The phase space PIC method for solving the

Boltzmann equation. Part I.

E.A. Malkov∗

and Mikhail S. Ivanov†

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

The paper describes a scheme for calculating the collision integral in high-accuracy

computations of rarefied gas flows. Numerical test computations show that the proposed

scheme allows finding numerical solutions of the Boltzmann equation in a wide range of

flow velocities, including extremely slow flows.

I. Introduction

The theoretical fundamentals of the deterministic particle method for solving the Boltzmann equation,which is an extension of the Particle-in-Cell (PIC) method to the velocity space, were described in.1 Resultson homogeneous relaxation of a Maxwellian gas calculated by this method, which can be called the phasespace PIC method, were also presented there. Preliminary calculations of one-dimensional rarefied gas flowshowed that the deterministic particle method in this case imposes more severe requirements to collisionintegral calculation than in the case of homogeneous relaxation. Some procedures substantially improvingthe collision integral calculation accuracy are considered in this paper. This improvement is validated by theresults for low-velocity flows calculated with a difference scheme of solving the Boltzmann equation. Theseprocedures also ensure a significantly shorter time needed to calculate the collision integral. The next partof the paper will describe the results calculated by the deterministic particle method with the use of thecollision integral calculation scheme presented in this paper.

II. Collision integral calculation scheme

II.A. Conservative interpolation to computational grid nodes

Following the approach described in2 -,4 we write the collision integral in a symmetric form5 convenient forconstructing conservative schemes for calculating this integral with conservation of mass, momentum, andenergy:

I(−→v ) =1

2

∫

R3×R3

d3v1d3v2

∫

S2

d2nf(−→v 1)f(−→v 2) ×

×[

δ(−→v −−→v1′) + δ(−→v −−→v2

′) − δ(−→v −−→v1) − δ(−→v −−→v2)]

|−→V |σ (1)

The grid approximation of the distribution function is written in the form

f(−→v ) =∑

α

fαδ(−→v −−→vα), (2)

where α indexes the computational grid node in the velocity space. To avoid cumbersome expressions withoutloss of generality, we assume that the differential scattering section is independent of the scattering angle.Let σ = 1

4π σ (/−→v α −−→v β/) δ (−→n −−→n α′)wα′ be a discrete approximation of the differential scattering section

∗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

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on a sphere constructed on a diameter −→v α − −→v β. The unit vectors −→n α′ define K nodes of appxoximation,and wα′ defines the node weights. Then, the collision integral for a discrete distribution function (2) iswritten as

I(−→v ) = −∑

α,β

fαfβσ(| −→v α −−→v β |)

[δ(−→v −−→vα) + δ(−→v −−→vβ)] −

1

4π

∑

α′[δ(−→v −−→vα′) + δ(−→v −−−→v−α′)] wα′

, (3)

where

−→v α′ =−→v α + −→v β

2+

|−→V |2

−→n α′,

−→v −α′ =−→v α + −→v β

2− |−→V |

2−→n α′, (4)

There are several approaches for choosing the nodes of the quadrature on a sphere. The authors ofthe paper3 proposed to choose points of intersection of the sphere with the computational grid nodes as thequadrature nodes. This approach, however, has a drawback: the number of such intersections at small valuesof | −→v α −−→v β | is too small to ensure acceptable accuracy of collision integral calculation. An advantage ofthis method is the absence of problems in distributing the collision integral fraction ∆I = fαfβσwα′ betweenthe computational grid nodes that are neighbors of the quadrature node −→v α′.

Later, it was proposed a procedure of the distribution to velocity grid nodes, which conserves not onlymass and momentum, but also energy.4 Thus, constraints on selection of quadrature nodes on a sphereare removed. Note that a distribution with similar properties was proposed earlier in the paper,2 and thecorresponding scheme of collision integral calculation was called a conservative scheme. Figure (1) shows

Figure 1. Fragment of the computational grid and quadrature nodes on a sphere

quadrature nodes on a sphere (indicated by crosses) for one pair of nodes of a rectangular computationalgrid. Icosahedron vertices are chosen in this illustration as quadrature nodes on a sphere. With such a choiceof nodes, their weights are identical and equal to wα′ = 4π/16. Figure (2) shows one of the quadrature nodeson a sphere, which is an internodal point with respect to the computational grid. The collision integralfraction related to the node on the sphere −→v α′ is distributed to the nearest node and to the neighboringnodes of the computational grid indicated in the figure by bold points in accordance with the formulas

s0 = 1 − d2x − d2

y − d2z, (5)

sext = −1

6(dx + dy + dz − d2

x − d2y − d2

z), (6)

sx = dx + sext, (7)

sy = dy + sext, (8)

sz = dz + sext, (9)

(10)

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where s0, sx, sy, sz, and sext are the fractions of the mass distribution over the corresponding nodes. Suchinterpolation to the computational grid nodes really yields a conservative scheme of collision integral calcu-lation.4

Figure 2. Distribution of mass to velocity grid nodes

II.B. Quadratures on a sphere

As there is a possibility of a conservative distribution of mass, momentum, and energy to velocity grid nodes,there is also a certain freedom in choosing the quadrature formula on a sphere. After analyzing variousquadratures, we found optimal formulas in terms of accuracy and speed of collision integral calculation.The following considerations were taken into account. The quadrature nodes are divided into diametricallyopposite pairs, and the node weights do not differ too much from each other, i.e., the nodal areas areapproximately identical, and the quadrature is homogeneous to the most possible extent. Several quadratureformulas invariant to groups of rotation of regular polyhedrons with inversion were chosen for testing. Theseare the formula of a icosahedron with 12 nodes, which are icosahedron vertices (FI-12), the formula of anoctahedron of the 5-th order of algebraic accuracy (the quadrature yields the exact value of the integral forall polynomials with powers up to 5) with 14 nodes (FO-14), and the formula of an octahedron of the 5-thorder of algebraic accuracy with 26 nodes (FO-26). For comparisons, we additionally tested the quadratureformula based on a grid of spherical coordinates with nodes at the centers of spherical rectangles with 32and 128 nodes (SP-32,128). The characteristics of these quadratures are listed in Tables 1-4.

Table 1. Parameters of the quadrature FO-14

Nodes Weights/4π Comments

(±1,0,0), (0,±1,0),(0,0,±1) 1/15 Octahedron vertices

(±1/√

3,±1/√

3,±1/√

3)) 3/40 Face centers

Table 2. Parameters of the quadrature FO-26

Nodes Weights/4π

Comments

(±1, 0, 0), (0, ±1,0),(0,0, ±1) 1/21 Octahedron vertices

(±1/√

3,±1/√

3,±1/√

3)) 9/280 Face centers

(± 1√2,0,± 1√

2), (0,± 1√

2,± 1√

2),

(± 1√2,± 1√

2,0)

4/105 Centers of octahedronedges

The results predicted by different quadrature formulas on a sphere are comparaed below, in the paragraphdealing with testing the algorithm of collision integral calculation.

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Table 3. Parameters of the quadrature FI-12

Nodes Weights/4π

Comments

(±X,0,±Z), (0,±Z,±X),(±Z,±X,0) 1/12 Icosahedron ver-tices φ = 1+

√5

2 ,X = 1√

1+φ2,

Z = Xφ

Table 4. Parameters of the quadrature SC-2nm

Nodes Weights/4π

Comments

(sin θi cosφj , sin θi cosφj , cos θi) sin θi φi = πn i, i = 0..2n − 1,

θj = πm (j + 0.5), j =

0..m

II.C. Pairs of nodes with identical interpolation parameters

There is no need to calculate the parameters s0, sx, sy, sz, and sext of interpolation of the quadrature gridnodes to the neighboring nodes of the computational grid for each pair of nodes of the computational grid−→v α,−→v β . These parameters are identical for all pairs of nodes with an identical distance between them (withidentical relative velocities). In calculating the collision integral, therefore, the cycle over the pairs of thecomputational grid nodes should be organized as follows.

for(ibase=0;ibase<n;ibase++)

for(jbase=0;jbase<n;jbase++)

for(kbase=0;kbase<n;kbase++)

The relative velocities are calculated for eachbase pair of computational grid nodes (0,0,0)and (ibase,jbase,kbase)

for(iknot=0;iknot<number_of_knots;iknot++)

Interpolation parameters and the nearestcomputational grid node (icell, jcell, kcell) arecalculated for each node of the quadrature ona sphere

for(ishift=0;ishift<n-ibase;ishift++)

for(jshift=0;jshift<n-jbase;jshift++)

for(kshift=0;kshift<n-kbase;kshift++)

//the fraction of the collision integral is calculated:

dmass=mass=df[kshift+jshift*n+ishift*n*n]*

df[kshift+kbase+(jshift+jbase)*n+(ishift+ibase)*n*n]*

weights[iknot];

st[kshift+jshift*n+ishift*n*n]-=dmass;

st[kshift+kbase+(jshift+jbase)*n+(ishift+ibase)*n*n]-=dmass;

//if the pair of base nodes is not in one coordinate plane,

//then the contribution of the adjoint pairs (Fig. ) is calculated

if(ibase&&jbase)

dmass=df[kshift+jshift*n+(ishift+ibase)*n*n]*

df[kshift+kbase+(jshift+jbase)*n+ishift*n*n]*

weights[iknot];

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st[kshift+jshift*n+(ishift+ibase)*n*n]-=dmass;

st[kshift+kbase+(jshift+jbase)*n+ishift*n*n]-=dmass;

if(ibase&&kbase)

dmass=df[kshift+(jshift+jbase)*n+(ishift+ibase)*n*n]*

df[kshift+kbase+jshift*n+ishift*n*n]*

weights[iknot];

st[kshift+(jshift+jbase)*n+(ishift+ibase)*n*n]-=dmass;

st[kshift+kbase+jshift*n+ishift*n*n]-=dmass;

if(jbase&&kbase)

dmass=df[kshift+kbase+jshift*n+(ishift+ibase)*n*n]*

df[kshift+(jshift+jbase)*n+ishift*n*n]*

weights[iknot];

st[kshift+kbase+jshift*n+(ishift+ibase)*n*n]-=dmass;

st[kshift+(jshift+jbase)*n+ishift*n*n]-=dmass;

The mass is interpolated to the neighboringnodes of the computational grid

II.D. Test calculations of the collision integral

To check the accuracy of calculations with different quadratures on a sphere, we used the exact solution of theBoltzmann equation for a Maxwellian gas (where the differential scattering section is inversely proportionalto the relative velocity of the colliding molecules), i.e., Bobylevs solution:6

f(t, v) =1

(2πτ)3/2exp

(

− v2

2τ

) [

1 +1 − τ

τ

(

v2

2τ− 3

2

)]

, (11)

τ(t) = 1 − 2

5e−t/6, t ≥ 0.

Correspondingly, the collision integral for function (11) is written as

st(B)(t, v) =e−t/6

15τ(2πτ)3/2

(

1

τ− 1

)

exp

(

− v2

2τ

)

[

(

v2

2τ− 3

2

)2

+3

2− v2

τ

]

(12)

The calculated collision integral for the function (11) is compared in Table 5 with the exact collisionintegral (12) at t = 0. The column entitled “Calculation accuracy shows the values of the expression

ǫ =

√

√

√

√

√

√

√

√

∑

i,j,k

(

sti,j,k − st(B)i,j,k

)2

∑

i,j,k

(

st(B)i,j,k

)2 . (13)

The best accuracy of collision integral calculation is reached by using the quadrature with 26 nodesinvariant to the octahedron group.

II.E. Correction of the collision integral

The method of collision integral calculation, the number and locations of quadrature nodes, and the final sizeof the computational domain affect of the collision integral calculation accuracy. For functions close to the

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Table 5. Comparison of collision integral calculations with the use of difference quadratures on a sphere

Quadrature Number ofnodes

Calculation accuracy

FI-12 12 0.250358

FO-14 14 0.235963

SC-16 16 0.282812

FO-26 26 0.205421

SC-128 128 0.206925

Maxwellian distribution, the values of errors induced by these factors can exceed typical values of the truecollision integral. Such a situation is illustrated in Fig. (3), which shows the central sections of the calculatedcollision integrals of the truncated Maxwellian function and Bobylevs distribution function at t = 10, whenthe latter is little different from the Maxwellian function.6 To finalize collision integral calculation, we usea correction based on the following considerations. The collision integral turns the Maxwell function to azero function; we require that the grid operator of collisions, independent of its presentation, should turnthe Maxwellian grid function to zero. To satisfy this requirement, we use a calibration implemented in thefollowing procedure.

• The collision integral is calculated in accordance with our grid presentation of the collision integral.

• The first and second moments of the distribution function are calculated.

• The collision integral for the Maxwellian grid function with parameters corresponding to the momentsobtained is calculated.

• Thus difference between the thus-obtained collision integrals is considered as the sought collision inte-gral.

Figure (4) shows the central sections of the corrected collision integral for Bobylevs distribution functionat t = 10, the central section of the same integral without correction, and exact collision integral obtainedon the basis of Bobylevs analytical solution (12).

Table 6. Effect of correction on the collision integral calculation accuracy

Accuracy of calculations without correction 3.99534

Accuracy of calculations after correction 0.438941

As is seen in these figures, the proposed correction allows the collision integral to be correctly calculatedfor distribution functions little different from equilibrium distribution functions.

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−6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6

−0.0004

−0.0003

−0.0002

−0.0001

0

0.0001

0.0002

0.0003

0.0004

The collision integral of the truncated Maxwell functionThe collision integral of the function (11) without correction

vx

vy

Figure 3. Collision integral of the truncated Maxwellian function and the calculated integral for Bobylevs function att = 10

−6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6

−0.0004

−0.0003

−0.0002

−0.0001

0

0.0001

0.0002

0.0003

0.0004

The exact collision integral (12)The collision integral of the function (11) with correction

The collision integral of the function (11) without correction

vx

vy

Figure 4. Corrected calculated Bobylevs integral at t = 10

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III. Test calculations of the slow Couette flow (difference method)

III.A. Difference scheme of the solution

To check the accuracy of the algorithms of collision integral calculation by the difference method, we calculatea slow Couette flow with the velocity distribution function little different from the Maxwellian distributionfunction. The plate temperatures are assumed to be identical: T1 = T2. The Ox axis is chosen to benormal to the plates. The left plate is at rest, and the right plate moves with a velocity W = 0.1

√

(2T1)(the Boltzmann constant and the molecule mass are assumed to be equal to unity) in the Oz direction. Thecollision model is based on hard spheres. The Knudsen number is assumed to be equal to 0.1. A scheme withsplitting in terms of physical processes and with alternation of calculations of homogeneous relaxation ineach spatial cell and collisionless transfer in physical space is used. The nodes of the computational domainΩ = [0, X ]× [vmin, vmax]3 are

vmin + (i + 0.5)hv,

vmin + (j + 0.5)hv,

vmin + (k + 0.5)hv,

(l + 0.5)h

i,j,k∈[0,I−1],l∈[0,L−1]

,

where hv = (vmax − vmin)/I, h = X/LAn explicit upwind difference scheme of the first-order approximation in space is chosen for the convective

term of the equation. For internal points l ∈ (0, L − 1), the scheme has the form

fn+1i,j,k;l = fn

i,j,k;l +τu

h

fni,j,k;l − fn

i,j,k;l−1, u ≥ 0

fni,j,k;l+1 − fn

i,j,k;l, u < 0(14)

where u = V min+(i+0.5)h. The scheme is stable if the CourantFriedrichs-Lewy condition max(|u|)i ·τ ≤ his satisfied.

Fully diffuse reflection is imposed on the solid boundary, i.e., the velocities of molecules reflected fromthe plates have the Maxwellian distribution

f (M)(−→v ) =ρ

(2πT )3/2e(−|−→v −

−→V |2/2T )

with the mean velocity−→V and temperature T parameters corresponding to the velocity and temperature

of the plate. The third parameter of the Maxwellian distribution, the density ρ, is found from the massconservation equation, i.e., the flow incoming onto the plate is equal in magnitude to the reflected flow:

∫

−→v ·−→n <0

−→v · −→n fΓ(v)d3v = −∫

−→v ·−→n ≥0

−→v · −→n f (M)(v)d3v, (15)

(−→n is the normal to the boundary). For the left boundary node (l = 0) scheme (4) is written as

fn+1i,j,k;0 = fn

i,j,k;0 +τu

h

fni,j,k;0 − Flb, u ≥ 0

fni,j,k;1 − fn

i,j,k;0, u < 0(16)

where

Flb =

I/2−1∑

i=0

I−1∑

j,k=0

fni,j,k;0 · (vmin + (i + 0.5)hv)

I−1∑

i=I/2

I−1∑

j,k=0

f(M)i,j,k · (vmin + (i + 0.5)hv)

· f (M)i,j,k . (17)

Correspondingly, for the right boundary node (l = L − 1), we have

fn+1i,j,k;L−1 = fn

i,j,k;L−1 +τu

h

fni,j,k;L−1 − fn

i,j,k;L−2, u ≥ 0

Frb − fni,j,k;L−1, u < 0

(18)

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where

Frb =

I−1∑

i=I/2

I−1∑

j,k=0

fni,j,k;L−1 · (vmin + (i + 0.5)hv)

I/2−1∑

i=0

I−1∑

j,k=0

f(M)i,j,k · (vmin + (i + 0.5)hv)

· f (M)i,j,k . (19)

Here, f(M)i,j,k is the projection of the Maxwellian function onto the computational grid in the velocity space.

For the stage of relaxation, we use the explicit scheme

fn+1i,j,k;l = fn

i,j,k;l + τIi,j,k;l, (20)

where Ii,j,k;l is the grid approximation of the collision integral (1).

III.B. Results

Figure (5) shows the temperature profile calculated for the Couette flow with W = 1, with the collisionintegral being calculated without the above-described correction. This profile coincides with the temperatureprofile obtained previously by solving a similar problem by the Direct Simulation Monte Carlo (DSMC)method.7

1

1.02

1.04

1.06

1.08

1.1

1.12

0 0.2 0.4 0.6 0.8 1

Tem

pera

ture

x/L

T

Figure 5. Temperature (without correction), W=1

A comparison of the DSMC results8 with the calculations of the Couette flow with the parameter W = 0.1without correction of the collision integral reveals absolute failure of the latter (Fig. 6), though the velocityprofile is predicted correctly, as is seen in Fig. (7).

0.999

1

1.001

1.002

1.003

1.004

1.005

1.006

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Tem

pera

ture

x/L

T DSMCT

Figure 6. Temperature (without correction), W=0.1

The data calculated with the proposed correction of collision integral calculation are compared with theDSMC calculations8 in Figs. (8 - 11). It is seen that the results are in good agreement, being accuratewithin statistical noise.

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0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Vel

ocity

x/L

Vz DSMCVz

Figure 7. Velocity in the direction parallel to the plates (without correction), W=0.1

0.9996

0.9997

0.9998

0.9999

1

1.0001

1.0002

1.0003

1.0004

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Den

sity

x/L

Density DSMCDensity

Figure 8. Density, W=0.1

1.0003

1.0004

1.0005

1.0006

1.0007

1.0008

1.0009

1.001

1.0011

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Tem

pera

ture

x/L

T DSMCT

Figure 9. Temperature, W=0.1

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0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Vel

ocity

x/L

Vz DSMCVz

Figure 10. Velocity in the direction parallel to the plates, W=0.1

−0.0002

−0.00015

−0.0001

−5e−05

0

5e−05

0.0001

0.00015

0.0002

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Sx

x/L

Sx DSMCSx

Figure 11. Heat flux normal to the plates, W=0.1

IV. Conclusions

Based on test calculations, quadrature formulas with a small number of nodes on a sphere were selected.These formulas allow the computation time to be significantly (by an order of magnitude) reduced. A specialorganization of the collision integral calculation cycle is also proposed, where the collision parameters andthe coefficients of mass interpolation from internodal points to velocity grid nodes are calculated only forbase n3 pairs of nodes (here n is the grid dimension in the velocity space) rather than for all n3(n3 − 1/2)nodes. This trick also leads to substantial reduction of the calculation time and eliminates rounding errorsthat could appear in calculating identical collision parameters for similar pairs of nodes. The most importantfeature of the proposed algorithm of collision integral calculation is its correction, resulting in a zero value ofthe numerical collision integral as a function of the equilibrium grid distribution function. The test solutionsof the Boltzmann equation presented in this paper reveal the necessity of this correction for calculations ofslow flows. In what follows, we are going to present the results for rarefied gas flows calculated by the phasespace PIC method with the use of the collision integral calculation scheme described here. Let us recall thatthe phase space PIC method reduces the solution of the integrodifferential equation to the following systemof ordinary differential equations:

d−→R p

dt=

−→V p,

d−→V p

dt=

1

m

−→F p(t,

−→R p

−→V p), (21)

(−→Rp and

−→V p are the locations and velocities of particles representing the distribution function. The vector

field−→F is determined from the condition

∂−→F f

∂−→v = −I(−→v ) (22)

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where I(−→v ) is the collision integral (1).

V. ACKNOWLEDGMENTS

The authors are sincerely grateful to Aigul Baimussayeva for the help with numerical simulations.

This study was supported within fundamental research Program 11 of the Russian Academy of Sciencesand Interdisciplinary research project 26 of Siberian Branch of the Russian Academy of Sciences.

References

1E.A. Malkov, M.S. Ivanov, ”Particle-in-Cell method for solving the Boltzmann equation“, 27-th International Symposiumon Rarefied Gas Dynamics. AIP Conference Proceedings, Volume 1333, pp. 940-945 (2011).

2F.G. Tcheremissine, “Conservative method of calculating the Boltzmann collision integral,” Dokl. Ross. Akad. Nauk,Vol. 355, No.1, pp.53-56 (1995).

3Z. Tan and P.L. Varghese, “The ∆ − ǫ method for the Boltzmann equation, J. Comp. Phys., Vol.110, p.325 (1994).4P.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. 225-232(2005).

5C. Cercigniani, ”Mathematical methods in Kinetic theory”, McMillan, 1969, 242p.6A.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)7Shevyrin A.A., Bondar Ye.A., and Ivanov M.S. Analysis of Repeated Collisions in the DSMC Method // AIP Conference

Proceedings. 24th Symp. on Rarefied Gas Dynamics. Melville, New York, 2005. Vol. 562. P. 565- 550.8A.A. Shevyrin, Private communication (2011)

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