = At/2 to be found at x at r = tc. Due to the Maxwellian
velocity distribution q(x x\ tc) has a particularly simple form that leads to
382
FIGURE 2. Schematic of a representative collision between particles in the continuous-time model. The discontinuous
jump in the trajectories represents the shift occuring along with the collision.
f(x;tc) =
(2)
where k is Botzmann's constant, and T is the temperature. The above expression reduces to a Maxwellian
distribution (centered on x = 0) if the cell size L -> 0. We can see that a finite cell size modifies the particle
distribution function and thus, strictly, couples to the finite timestep truncation error. In the interest of
simplicity (the effect of finite cell size can be fully quantified if the algebraic complexity is undertaken) and
because the error due to a finite cell size in the limit At —>• 0 is known [1], we will take L —>• 0. In this limit
the relative distance in the y direction between a collision pair at collision can be written as
= (tc- - )
(3)
where g = vi — Vj is the pre collision relative velocity of the particles.
Figure 2 indicates that for this model to be dynamically identical to the DSMC implementation, colliding
particles need to be shifted after their collisions by an amount that corrects for their ballistic motion with
post-collision velocities instead of the pre-collision velocities for the appropriate amount of time (tc — At/2).
that are the same as for DSMC
The shift ensures that the molecules have final positions (#jf ,2//) and
(Fig. 1). The amount each molecule needs to be shifted is given by
A
-»
___ /,
___
\(
~*
___ ~M
f/l^
This shift introduces mass, momentum, and energy fluxes that contribute to "enhanced" transport coefficients
as shown below.
Ill
CALCULATION OF TRANSPORT COEFFICIENTS
We now present the calculation of the diffusion coefficient of the DSMC-equivalent continuous-time model
as a function of the timestep. The derivation of the coefficient of viscosity is given in [5].
383
Application of the Green-Kubo theory implies that steady problems are considered, at least at the autocorrelation decay timescale. This is not a very restrictive assumption since typical hydrodynamic evolution
takes place at timescales much longer than the latter timescale. It is thus reasonable to assume that the results
derived below will also hold for problems evolving at times long compared to the molecular relaxation timescale.
The diffusion coefficient can be calculated using the Einstein formula
where a long time limit is assumed. In addition to the ideal gas (Enskog) contribution to (xt(i) — #$(())), the
continuous- time model reveals that the shift (eq. (4)) for both colliding particles required for every collision,
leads to an extra contribution proportional to Y^c^xc = Sc(^ — ^)Ai£c, where the sum is over all collisions
in time t. The cross term between the Enskog and shift contribution vanishes because DSMC is "centered" in
time
(6)
where the (}c denotes average over collisions. The extra contribution is thus
*- 5
where M is the number of collisions in time t. For uniformly distributed collisions in time
Additionally,
which leads to
D' = --—At2
9n m
v(10)
;
where n is the number density, and F = 2a2n2^/7rkT/m is the collision rate per unit volume. The resulting
expression for the diffusion coefficient D is
8r^V™ l i T ^ ^ ^ "
(U)
where c0 = -\/2kT/m is the most probable speed.
The calculation of other transport coefficients follows along the same lines [5]. The expression for the viscosity
including the timestep contribution is [5]
16(J2 V
7T
V
iSOTT
A2
A similar calculation for the thermal conductivity K yields
75k
[kT (
+
64 (c0A£)2
6757T A2
384
(12)
Collisionless Limit
10'
10"
10"
10"
10U
10'
FIGURE 3. Error in coefficient of diffusion as a function of timestep At (from [7]). Circles denote the normalized
error in particle flux (E°) in the simulations of Garcia and Wagner [7], and the solid line is the prediction of (11).
Collisionless Limit
10'
10"
10U
10
At
FIGURE 4. Error in coefficient of viscosity as a function of timestep At (from [7]). Circles denote the normalized
error in momentum flux (E%) in the simulations of Garcia and Wagner [7], and the solid line is the prediction of (12).
385
Collisionless Limit
10'
10°
10"
10U
10'
FIGURE 5. Error in thermal conductivity as a function of timestep At (from [7]). Circles denote the normalized error
in energy flux (E%) in the simulations of Garcia and Wagner [7], and the solid line is the prediction of (13).
IV
COMPARISON WITH NUMERICAL SIMULATIONS
Garcia and Wagner [7] performed steady state simulations in a variety of configurations to measure the
truncation error as a function of the timestep. In those simulations the cell size was taken to be L = A/5 so
that the cell contribution is negligible, and the timestep was varied from At = A/(2c 0 ) to At = 16A/c0. The
error is defined as the normalized deviation in the flux corresponding to the transport coefficient with respect
to the exact result. The exact result is taken to be a very accurate simulation with At = A/(8co).
Figures 3, 4, and 5 show the comparison between the numerical results of Garcia and Wagner and the
theoretical prediction of equations (11), (12), and (13) for the diffusion coefficient, the viscosity coefficient,
and the thermal conductivity respectively. The agreement is very good for At —»• 0. For At ^> A/CO the error
deviates from the quadratic timestep dependence. Garcia and Wagner [7] point out that this is due to the
upper bound set on the transport coefficients by the collisionless limit that is indicated on the same graph.
V
CONCLUDING REMARKS
We have presented a formulation that allows the calculation of the transport coefficient dependence on
timestep. The calculations predict that the error in the transport coefficients is of order At 2 , and that for
At ~ A/CO the truncation error is approximately 5%, which confirms the empirical observations that accurate
solutions require a timestep that is a small fraction of the mean free time. The theory presented relies on the
application of the Green-Kubo theory and is thus valid for problems that appear steady at the autocorrelation
decay timescale.
The results of this paper have been verified by extensive numerical simulations [7]; the agreement between
simulations and theory is very good for At —»• 0. For At ^> A/CO the error approaches the value set by the
collisionless limit. Interestingly, Garcia and Wagner show that the error in transient calculations is of order
At 2 , but no direct comparison between transport coefficients and equations (11-13) was presented.
386
VI
ACKNOWLEDGEMENTS
The author would like to thank Alej Garcia and Wolfgang Wagner for making their simulation results
available, and Frank Alexander for helpful discussions.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
Alexander, F. J., Garcia, A. L., and Alder, B. J., Phys. Fluids, 10, 1540 (1998).
Wainright, T., J. Chem. Phys., 40, 2932 (1964).
Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon, Oxford (1994).
Alexander, F. J., and Garcia, A. L., Comp. Phys., 11, 586 (1997).
Hadjiconstantinou, N. G., to appear in Phys. Fluids.
Wagner, W., J. Stat. Phys., 66, 1011, 1992.
Garcia, A. L., and Wagner, W., to appear in Phys. Fluids.
387