Particle-in-Cell method for solving the Boltzmann

equation

E.A. Malkov ∗ and Mikhail S. Ivanov †

Khristianovich Institute of Theoretical and Applied Mechanics (ITAM), Novosibirsk 630090, Russia

A method of the numerical solution of the Boltzmann equation, which belongs to the

family of deterministic particle-in-cell methods, and results of its application are described.

Such a method is based on the possibility of presenting the collision integral of the Boltz-

mann equation in divergent form.

I. Introduction

The Boltzmann equation for a single-particle distribution function f = f(t,−→r ,−→v ), which describes therarefied gas dynamics, has the form

L(f) ≡ ∂f

∂t+−→v ∂(f)

∂−→r = St(f), (1)

with the right-hand side containing a nonlinear integral operator, which is the collision integral

St(f) =

∫

R3

d3v1

∫

S2

d2n(f ′f ′

1 − ff1)|−→V |σ(V, cos(θ)), (2)

where

f = f(t,−→r ,−→v ), f1 = f(t,−→r ,−→v1),f ′ = f(t,−→r ,−→v ′), f ′

1 = f(t,−→r ,−→v1 ′),

−→v ′ =−→v +−→v1

2+

|−→V |2

−→n ,

−→v1 ′ =−→v +−→v1

2− |−→V |

2−→n , (3)

−→V = −→v −−→v1 , −→n is the unit vector (collision parameter) defining the direction of the relative velocity after thecollision, and σ = σ(V, cos(θ)) is the differential scattering cross section (θ is the angle between the relativeparticle velocity vectors before and after the collision).

The structure of Eq. (1) reveals the assumptions underlying the physical model of rarefied gases. Inde-pendence and additivity of the transfer process in the physical space due to spatial gradients of the phasedensity and the process of binary molecular collisions described by the nonlinear collision integral are implied.

There are several differential and integral forms of presentation of the Boltzmann equation. It seemsof interest to use its divergent form to construct conservative numerical methods of rarefied gas dynamics,in particular, the particle-in-cell (PIC) method of the nonlinear Boltzmann equation. One of the firstpublications in this field available for the authors was the paper.1 In that paper, a force field whose effectis equivalent to the collision integral in the Boltzmann equation was introduced to derive a kinetic equationin divergent form. Somewhat later, the divergent form of the Boltzmann equation was used in2, 3 to solveproblems of semiconductor physics. Two new independent approaches to presenting the Boltzmann collisionintegral in divergent form have been recently developed in.4–7 Based on results obtained in,6, 7 a new methodcalled the Kinetic Force Method for numerical simulation of rarefied gas problems was proposed in.8

∗Leading researcher, Khristianovich Institute of Theoretical and Applied Mechanics (ITAM), Institutskaya 4/1, Novosibirsk630090, Russia. Email: malkov@itam.nsc.ru

†Professor, Head of CFD Lab, Khristianovich Institute of Theoretical and Applied Mechanics (ITAM), Institutskaya 4/1,Novosibirsk 630090, Russia, AIAA Fellow

1American Institute of Aeronautics and Astronautics

10th AIAA/ASME Joint Thermophysics and Heat Transfer Conference28 June - 1 July 2010, Chicago, Illinois

AIAA 2010-4503

Copyright © 2010 by E.A. Malkov and M.S. Ivanov. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

II. Particle-in-cell method for solving the Boltzmann equation

In accordance with,1 Eq. (1) can be written in the form

L(f) ≡ ∂f

∂t+

∂(−→v f)

∂−→r +∂(−→Q)

∂−→v = 0, (4)

∂−→Q

∂−→v = −St(f), (5)

where−→Q(t,−→r ,−→v ) =

−→F (t,−→r ,−→v )f(t,−→r ,−→v )/m is the vector flux of molecules in the velocity space. Being

written in such a manner, the Boltzmann equation has the form of a differential conservation law anddescribes a compressible flux of molecules in the phase space under the action of the vector field −→v +

−→F /m.

The time evolution of the distribution function f(t,−→r ,−→v ) can be interpreted as a result of two additiveprocesses: transfer in the actual space and transfer in the space of velocities, which is caused by changes invelocities induced by the collective force

−→F . To find this force, we have to solve Eq. (5). At fixed boundary

conditions, this equation admits numerous solutions that differ from each other by the vortex component.For this reason, versatile divergent presentations of the collision integral are possible. As in,1 we assumethat the flux

−→Q is vortex-free. Then, it can be presented as a gradient of a certain scalar field Φ:

−→Q = − ∂Φ

∂−→v (6)

Then, Eq. (5) reduces to Poissons equation

∂

∂−→v · ∂Φ

∂−→v = St(t,−→r ,−→v ), (7)

Equation (7) is solved in the space of velocities with fixed values of time at fixed points of the physicalspace. Correspondingly, we skip writing the dependences of the functions on the variables t and −→r , whichhave the meaning of parameters in this context. The solution of Eq. (7), which satisfies the natural boundarycondition Φ(−→v ) −→ 0 as |−→v | −→ ∞, has the form

Φ(−→v ) =1

4π

∫

R3

St(−→v ′)

|−→v −−→v ′|d3v′. (8)

Correspondingly, we obtain

−→Q(−→v ) =

∂Φ

∂−→v = − 1

4π

∫

R3

St(−→v ′)(−→v −−→v ′)

|−→v −−→v ′|3 d3v′ (9)

The formulation of Eq. (4) with the flux determined by Eq. (9) does not imply a particular form of thecollision term; therefore, we can speak about the divergent form of kinetic equation extending it to the modelkinetic equations as well.

The Boltzmann equation written in divergent form is similar in many aspects to the Vlasov kineticequation, which describes the collisionless plasma, or to the collisionless stellar dynamics (CSD) equation,9

which describes the motion of starts in galaxies. These collisionless kinetic equations can be written in theform

Df

Dt= 0, (10)

where D/Dt is the Lagrangian derivative in the phase space. The phase density remains unchanged alongthe trajectories, i.e., the phase density is a Lagrangian invariant. In the Boltzmann equation (1), the fluxdivergence is not equal to zero because of collisions (the collision integral is not equal to zero); therefore, itcannot be presented in the form (10). Equation (4) describes a compressible phase medium. The equationsof characteristics include an equation for the change in the phase density along the trajectories. Despitethis difference between the Boltzmann equation and collisionless kinetic equations, it is possible to use theparticle-in-cell numerical method successfully applied to model processes in the collisionless plasma andstellar systems.10, 11 In contrast to the PIC method in plasma physics or collisionless stellar dynamics,

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particles in cells of the phase space should be considered here. Let us present the distribution function as asum of the form

f(t,−→r ,−→v ) =

N∑

p=1

npK(−→r ,−→v ,−→R p(t),

−→V p(t)), (11)

where−→Rp(t) and

−→V p(t) are three-dimensional vector functions of time. Following the ideology of PIC

methods, the individual terms of this sum can be interpreted as particles whose shape, size, and locationare determined by a function K called the particle kernel; the constants np define the particle weight. Inaddition to the requirement to satisfy the conservation law, which, in the case of kernel normalization

∫

R3×R3

Kd3rd3v = 1, (12)

has the form∫

R3×R3

f(t,−→r ,−→v )d3rd3v =

N∑

p=0

np, (13)

the function K is subjected to additional requirements: non-negativity and symmetry of the form

K(−→r ,−→v ,−→R,

−→V ) = K(

−→R,

−→V ,−→r ,−→v ), (14)

∂K/∂−→R = −∂K/∂−→r , ∂K/∂

−→V = −∂K/∂−→v (15)

Note that the function K can be only piecewise-smooth rather than smooth. Therefore, the function (11)can satisfy Eq. (4) only in the weak sense. Let us write the weak solution condition:12

N∑

p=1

np

∫

[K(−→r ,−→v ,−−−→Rp(t),

−−−→Vp(t))

∂φ

∂t+

K(−→r ,−→v ,−−−→Rp(t),

−−−→Vp(t))−→v

∂φ

∂−→r +

K(−→r ,−→v ,−−−→Rp(t),

−−−→Vp(t))

−→F (t,−→r ,−→v )

m

∂φ

∂−→v ]dtd3rd3v = 0 (16)

for any smooth finite function φ(t,−→r ,−→v ). Taking the integrands by parts and using the properties offiniteness of the test functions φ(t,−→r ,−→v ) and the properties of kernel (14)-(15), we obtain

N∑

p=1

np

∫

[K(−→r ,−→v ,−−−→Rp(t),

−−−→Vp(t))

∂φ

∂−→r [d−→Rp

dt−−→

V p(t)] +

K(−→r ,−→v ,−−−→Rp(t),

−−−→Vp(t))

∂φ

∂−→v [d−→V p

dt−

−→F p(t)

m]]dtd3rd3v = 0 (17)

As the test function φ(t,−→r ,−→v ) is arbitrary, it follows that expansion (11) satisfies Eq. (4) if the expressionin square brackets in Eq. (17) is equal to zero, i.e., the following conditions having the form of a dynamicsystem in the phase space (“particle motion equations) are met:

d−→Rp

dt=

−→V p,

d−→V p

dt=

1

m

−→F p(t,

−→Rp,

−→V p). (18)

Numerical PIC methods imply the use of an Eulerian grid in the phase space. The force field−→F (t,−→r ,−→v )

is calculated at the nodes of this grid, with subsequent interpolation of the field to the phase space pointswhere the particles are located. The number of the Eulerian grid nodes in the phase space is substantiallysmaller than the number of particles N in most cases. To calculate the force in the computational grid nodes,one has to calculate the values of the distribution function in these nodes. This is made by the “particle-to-cell interpolation otherwise called the distribution of particle attributes.10 In our case, the number ofmolecules (or the fraction of molecules, depending on normalization) associated with the p-th particle, equalto the weight np in expansion (11), is distributed among the nodes. Figure 1 shows a schematic image

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Figure 1. Computational grid in the phase space

of the computational grid in the phase space with nodes indicated by the pair of indices (I, α): I ensuresindexation of the nodes in the physical space (which is three-dimensional in the general case), and α providesindexation in the three-dimensional velocity space. Let the weight of the particle indicated by the black colorin Fig. 1 be np. First, we use the nearest grid point method (NGP10) and assign this weight to the node(I, α). Otherwise, using the particle-in-cell (PIC), cloud-in-cell (CIC) or other methods of interpolation, wedistribute this weight among the nodes in the physical space so that

∑

I

∆nI,α = np (19)

After that, with a fixed index I, each fraction ∆nI,α is distributed among the nodes of the velocity grid(I, α), (I, α+ 1), (I, α− 1), etc., depending on the interpolation method used. After this procedure, not onlythe conservation law (13) should be valid, but also the momentum and energy should be conserved. Notethat this requirement cannot be satisfied in the traditional PIC methods used in plasma physics,13 becausethey deal with the physical rather than phase space.

In our computations, we used the interpolation proposed in14 for conservative correction of collisionintegral calculations (note that the interpolation with such properties for correcting collision integral cal-culations was first proposed in15). Figure 2 illustrates the 7-point interpolation in the velocity space. Let(dx, dy, dz) be the vector of the particle location in the velocity space with the origin chosen in the nearestnode to the particle, which has the index α. Then, the fraction ∆nI,α is distributed between the nodes−αx, αx,−αy, αy,−αz, αz and the nearest node as follows:

s(−αx) = −(dx+ dy + dz − dx2 − dy2 − dz2)/6.0, (20)

s(−αy) = −(dx+ dy + dz − dx2 − dy2 − dz2)/6.0, (21)

s(−αz) = −(dx+ dy + dz − dx2 − dy2 − dz2)/6.0, (22)

s(αx) = dx+ s(−αx), (23)

s(αy) = dy + s(−αy), (24)

s(αz) = dz + s(−αz), (25)

s(0) = 1− dx2 − dy2 − dz2. (26)

With such a distribution, the momentum and energy are conserved. To obtain the phase density in thecomputational grid nodes, we divide the distributed fractions of particles by the phase cell volume.

After the particle-to-grid interpolation procedure, we calculate the collision integral in the computationalgrid nodes. Two approaches are used for collision integral calculations: based on the adaptive Monte Carlomethod16 of estimating multidimensional integrals with the function values at points between the nodes being

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found by means of trilinear interpolation and the regular method of rectangles with molecular velocities afterthe collision being also found by Eqs. (20)-(26). The next stage of computations by the proposed method iscalculating expression (8) whose grid approximation in the velocity space has the form

Φα =1

4π

∑

α′

Stα′

hv

√

(αx − α′

x)2 + (αy − α′

y)2 + (αz − α′

z)2

. (27)

Figure 2. Interpolation of particles in the velocity space

Expression (27) has the form of a convolute, and the convolute theorem can be used for its calculation.Then, the calculation of the sum of sequence (27) (estimate of the triple integral (8)) reduces to obtaininga discrete Fourier image of the sequence

Φβ =√LMN ˆStβGβ , (28)

where L,M,N are the dimensions of the regular grid in the velocity space, ˆStβ is the Fourier image of the grid

approximation of the collision integral, and Gβ is the Fourier image of the sequence Gβ = 1/√

β2x + β2

y + β2z ,

and to applying the inverse Fourier transform to Φβ . The grid approximation of the vector field−→F (t,−→r ,−→v )

is found by using finite differences. As the Fast Fourier Transform is highly efficient, the time needed tocalculate the force field is negligibly small, as compared to the time needed to calculate the collision integral.Therefore, force field calculations require minor computer resources. The computations were performed withthe FFTW software package17 implementing the Fast Fourier Transform. The last stage of the computationalcycle of solving the Boltzmann equation by the PIC method involves numerical integration of the system ofequations of motion (18). The values of

−→F p(t,

−→Rp,

−→V p) are found by interpolating of the grid vector function−→

F I,α at the phase space point (−→Rp,

−→V p). The interval τ of integration of system (18) is chosen so that the

length of the phase trajectories during this time remains the phase cell size. Summarizing the description ofthe PIC method used to solve the Boltzmann equation, we show the diagram (Fig. 3), which illustrates thealgorithm of rarefied gas flow computations.

III. Numerical example

The main difficulties of the PIC method described above are associated with computing the particletrajectories in the velocity space. Therefore, we chose the test problem of uniform relaxation as the first

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Figure 3. Diagram of actions in calculating rarefied gas dynamics by the PIC method

step. We computed homogeneous relaxation of a gas consisting of the Maxwell molecules with the initialdistribution function corresponding to Bobylevs exact solution at t = 0:18

f(t, v) =1

(2πτ)3/2exp

(

− v2

2τ

)[

1 +1− τ

τ

(

v2

2τ− 3

2

)]

, (29)

τ(t) = 1− 2

5e−t/6, t ≥ 0.

In the case of homogeneous relaxation, the equations of the problem have the form

∂f

∂t+

∂(−→F f)

∂−→v = 0, (30)

−→F f = − ∂Φ

∂−→v , (31)

Φ(−→v ) =1

4π

∫

R3

St(−→v ′)

|−→v −−→v ′|d3v′., (32)

St(f) =

∫

R3

d3v1

∫

S2

d2n(f ′f ′

1 − ff1)|−→V |σ(V, cos(θ)), (33)

and the equations of motion of the particles are

d−→V p

dt=

1

m

−→F p(t,

−→V p). (34)

As compared to the general case, the computational cycle lacks only the interpolation to the grid nodes ofthe physical space. Figure 4 shows the normalized moments < v2n >N = < v2n >/(2n+ 1)!! of Bobylevsdistribution function of the 2-nd, 4-th, 6-th, and 8-th orders and their theoretical values (solid curves). Thecomputational domain size is [−6, 6]3, the computational grid has 32× 32× 32 nodes, and the time step inthe computational cycle is τ = 0.05.

In calculating the collision integral, we used the idea proposed in19 with small modifications. The ideais to reject nodes with small values of the phase density in calculating the collision integral. The authors19

estimated the error of calculating gas-dynamic moments depending on the threshold value of the phasedensity, assuming that the distribution is little different from the equilibrium one. We propose anothercriterion for node rejection. The following procedure is used:

• the nodes are sorted in decreasing order of the distribution function;

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0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 2 4 6 8 10

Time

Normalized high-order momenta of the distribution function

<v2>/3

<v4>/15

<v6>/105

<v8>/945

Figure 4. Homogeneous relaxation computed by the PIC method

• partial sums of this decreasing sequence are found and compared with the total sum

n∑

i=0

fiv2pi /

N∑

i=0

fiv2pi < ǫ, (35)

where N is the total number of grid nodes and i is the linear index of the node.

Nodes with linear indices i < n are left for collision integral calculations. Despite the apparent complexity ofsuch an algorithm, it can be effectively implemented with the use of the Standard Template Library (STL)of the C++ language. A fragment of the code with our implementation of this algorithm is given below:

CFunction3D df;

..............................

vector<long> fat_points;

double eps;

int p_for_eps;

................................

fat_points.clear();

multimap<double, long> ordered_points;

multimap<double, long>::const_iterator iter;

double S=0.0;

for(size_t i=0;i<I;i++)

for(size_t j=0;j<J;j++)

for(size_t k=0;k<K;k++)

S+=df(i,j,k)*

pow((vxmin+(i+0.5)*hvx)*(vxmin+(i+0.5)*hvx)+

(vymin+(j+0.5)*hvy)*(vymin+(j+0.5)*hvy)+

(vzmin+(k+0.5)*hvz)*(vzmin+(k+0.5)*hvz),p_for_eps);

for(size_t i=0; i<df.storage.size();i++)

ordered_points.insert(pair<double,long>(df.storage[i],i));

double s=0.0;

for( (iter=ordered_points.end())--;

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(iter!=ordered_points.begin()) &&

s/S<1.-eps;--iter){

long Idx=iter->second;

int i=(int)(Idx/J/K);

int j=(int)((Idx-i*J*K)/K);

int k=(int)(Idx-i*J*K-j*K);

double vn=pow((vxmin+(i+0.5)*hvx)*(vxmin+(i+0.5)*hvx)+

(vymin+(j+0.5)*hvy)*(vymin+(j+0.5)*hvy)+

(vzmin+(k+0.5)*hvz)*(vzmin+(k+0.5)*hvz),p_for_eps);

s+=(iter->first*vn);

fat_points.push_back(iter->second);

}

The cost of execution of this algorithm are multiply compensated owing to a decrease in the number of nodesin calculating the collision integral. Let us give the number of remaining significant nodes on the 32×32×32computational grid (the total number of nodes is 32,768) at the initial time as a function of the values of ǫand p:

Variant ǫ p Number of nodes Efficiency

I 0.01 0 2500 x200

II 0.01 1 3500 x70

III 0.01 2 7500 x20

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 2 4 6 8 10

Time

Normalized high-order momenta of the distribution function

<v2>/3

<v4>/15

<v6>/105

<v8>/945

<v10>/10395

<v12>/135135

Figure 5. Variant I

Figures 5-7 show the normalized moments < v2n >N = < v2n >/(2n+ 1)!! of Bobylevs distributionfunction of the 2-nd, 4-th,... 12-th orders and their theoretical values (solid curves), which illustrate thedependence of the model computation accuracy on the number of nodes of the computational grid in thevelocity space.

The basic conclusion of the present study is the fact that the modified PIC method used for numericalsimulations of rarefied gas dynamics provides adequate results in homogeneous relaxation computations.The computations also show that the interpolation method proposed in14 and used here does not distortthe evolution of higher-order moments. Therefore, we can start solving spatially inhomogeneous problems,which are expected to reveal the capabilities of the method to the full extent.

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0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 2 4 6 8 10

Time

Normalized high-order momenta of the distribution function

<v2>/3

<v4>/15

<v6>/105

<v8>/945

<v10>/10395

<v12>/135135

Figure 6. Variant II

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 2 4 6 8 10

Time

Normalized high-order momenta of the distribution function

<v2>/3

<v4>/15

<v6>/105

<v8>/945

<v10>/10395

<v12>/135135

Figure 7. Variant III

IV. ACKNOWLEDGMENTS

The authors are sincerely grateful to Prof. V.L. Saveliev whose activities stimulated our interest in usingthe divergent form of the Boltzmann equation.

References

1B.V Filippov and V.B. Khristinich, “Kinetic equations of rarefied gas dynamics in divergent form,” in: Dynamic Pro-cesses in Gases and Plasma. Physical Mechanics (collected scientific papers) [in Russian], Izd. Leningr. Univ., Leningrad, No.4, pp. 7–18 (1980).

2Motta S., Wick J., A New Numerical Method for Kinetic Equations in Several Dimensions, Computing, Vol. 46, 1991,pp. 223-232.

3Wick J., Numerical approaches to the kinetic Semiconductor Equation, Computing, Vol. 52, 1994, pp. 39-49.4S. Villani, “Conservative forms of Boltzmann’s collision operator: Landau revisited,” Math. Mod. An. Num., Vol. 33,

No. 1, pp. 209-227 (1999).5V.L. Saveliev and K. Nanbu, “Collision group and renormalization of the Boltzmann collision integral,” Phys. Rev. E,

Vol.65 (2002).

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6Saveliev V.L., Nanbu K., Exact Forms of Representation of Boltzmann Collision Integral as a Divergence of the Flowin Velocity Space, Proceedings of the 24th International Symposium on Rarefied Gas Dynamics, Vol. 762, Tohoku University,Sendai, Japan, 2005, pp. 114-119.

7Saveliev V.L., “New forms of the Boltzmann Collision Integral, Proceedings of the 25th International Symposium onRarefied Gas Dynamics, Saint-Petersburg, Russia, 2007, pp. 131-136.

8V.L. Saveliev and S.A. Filko, “Kinetic force method for numerical modeling 3D-relaxation in Homogeneous rarefied gas,AIP Conf. Proc., December 31, 2008, Vol. 1084, pp. 513-518 (2008).

9Binney J., Tremaine S. Galactic Dynamics. Princeton University Press, 1987. 733p.10R.W. Hockney and J.W. Eastwood, Computer Simulation Using Particles, CRC Press (1988).11D.E. Potter, Computational Physics, Wiley (1973).12R.D. Richtmyer, Principles of Advanced Mathematical Physics, Springer Verlag, New York (1978).13A.B. Langdon, “Effects of the spatial grid in simulation plasma, J. Comp. Phys., Vol.6, pp. 247-267 (1970).14P.L. Varghese, “Arbitrary post-collision velocities in a discrete velocity scheme for the Boltzmann equation, in: Rarefied

Gas Dynamics: Proc of the 25th Intern. Symposium, edited by M.S. Ivanov and A.K. Rebrov, Novosibirsk, Russia, pp. 227-232(2007).

15F.G. Tcheremissine, “Conservative method of calculating the Boltzmann collision integral,” Dokl. Ross. Akad. Nauk,Vol. 357, No.1, pp.53-56 (1997).

16G.P. Lepage, “A new algorithm for adaptive multidimensional integration,” J. Comp. Phys., Vol.27, pp. 192-203 (1978).17FFTW library http://www.fftw.org/18A.V. Bobylev, “Exact solutions of the nonlinear Boltzmann equation and theory of relaxation of a Maxwellian gas,”

Teor. Mat. Fiz., Vol. 60, No. 2, pp. 280–310 (1984)19Z. Tan and P.L. Varghese, “The ∆− ǫ method for the Boltzmann equation, J. Comp. Phys., Vol.110, p.327 (1994).

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