Chapter 4. Accurate numerical methods for the Boltzmann ... · Accurate numerical methods for the...

Chapter 4.

Accurate numerical methods for the

Boltzmann equation

Francis Filbet1 and Giovanni Russo2

1 Mathematiques et Applications, Physique Mathematique d’Orleans (MAPMO),CNRS-Universite d’Orleans, B.P. 6759, 45067 Orleans, [email protected]

2 Universita di Catania, Viale Andrea Doria 6 95125 Catania, [email protected]

Summary. We present accurate methods for the numerical solution of the Boltz-mann equation of rarefied gas. The methods are based on a time splitting technique.On the one hand, the transport is solved by a third order accurate (in space) Pos-itive and Flux conservative (PFC) method. On the other hand, the collision stepis treated by a Fourier approximation of the collision integral, which guaranteesspectral accuracy in velocity, coupled with high order integrators in time preservingstationary states. Several space dependent numerical tests in 2D and 3D illustratethe accuracy and robustness of the methods.

1 Introduction.

In a microscopic description of rarefied neutral gas, the gas particles move bya constant velocity until they undergo binary collisions. In a kinetic picture,the properties of the gas are described by a density function in phase space,f(t, x, v), called the distribution function, which gives the number of particlesper unit volume in phase space at time t. The distribution function satisfiesthe Boltzmann equation, an integro-differential equation, which describes theeffect of the free flow and binary collisions between the particles. In absenceof external forces the time evolution of a single component mono atomic gas,the Boltzmann equation reads to (Cf.[6, 27])

∂f

∂t+ v · ∇xf =

1

knQ(f, f), x, v ∈ Rd (1)

where d > 1 denotes the dimension of the velocity space, the number kn > 0is called Knudsen number and is proportional to the mean free path betweencollisions. In the right hand side, Q(f, f) is the so-called collision operatorgiven by

124 Francis Filbet and Giovanni Russo

Q(f, f)(v) = Q+(f, f) − L[f ]f (2)

with

Q+(f, f) =

∫

Rd

∫

Sd−1

B(|v − v∗|, θ)f(v′)f(v′∗) dω dv∗, (3)

L[f ] =

∫

Rd

∫

Sd−1

B(|v − v∗|, θ)f(v∗) dω dv∗. (4)

In the above integrals, v and v∗ are the velocities after the collision of two par-ticles which had the velocities v′ and v′∗ before the encounter. The deflectionangle θ is the angle between v − v∗ and v′ − v′∗.

Here the pre-collision velocities are parameterized by

v′ =1

2(v + v∗ + |v − v∗|ω), v′∗ =

1

2(v + v∗ − |v − v∗|ω), (5)

where ω is a unit vector of the sphere Sd−1.The quantities Q+(f, f) and L[f ]f are the gain and loss term, respectively.

The precise form of the kernel B, which characterizes the details of the binaryinteractions, depends on the physical properties of the gas. In the case ofinverse k-th power forces between particles, the kernel has the form

B(|v − v∗|, θ) = bα(θ)|v − v∗|α, (6)

where α = (k − 5)/(k − 1). In particular, we will consider the variable hardsphere (VHS) model [2] i.e. bα(θ) = Cα where Cα is a positive constant. Thecase α = 0 is referred to as Maxwellian gas whereas the case α = 1 yields theHard Sphere gas. Note that in the case of Maxwellian gas the coefficient ofthe loss term, L[f ], does not depend on v. Boltzmann’s collision operator hasthe fundamental properties of conserving mass, momentum and energy

∫

Rd

Q(f, f)

1v

|v|2

dv = 0, (7)

and satisfies the well-known Boltzmann’s H-theorem∫

Rd

Q(f, f) log(f)dv ≤ 0. (8)

Boltzmann H-theorem implies that any equilibrium distribution function, i.e.any function f for which Q(f, f) = 0, has the form of a locally Maxwelliandistribution

M(ρ, u, T )(v) =ρ

(2πT )d/2exp(−|u− v|2

2T), (9)

where ρ, u, T are the density, mean velocity and temperature of the gas

ρ =

∫

Rd

f(v)dv, u =1

ρ

∫

Rd

v f(v)dv, T =1

3ρ

∫

Rd

|u− v|2 f(v)dv. (10)

Accurate numerical methods for the Boltzmann equation 125

Among the different approaches for the approximation of the Boltzmannequation, we may distinguish between deterministic and Monte Carlo meth-ods. The first usually provide accurate oscillations-free solutions, but they aremuch more expensive than Monte Carlo methods with the same number ofdiscrete degrees of freedom. For example, if we denote by n the number of pa-rameters which characterize the density with respect to the velocity variablesin a space homogeneous calculation, the computational cost of a conventionaldeterministic method for the evaluation of the collisional integral is muchlarger than n2.

As a consequence most numerical computations are based on probabilisticMonte-Carlo techniques at different levels. Examples are the Direct Simula-tion Monte Carlo method (DSMC) by Bird [2] and the modified Monte Carlomethod by Nanbu and Babovsky [14, 1]. For a detailed description of suchmethods we refer to previous chapters of this book.

Probabilistic particle methods present different advantages: the computa-tional cost is strongly reduced and approximatively can be considered of theorder of the number of points n. Moreover, the computer memory require-ment is highly reduced, since the particles concentrate where the function isnot small, and memory is not wasted representing a function which is virtu-ally zero in most phase space. For these reasons, particle methods have nocompetitor for situations very far from thermodynamical equilibrium.

However, deterministic methods can be much more accurate, and can becompetitive with Monte Carlo methods for problems in which the solutionis not very far from thermodynamical equilibrium, and high accuracy is re-quired. In the framework of deterministic approximations, the most popularclass of methods is based on the so called discrete velocity models (DVM) ofthe Boltzmann equation. All these methods [4, 26, 12, 23] make use of regulardiscretizations on hypercubes in the velocity field and construct a discrete col-lision mechanics on the nodes of the hypercube in order to preserve the mainphysical properties. Although the numerical results have shown that theseschemes are able to avoid fluctuations, their computational cost is high (ingeneral O(nan

2), where na is the number of parameters used for the angularintegration, typically in such methods na ≈ O(n1/3)) and, due to the particu-lar choice of the integration points imposed by the conservation properties, theorder of accuracy is lower than that of a standard quadrature formula applieddirectly to the collision operator. Hence we observe that the requirement ofmaintaining at a discrete level the main physical properties of the continuousequation makes it extremely difficult to obtain high order accuracy. More-over, even if conservation properties are not imposed from the beginning, anaccurate scheme would provide an accurate approximation of the conservedquantities.

In [35], Pareschi and Perthame developed a discretization of the collisionoperator based on expanding in Fourier series the distribution function withrespect to the velocity variable. The resulting spectral approximation can beevaluated with a computational cost of O(n2) which is lower than that of


previous deterministic methods. Bobylev and Rjasanow [3] used a Fouriertransform approximation of the distribution function, and they were able toobtain exact conservation by a suitable modification of the evolution equationsfor the Fourier coefficients. The method proposed is second order accurate.On the other hand, Pareschi and Russo [20] developed a scheme based on theapproximation of the distribution function by a periodic function in phasespace, and its discretization by truncated Fourier series. Evolution equationsfor the Fourier modes are explicitly derived for the Variable Hard Sphere(VHS) model. The method provides spectral accuracy in the velocity domain,which is the highest accuracy achieved by a numerical method for the Boltz-mann equation, and the computational complexity of the collisional operatoris O(n2). The method preserves mass, and approximates with spectral accu-racy momentum and energy. For a more detailed description of the spectralapproach to the Boltzmann equation and to other kinetic equations see forexample [17].

Here, we are interested in the construction of an accurate method for thespace non homogeneous Boltzmann equation [10]. The discretization of thetransport step has to be done carefully because it induces physical oscilla-tions in the velocity space. In this chapter we construct a fractional stepdeterministic scheme for the time dependent Boltzmann equation, which isbased on five main ingredients

Fractional step in time allows to treat separately the transport and the colli-sion.

Fourier-Spectral method for the evolution of the collision step allows a veryaccurate discretization in velocity domain, at a reasonable computationalcost [20].

Positive and Flux Conservative (PFC) finite volume method for the free trans-port [8] provides a third order (in space) accurate scheme for the evolutionof distribution function during the transport step. The scheme is conser-vative, and preserves positivity. It is much less dissipative than Essen-tially Non Oscillatory (ENO) and Weighted Essentially Non Oscillatory(WENO) schemes usually used for hyperbolic systems of conservation laws[9, 24]. We also refer to [7] for the implementation of different boundaryconditions.

Positive time discretization. A suitable time discretization of the collisionalequation is used, which allows a large stability time step, even for problemswith considerably small Knudsen number. The time discretization methodfor the collision step is based on a modified Time Relaxed scheme [21].

Multiple resolution. A different resolution will be used in velocity space in thetransport and in the collision step. Considering that the collision step ismore expensive, and more accurate (spectral accuracy) than the transportstep, it is convenient to use more points in velocity space during thetransport step.


In the next section we give a general setup to solve kinetic equations innon homogeneous situations. Then, we describe the PFC method for the freetransport and the spectral method for the evolution of the collision step.Several numerical issues are discussed and time dependent and stationaryproblems are proposed. Finally, in the last section we draw conclusions.

2 The general framework.

Let us consider the initial-boundary value problem for the Boltzmann trans-port equation

∂f

∂t+ v · ∇xf =

1

knQ(f, f) (11)

f(0, x, v) = f0(x, v)

where x ∈ Ω ⊂ Rd, v ∈ Rd, t ∈ [0, T ]. Boundary conditions will be specifiedin the section on numerical results and we refer to [7] for their implementa-tion. We discretize time into discrete values tn, and we denote by fn(x, v) anapproximation of the distribution function f(tn, x, v). As it is usually donefor a kinetic equation like (11), a simple first order time splitting is obtainedconsidering, in a small time interval ∆t = [tn, tn+1], the numerical solution ofthe transport step

∂f∗

∂t+ v · ∇xf

∗ = 0,

f∗(0, x, v) = fn(x, v),(12)

and the space homogeneous collision step

∂f∗∗

∂t=

1

knQ(f∗∗, f∗∗),

f∗∗(0, x, v) = f∗(∆t, x, v),(13)

We shall denote by S1(∆t) and S2(∆t) the solution operators correspondingrespectively to the transport and collision step, i.e. we can write

f∗(∆t, x, v) = S1(∆t)fn(x, v),

f∗∗(∆t, x, v) = S2(∆t)f∗(∆t, x, v).

The approximated value at time tn+1 is then given by

fn+1(x, v) = f∗∗(∆t, x, v) = S2(∆t)S1(∆t)fn(x, v). (14)

We assume that S1 and S2 represent either exact or at least second orderevolution operators in time of transport and collision step, respectively.

A second order scheme for non stiff problems can be easily derived simplyby symmetrizing the first order scheme [48]


fn+1 = S1(∆t/2)S2(∆t)S1(∆t/2)fn, (15)

provided every step is solved with a method at least second order accurate intime [16]. Although higher order splitting strategies are available, in practicethey are seldomly used because of stability problems. We remind that secondorder accuracy for such complex problems is considered “high order” in thisfield.

In the next two sections we discuss transport and collision steps. As weshall see, the grid step size in time, space and velocity are not directly relatedby strict stability requirements, and therefore one can benefit from high orderaccuracy whenever possible.

3 Discretization of the transport step.

In this section, we discuss the numerical resolution of the Vlasov equationwhich characterizes the transport step (12)

∂tf + ∇x (v f) = 0, ∀(t, x) ∈ R+ × Rd. (16)

Then, the solution of the transport equation at time tn+1 reads

f(tn+1, x) = f(tn, x− v ∆t), ∀x ∈ Rd.

For simplicity, let us restrict ourselves to a one dimensional problem. We intro-duce a uniform mesh, characterized by a finite set of mesh points xi+1/2i∈Ion the computational domain. We will use the notation ∆x = xi+1/2−xi−1/2,Ci = [xi−1/2, xi+1/2] and xi the center of Ci. Assuming the values of the dis-tribution function are known at time tn = n∆t on cells Ci, we compute thenew values at time tn+1 by integration of the distribution function on eachsub-interval. Thus, using the explicit expression of the solution, we have

∫ xi+1/2

xi−1/2

f(tn+1, x)dx =

∫ xi+1/2−v ∆t

xi−1/2−v ∆tf(tn, x)dx,

then, setting

Φi+1/2(tn) =

∫ xi+1/2

xi+1/2−v ∆tf(tn, x)dx,

we obtain the conservative form∫ xi+1/2

xi−1/2

f(tn+1, x)dx =

∫ xi+1/2

xi−1/2

f(tn, x)dx + Φi−1/2(tn) − Φi+1/2(t

n). (17)

The evaluation of the average of the solution over [xi−1/2, xi+1/2] allows toignore fine details of the exact solution which may be costly to compute. Themain step is now to choose an efficient method to reconstruct the distribution


function from the cell average on each cell Ci. We will consider a reconstructionvia primitive function preserving positivity and maximum values of f [8]. LetF (tn, x) be a primitive of the distribution function f(tn, x), if we denote by

fni =1

∆x

∫ xi+1/2

xi−1/2

f(tn, x)dx,

then F (tn, xi+1/2) − F (tn, xi−1/2) = ∆xfni and

F (tn, xi+1/2) = ∆x

i∑

k=0

fnk =: wni .

First we construct an approximation of the primitive on the small interval[xi−1/2, xi+1/2] using the stencil xi−3/2, xi−1/2, xi+1/2, xi+3/2

Fh(tn, x) = wni−1 + (x− xi−1/2)f

ni +

1

2∆x(x− xi−1/2)(x− xi+1/2)[f

ni+1 − fni ]

+1

6∆x2(x− xi−1/2)(x− xi+1/2)(x− xi+3/2)[f

ni+1 − 2 fni + fni−1],

where we use the relation wni − wni−1 = ∆xfni . Thus, by differentiation, weobtain a third order accurate approximation of the distribution function onthe interval [xi−1/2, xi+1/2]

fh(tn, x) =

∂ Fh∂x

(tn, x) = fni +

+1

6∆x2

[2 (x− xi)(x− xi−3/2) + (x− xi−1/2)(x− xi+1/2)

](fni+1 − fni )

− 1

6∆x2

[2 (x− xi)(x− xi+3/2) + (x− xi−1/2)(x− xi+1/2)

](fni − fni−1).

Unfortunately, this approximation does not preserve positivity of the distri-bution function f . Then, in order to satisfy a maximum principle and to avoidspurious oscillations we introduce slope correctors

fh(tn, x) = fni + (18)

+ε+i

6∆x2

[2 (x− xi)(x− xi−3/2) + (x− xi−1/2)(x− xi+1/2)

](fni+1 − fni )

− ε−i6∆x2

[2 (x− xi)(x− xi+3/2) + (x− xi−1/2)(x− xi+1/2)

](fni − fni−1),

with

ε±i =

min(1; 2 fni /(f

ni±1 − fni )

)if fni±1 − fni > 0,

min(1;−2 (f∞ − fni )/(fni±1 − fni )

)if fni±1 − fni < 0,

(19)


where f∞ = maxj∈I

fnj is a local maximum.

The theoretical properties of this reconstruction can be summarized bythe following

Proposition 1. The approximation of the distribution function fh(x), definedby (18)-(19), satisfies

• The conservation of the average: for all i ∈ I,∫ xi+1/2

xi−1/2fh(x)dx = ∆xfi.

• The maximum principle: for all x ∈ (xmin, xmax), 0 ≤ fh(x) ≤ f∞.

Moreover, if we assume the Total Variation of the distribution function f(x)is bounded, then we obtain the global estimate:

∫ xmax

xmin

|fh(x) − fh(x)| dx ≤ 4TV (f)∆x,

where fh denotes the third order approximation of f without slope corrector.

Proof. Let us consider x ∈ Ci = [xi−1/2, xi+1/2] and denote by

α(x) =1

∆x2

[2 (x− xi)(x− xi−3/2) + (x− xi−1/2)(x− xi+1/2)

],

β(x) =1

∆x2

[2 (x− xi)(x− xi+3/2) + (x− xi−1/2)(x− xi+1/2)

].

It is easy to check that

∫ xi+1/2

xi−1/2

α(x)dx =

∫ xi+1/2

xi−1/2

β(x)dx = 0,

then the conservation of the average immediately follows. To obtain the preser-vation of positivity, assuming the values fj are positive, we observe that inthe cell Ci, the function α(x) is increasing whereas β(x) decreases and α(x),β(x) ∈ [−1, 2]. Then, we split fh(x) as the sum of h(x) and g(x) with

h(x) =1

3

[fi +

α(x)

2ε+i (fi+1 − fi)

], g(x) =

1

3

[2 fi −

β(x)

2ε−i (fi − fi−1)

].

The function h(x) ( resp. g(x) ) is only a combination of fi and fi+1 ( resp.fi−1 and fi ), then from the value of ε+i ( resp. ε−i ), it is easy to prove thath(x) ( resp. g(x) ) is positive. Using a similar decomposition, we also provethat fh(x) is bounded by f∞.Now, we prove the global estimate on the positive reconstruction:


∫ xmax

xmin

|fh(x) − fh(x)|dx

=∑

i

∫ xi+1/2

xi−1/2

| α(x) (1 − ε+i )[fi+1 − fi] + β(x) (1 − ε−i )[fi − fi−1] | dx

≤ 2∆x∑

i

(1 − ε+i )|fi+1 − fi| + 2∆x∑

i

(1 − ε−i )|fi − fi−1|

≤ 4∆x∑

i

|fi+1 − fi| ≤ 4∆xTV (f).

¤

Remark 1. If the solution is smooth, we can check numerically that the schemeis third order. But, the numerical analysis of such a nonlinear scheme is reallydifficult to perform.

4 Spectral approximation of the collision operator.

We consider now the space homogeneous Boltzmann equation in each cell,

∂f

∂t= Q+(f, f) − L[f ]f (20)

with Q+ and L given by equations (3) and (4). To keep notation simple, wehave fixed kn = 1. A simple change of variables permits to write

Q+(f, f) =

∫

Rd

∫

Sd−1

B(|g|, θ)f(v′)f(v′∗) dω dg, (21)

L(f) =

∫

Rd

∫

Sd−1

B(|g|, θ)f(v − g) dω dg, (22)

where g = v − v∗ and then

v′ = v − 1

2(g − |g|ω), v′∗ = v − 1

2(g + |g|ω). (23)

First, from the conservation of the momentum and the total energy, (v′∗)2 +

(v′)2 = v2∗ + v2, we get the following result [35]:

Lemma 1. Let Supp(f(v)) ⊂ B(0, R) then

i) Supp(Q(f, f)(v)) ⊂ B(0,√

2R),ii)

Q(f, f)(v) =

∫

B(0,2R)

∫

Sd−1

B(|g|, θ)[f(v′)f(v′1) − f(v)f(v − g)] dω dg,

with v′, v′∗, v − g ∈ B(0, (2 +√

2)R).


2)R(2+

0-

R

V V 3V2V

Fig. 1. Restriction of the distribution function on the periodic box [−π, π]× [−π, π]in two dimensions.

As a consequence of the above lemma, in order to write a spectral approx-imation to (20) we can consider the distribution function f(v) restricted on[−V, V ]d with V ≥ (2 +

√2)R), assuming f(v) = 0 on [−V, V ]d \ B(0, R), and

extend it by periodicity to a periodic function on [−V, V ]d. In view of Fig. 1,the shortest period can be restricted to [−V, V ], with V ≥ (3 +

√2)R/2.

If the distribution function is well approximated by a function of com-pact support in velocity space, then the above approximation will provide anaccurate evaluation of the collision integral.

To simplify the notation let us take V = π and hence R = λπ with λ =2/(3 +

√2). Hereafter, we used just one index to denote the tree-dimensional

sums with respect to the vector k = (k1, .., kd) ∈ Zd, hence we set

N∑

k=−N=

N∑

k1,..,kd=−N.

The approximate function fN is represented as the truncated Fourier series

fN (v) =N∑

k=−Nfke

ik·v, (24)

fk =1

(2π)d

∫

[−π,π]df(v)e−ik·v dv.

In a Fourier-Galerkin method the fundamental unknowns are the coefficientsfk, k = −N, . . . , N . We obtain a set of ODEs for the coefficients fk by requir-ing that the residual of (20) be orthogonal to all trigonometric polynomialsof degree ≤ N . Hence for k = −N, . . . , N

∫

[−π,π]d

(∂fN∂t

+ fN L(fN ) −Q+(fN , fN )

)e−ik·v dv = 0. (25)


By substituting expression (24) in (22) and (21) we get respectively

fN L(fN ) =

N∑

l=−N

N∑

m=−Nfl fmB(m,m)ei(l+m)·v,

and

Q+(fN , fN ) =

N∑

l=−N

N∑

m=−Nfl fmB(l,m)ei(l+m)·v,

where the kernel modes B(l,m) are given by

B(l,m) =

∫

B(0,2λπ)

∫

Sd−1

B(|g|, θ)e−ig· (l+m)2 −i|g|ω· (m−l)

2 dω dg. (26)

Using the orthogonality property we get from (25)

∂fk∂t

=

N∑

m=k−Nfk−m fm(B(k −m,m) − B(m,m)), (27)

with the initial condition

fk(0) =1

(2π)d

∫

[−π,π]df0(v)e

−ik·v dv. (28)

The evaluation of the right hand side of (71) requires exactly O(N 2d) opera-tions. We emphasize that the usual cost for a method based on N d parametersfor f in the velocity space is O(N 2dM) where M is the numbers of angle dis-cretizations. The loss term on the right hand side is a convolution sum andthus transform methods allow this term to be evaluated only in O(N d log N)operations. Hence the most expensive part of the computation is representedby the gain term.

4.1 Analysis of the kernel modes.

In this section we study the main characteristics of the kernel modes and inparticular we give an explicit representation of them for the VHS model.

Let us start from equation (26). In the VHS model, the kernel does notdepend on the angle θ: B = Cα|g|α. One has

B(l,m) = Cα

∫

B(0,2λπ)

|g|α exp

(−ig · l +m

2

)I2(|g|, l −m), dg (29)

where

I2(|g|, l −m) =

∫

Sd−1

exp

(i|g|ω · l −m

2

)dω. (30)

We shall consider separately 3D and 2D collision model. The 3D case is theimportant one for practical application. However, a two dimensional collisionalmodel will be considered for test problems.


2D case.

For the computation in 2D we start from (29) and (30). We will consider onlythe VHS model.

In this case it is

I2 =

∫

S

exp(i|q| · ω) dω =

∫ 2π

0

exp(ir cos θ) dθ

= 2

∫ π

0

cos(r cos θ) dθ = 2πJ0(r), (31)

where r = |q| = |g||l − m|/2, and J0 is the Bessel function of order 0. Byinserting the result in the expression (29) for B(l,m), one obtains [20]

B(l,m) = Cα2π

∫

B(0,2λπ)

|g|α exp(−ig · (l +m)/2)J0(|l −m||g|/2) dg.

Making use of polar coordinates, the expression for the coefficients becomes

B(l,m) = Cα2π

∫ 2πλ

0

ρ1+α

(∫ 2π

0

cos(|l +m|ρ/2) cos θ dθ

)J0(|l −m|ρ/2) dρ

= Cα4π2

∫ 2πλ

0

ρ1+αJ0(|l +m|ρ/2)J0(|l −m|ρ/2) dρ

= Cα4π2(2πλ)2+α∫ 1

0

r1+αJ0(ξr)J0(ηr) dr (32)

where ξ = |l +m|λπ, η = |l −m|λπ. Taking now Cα = (4π2(2πλ)2+α)−1, theexpression of B(l,m) becomes

B(l,m) = Fα(ξ, η)

with

Fα(ξ, η) =

∫ 1

0

r1+αJ0(ξr)J0(ηr) dr. (33)

From (33) it is easy to prove that an analogue of proposition 2 holds also inthe two dimensional case. Note that also in this case each kernel mode can becomputed as a 1-D integral and stored in an array.

3D case.

Let q = |g|(l −m)/2. Then I2 is computed as follows

I2(|g|, l −m) =

∫

S2

eiq·ω dω = 2π

∫ π

0

ei|q| cos θ sin θ dθ

= 2π

∫ 1

−1

ei|q|µ dµ = 4π Sinc(|q|)

= 4π Sinc

( |g||l −m|2

)(34)


where

Sinc(x) ≡ sinx

x.

Let p = (l+m)/2. Then, taking into account the previous result, one obtains[20]

B(l,m) = Cα4π

∫

B(0,2λπ)

|g|α Sinc(|l −m||g|/2) exp(−ip · g) dg

Making use of spherical coordinates, with ρ = |g|, one has

B(l,m) = Cα8π2

∫ 2πλ

0

ρ2+α Sinc(|l −m|ρ/2) dρ∫ π

0

exp(−i|p|ρ cos θ) sin θ dθ

= Cα16π2

∫ 2πλ

0

ρ2+α Sinc(|l −m|ρ/2) Sinc(|l +m|ρ/2) dρ. (35)

With the change of variables ρ = 2λπr the coefficient B(l,m) can be writtenas

B(l,m) = Cα16π2(2λπ)3+α∫ 1

0

r2+α Sinc(ξr) Sinc(ηr) dr

where ξ = |l+m|λπ, η = |l−m|λπ. To simplify notations let us assume that

Cα = (16π2(2λπ)3+α)−1.

In this case the coefficient can be written as

B(l,m) = Fα(ξ, η)

where

Fα(ξ, η) =

∫ 1

0

r2+α Sinc(ξr) Sinc(ηr) dr. (36)

From (36) it is easy to prove the following

Proposition 2. Let Fα(ξ, η) be defined by (36) then

i) Fα(ξ, η) = Fα(η, ξ),ii) if α > −3 then |Fα(ξ, η)| ≤ Fα(0, 0) = (3 + α)−1,iii) if α > −1 then |Fα(ξ, η)| ≤ [ξη(1 + α)]−1.

For integer values of α, Fα has an explicit analytical expression. We givehere the expressions for α = 0 (Maxwellian gas) and α = 1 (Hard Sphere gas)

F0(ξ, η) =p sin(q) − q sin(p)

2ξηpq(37)

F1(ξ, η) =p2(q sin(q) + cos(q)) − q2(p sin(p) + cos(p)) − 4ξη

2ξηp2q2(38)

where p = (ξ + η), q = (ξ − η).


Storage of Fourier coefficients.

Note that, since the five-fold integral (26) which defines the B(l,m) has beenreduced to a one-dimensional integral (36), for non-integer values of α, thevalue of the coefficients can be easily computed numerically by an accuratequadrature formula, and stored in an array at the beginning of the calculation.

In 3D, the storage of coefficients B(l,m) is of order O(n6), where n isthe number of half modes for each direction. But, it can be easily reduced toO(n4). Indeed, the matrix B(l,m) only depends on |k| = |l+m| and |l−m|,it is then replaced by the smaller matrix B(i, j), where integers i, j are givenby

0 ≤ i = |k|2 ≤ 3n2, 0 ≤ j = |l −m|2 ≤ 12n2.

4.2 Properties of the spectral method.

We state here the main theoretical results of the Fourier-Spectral method,concerning consistency and spectral accuracy. For any function f(v), let fN (v)denote the truncated Fourier series of f , and let PN : L2([−π, π]d) → IPN

denote the projection operator, with

IPN = spaneik·v | −N ≤ kj ≤ N, j = 1, .., d

.

Then the following results hold [20]

Proposition 3. Let f ∈ L2([−π, π]d), f ≥ 0, ∀ v ∈ [−π, π]d, and let us define

ρρuρe

:=

∫

[π,π]df

1v

|v|2

dv. (39)

and let us denote by ρN , ρuN , and ρeN the moments of fN , then the followingrelations hold

ρ = ρN ,

|ρu− ρuN | ≤ C1

N||f ||2,

|ρe− ρeN | ≤ C2

N2||f ||2.

The estimates given above can be strongly improved if f is smooth. Iff ∈ Hr

p([−π, π]d), where r ≥ 0 is an integer and Hrp([−π, π]d) is the subspace

of the Sobolev space Hr([−π, π]d), which consists of periodic functions [5], foreach ϕ ∈ L2([−π, π]d) we have

| < f,ϕ > − < f,ϕN > | ≤ ||ϕ||2 ||f − fN ||2 ≤ C

Nr||ϕ||2||f ||Hr

p,

where || · ||Hrp

denotes the norm in Hrp([−π, π]d), and < f, g > denotes the

scalar product in L2p. This inequality shows that the projection error on


the moments decay faster than algebraically when the solution is infinitelysmooth.

We state a consistency result in the L2-norm for the approximation of thecollision operator Q(f, f) with QN (fN , fN ) [20, 43],

Theorem 1. Let f ∈ L2([−π, π]d), and B(u, θ) = Cα|u|α, with α > 0 then

||Q(f, f) −QN (fN , fN )||2 ≤ C

(||f − fN ||2 +

||Q(fN , fN )||Hrp

Nr

), ∀ r ≥ 0,

(40)where C depends on ||f ||2.

Proof. First, let us split the error in two parts

||Q(f, f) −QN (fN , fN )||2 ≤ ||Q(f, f) −Q(fN , fN )||2+ ||Q(fN , fN ) −QN (fN , fN )||2

On the one hand, observing that Q(fN , fN ) ∈ IP 2N and hence Q(fN , fN ) isperiodic and infinitely smooth [5]

||Q(fN , fN ) −QN (fN , fN )||2 ≤ CrNr

||Q(fN , fN )||2, ∀r ≥ 0. (41)

On the other hand using the symmetry of the Boltzmann operator, we get

Q(f, f) −Q(fN , fN ) = Q(f + fN , f − fN ). (42)

Now, let us prove that

||Q(f + fN , f − fN )||2 ≤ C||f + fN ||1 ||f − fN ||2. (43)

We use a duality argument : for each function ϕ ∈ L2([−π, π]d), we get formthe Holder inequality∣∣∣∣∫Q(f, g)ϕdv

∣∣∣∣ = Cα

∣∣∣∣∫ ∫

|v − v∗|αg(v∗) f(v) (ϕ(v′) − ϕ(v)) dωdv?dv

∣∣∣∣

≤ Cα||g||1 supv∗∈[−π,π]d

∫|v − v∗|αf(v)

∣∣∣∣∫

Sd−1

(ϕ(v′) − ϕ(v))dω

∣∣∣∣ dv.

Let us fix v?, then from the Cauchy-Schwartz inequality we obtain∫

|v − v∗|αf(v)

∣∣∣∣∫

Sd−1

(ϕ(v′) − ϕ(v))dω

∣∣∣∣ dv

≤ ||f ||2∥∥∥∥|v − v∗|α

∫

Sd−1

(ϕ(v′) − ϕ(v))dω

∥∥∥∥2

.

Using the invariance by translation, it is enough to prove this estimate forv∗ = 0. Moreover, the function v → |v|α defined in [−π, π]d is bounded, andthus there exists a constant C, independant of N , such that


∥∥∥∥|v|α∫

Sd−1

(ϕ(v′) − ϕ(v))dω

∥∥∥∥L2

≤ C ||ϕ||2. (44)

Finally, we have shown that there exists a constant C, independant of N ,such that for each smooth function ϕ

∣∣∣∣∫Q(f, g)ϕdv

∣∣∣∣ ≤ C ||g||L1 ||f ||2||ϕ||2,

which proves the inequality (43) with g = f + fN and f = f − fN . Gatheringinequalities (41) and (43), we conclude the proof. ¤

The previous estimate states that the rate of convergence in the L2-normof QN (fN ) to Q(f) depends only on the speed of convergence of fN to f .Hence if fN is spectrally accurate so it is QN (fN ). The following corollarystates the spectral accuracy of the approximation of the collision operator

Corollary 1. Let f ∈ Hr([−π, π]d), r ≥ 0 then

||Q(f) −QN (fN )||2 ≤ C

Nr(||f ||Hr + ||Q(fN )||Hr ) . (45)

4.3 Time discretization of the collision operator.

Here we focus on the time evolution of the collision step. Let ∆t denote thetime step of the transport phase. The goal is to solve, in each cell, the spacehomogeneous Boltzmann equation

∂f∗

∂t=

1

knQ(f∗, f∗)

f∗(0, v) = fn(v)

where, for simplicity, we drop the space dependence. One could use any secondorder time discretization, such as a Runge-Kutta method, with the same timestep, ∆t, used for the convection step, for the ordinary differential system ofthe Fourier modes, (71). If the time step is too large (for accuracy or stabilityreasons), then a smaller time step, ∆tc < ∆t, can be used during this phase.Since each cell is independent, ∆tc may depend on the cell. If ∆tc << ∆t,then a multi-step scheme can be used to improve efficiency and accuracy.With standard methods such as Runge-Kutta or multi-step, it is difficult tocontrol positivity of the solution. Here we propose a time discretization whichprovides essential positivity of the distribution function, and allows the use ofrather large time steps, even in regimes in which the Knudsen number is quitesmall. The schemes that we use are based on a variation of time relaxed (TR)schemes [21], which have been effectively used in the development of MonteCarlo methods suitable for a very wide range of Knudsen number, and for the


space non-homogeneous Boltzmann equation [10]. We briefly recall here theidea behind the TR schemes.

Let us consider an equation of the form

∂f

∂t=

1

kn[P (f, f) − µf ] , (46)

f(0, v) = f0(v),

where µ 6= 0 is a constant and P a positive bilinear operator.The Boltzmann equation for Maxwell molecules has the above form, with

Q+(f, f) = P (f, f), and L[f ] = µ.Let us replace the time variable t and the function f = f(t, v) using the

equationsτ = (1 − e−µt/kn), F (τ, v) = f(t, v)eµt/kn . (47)

Then F is easily shown to satisfy

∂F

∂τ=

1

µP (F, F ), (48)

with F (τ = 0, v) = f0(v).Now, the solution to the Cauchy problem for (48) can be sought in the

form of a power series

F (τ, v) =

∞∑

k=0

τkfk(v), fk=0(v) = f0(v), (49)

where the functions fk are given by the recurrence formula

fk+1(v) =1

k + 1

k∑

h=0

1

µP (fh, fk−h), k = 0, 1, . . . (50)

Making use of the original variables we obtain the following formal represen-tation of the solution to the Cauchy problem (20), called Wild sum expansion[29].

f(t, v) = e−µt/kn

∞∑

k=0

(1 − e−µt/kn

)kfk(v). (51)

The coefficients fk have the property that

limk→∞

fk(v) = M(v), (52)

where M(v) is the Maxwellian, satisfying

Q(M,M) = 0.

Representation (51) and property (52) suggest the use of a truncation of series(51) as a numerical scheme for time discretization, namely


fn+1(v) = (1 − τ)

m∑

k=0

τkfk(v) + τm+1M(v), (53)

with fk(v) computed from fn(v). Such scheme is of order m in µ∆t/kn, andhas the following properties [21]

Proposition 4. The Time-Relaxed scheme given by (53) satisfies

i) conservation: if P (f, g) is a non negative bilinear operator such that thereexist some functions φ(v) with the following property

∫

Rd

P (f, f)φ(v) dv = µ

∫

Rd

fφ(v) dv, (54)

and the initial condition f0 is a non negative function, then fn+1 is non-negative for any µ∆t/kn, and satisfies

∫

Rd

fn+1φ(v) dv =

∫

Rd

fnφ(v) dv; (55)

ii) asymptotic preservation (AP):for any m ≥ 1, we have

limµ∆t/kn→∞

fn+1 = M(v). (56)

Proof. The result is straightforward using the construction of the scheme.¤

The above time discretization can be generalized using different weight func-

tions to combine the influence of the high order coefficients appearing in theWild sum (51). In general such schemes can be written as

fn+1 =

m∑

k=0

Akfk +Am+1M, (57)

where the coefficients fk are given by (50) using f = fn(v). The weightsAk = Ak(τ) are nonnegative functions that satisfy some consistency condition.

Proposition 5. If Ak(τ) ≥ 0 satisfy

i) consistency:

limτ→0

A1(τ)/τ = 1, limτ→0

Ak(τ)/τ = 0, k = 2, . . . ,m+ 1, (58)

ii) conservation:m+1∑

k=0

Ak = 1 τ ∈ [0, 1], (59)


iii) asymptotic preservation (AP):

limτ→1

Ak(τ) = 0, k = 0, . . . ,m, (60)

then (57) is a consistent discretization of problem (46) that satisfies proposi-tion 4.

A choice of functions which satisfies the previous requirements is given by

Ak = (1 − τ)τk, k = 0, . . . ,m, Am+1 = τm+1, (61)

which correspond to the scheme (53). A better choice of parameters is [19]

Ak = (1−τ)τk, k = 0, . . . ,m−1, Am = 1−m∑

k=0

Ak−Am+1, Am+1 = τm+2,

(62)which corresponds to take fm+1 = fm, fk = M , k ≥ m+ 2 in (51). However,other choices are possible and it is an open problem the determination ofthe optimal set of functions Ak that satisfies the previous requirements andguarantees the most accurate approximation.

The Boltzmann equation for Maxwell molecules has the form (46), withP (f, f) = Q+(f, f). In order to apply the same discretization to a more generalB.E., we can proceed as follows. We write the Boltzmann equation in the form(46), with

P (f, g) = Q+(f, g) +1

2(µ(f + g) − L[f ]g − L[g]f) ,

where the operator P is written in a symmetric form. If we choose

µ ≥ L[f ](v) ∀v ∈ Rd, (63)

then P (f, f) is a positive symmetric operator. However, in general L(v) isan unbounded function, and therefore a constant µ satisfying (63) does notexist. Even if we consider that the discrete velocities lie in a bounded domainΩv = [−V, V ]3, a choice of µ satisfying (63) may lead to excessive numericalviscosity, as is evident from standard truncation analysis.

Ideally, one should choose the smallest value of µ that guarantees positivityof operator P . This should be obtained, for a given function f(v), by imposingthat

minv∈Ωv

[Q+(v) + (µ− L(v))f(v)] = 0. (64)

A first order (non AP) TR scheme has the structure

fn+1(v) = A0(τ)fn(v) +A1(τ)f1(v),

with f1 = P (fn, fn)/µ. Because of positivity of the coefficients A0(τ) andA1(τ), if P (f, f) is positive, then the scheme is positive.


Condition (64) is not practical, since the region of phase space Ωv nearthe edge is not physically representative, because of the approximation of thedistribution function by a periodic function in velocity. A better choice isobtained by computing a critical constant µc as

µc = maxv∈Ωc

(L(v) − Q+

f

), (65)

where Ωc ⊂ Ωv is a smaller region, for example, Ωc = [−V/2, V/2]3. Then theconstant µ is computed as µ = Cµc, where C is a safety factor of order one.In all the calculations we used the value C = 3/2, which we found a goodcompromise between numerical positivity and numerical viscosity.

These practical criteria deserve further analysis. We have to keep in mindthat the spectral scheme itself does not preserve positivity rigorously [20].However, the lack of positivity is very small, and can be neglected for allpractical purposes. Even if positive spectral scheme can be obtained, as shownin [20] and [19], they are not practical, because of the lack of accuracy andexcessive smoothing.

Accuracy requires a small value of µ∆t/kn. If µ is kept constant, indepen-dently of kn, then the time step becomes exceedingly small for small valuesof kn, and the method becomes inefficient. The approach that we outlinedabove allows rather large time steps, even for small values of kn. The reasonfor this is that when kn is small, then gain and loss terms balance each other,and therefore the quantity µ computed as above becomes small. It appears infact that for small values of kn, µ scales with kn, and the ratio µ/kn remainsbounded.

With these considerations in mind, we maintain the same criterion for theevaluation of the optimal µ, even for higher order TR schemes.

A second order (non AP) TR scheme has the structure

fn+1(v) = A0(τ)fm(v) +A1(τ)f1(v) +A2(τ)f2(v),

with f2(v) = P (fn, f1). The constant µ is computed as above, µ = 3/2µc,with µc given by (65). For all practical purpose the scheme can be consideredpositive, although it is not rigorously positive.

To conclude this section, let us mention that in [10], we present two algo-rithms to reduce the computational cost and to improve accuracy. On the onehand, a parallel algorithm based on the fractional step approach is given. Onthe other hand, a simple multi-resolution is proposed to solve the transportand collision steps using different grids.

5 Numerical tests.

In this section, we present test cases showing the effectiveness of the spectral-PFC method to get an accurate solution of the Boltzmann equation. We first


give two simple numerical tests in the 3D homogeneous case (without x) inorder to illustrate the spectral accuracy of the method. In space dependenttests we used 2D and 3D models of the Boltzmann equation in velocity spaceand present results to compare the scheme with the well known Monte-Carlomethod for the Boltzmann equation. We refer to [10] for more numericalresults (Riemann problem and stationary shock waves for a model Boltzmannequation 2D in velocity). Finally, we present a comparison with the MonteCarlo method for the evalutation of a stationary shock.

5.1 3D space homogeneous case: spectral accuracy.

We consider 3D Maxwellian molecules (i.e. α = 0), with C0 = 1/(4π). Thisproblem has an exact solution given by

f(t, v) =exp(−v2/2S)

2S (2πσ)3/2

[5S − 3 +

1 − S

Sv2

],

where S = 1− exp(−(t+ t0)/6), t ≥ 0, t0 = 5.6 > 6 log(5/2). This test is usedto check spectral accuracy, by comparing the error at a given time, when usingn = 8, 16, and 32 Fourier modes for each dimension, to check the accuracy inthe conservation of energy and to observe the evolution of the fourth moment.Because of the symmetry of the problem, the moments of order 1 and 3 areconserved, within round-off error. In figure 2 we report the L1 relative errorvs time, for different number of modes, and the fourth order moment.

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 1 2 3 4 5 6 7 8

N = 8x8x8N = 16x16x16N = 32x32x32

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

0 5 10 15 20 25 30

4th order moment of f(t,v)

Fig. 2. 3D homogeneous case I: evolution of the numerical L1 relative error and thefourth order moment of f(t, v).


5.2 3D space homogeneous case: convergence to equilibrium.

We present a first result for the 3D Boltzmann equation without the transportpart and consider Maxwellian (α = 0) and hard-sphere (α = 1) molecules,with Cα = 1/(2π). The initial condition is sum of two Gaussians

f(v, 0) =1

2(2πσ2)3/2

[exp

(−|v − v1|2

2σ2

)+ exp

(−|v − v2|2

2σ2

)],

with σ2 = 0.2, v1 = (1, 1, 1/4), v2 = (−1,−1,−1/4) and the final time istmax = 2. This test is used to check the evolution of the distribution functionand to observe the relaxation to equilibrium.

We first define the directional temperature

Tα(t) =1

ρ

∫

R3

(vα − uα)2f(t, v)dv, α ∈ x, y, z

where ρ and u are given by (53) and the entropy

H(t) =

∫

R3

f(t, v) log(f(t, v))dv.

In Fig. 3, the relaxation of Tα(t), with α ∈ x, y, z and H(t) for Hard-sphereand Maxwellian molecules are presented starting from the same initial data.Finally, the evolution of the distribution function f is given in Fig. 4.

5.3 Riemann problem: time dependent solutions.

This test deals with the numerical solution of the non homogeneous 1D× 2DBoltzmann equation for hard sphere molecules (α = 1). We present someresults for one dimensional Riemann problem and compare them with thenumerical solution obtained by the Monte-Carlo scheme. Let us note that theaccuracy of the Monte Carlo solution is improved by performing averages ofthe solution itself by repeating the calculation several times with differentseeds in the random number generator, and averaging the solution over thedifferent runs. Then, we have computed an approximation for different Knud-sen numbers, from rarefied regime up to the fluid limit. The solution in thehydrodynamic limit is also compared with the numerical solution of Euler sys-tem, which is obtained by Nessyahu-Tadmor scheme [15] using a large numberof points (nx=1600). The initial data is given by

(ρl, ul, Tl) = (1, 0, 1) if 0 ≤ x ≤ 0.5,

(ρr, ur, Tr) = (0.125, 0, 0.25) if 0.5 < x ≤ 1,

In Fig. 3 we plot the results obtained in the rarefied regime (kn=10−2)using the Spectral-PFC scheme and the Time Relaxed Monte Carlo (TRMC)


0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 0.5 1 1.5 2

TxTyTzT

-4.2

-4

-3.8

-3.6

-3.4

-3.2

-3

-2.8

-2.6

-2.4

0 0.5 1 1.5 2

Entropy - VHS Molecules

(1) (2)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 1 2 3 4 5

TxTyTzT

-4.2

-4

-3.8

-3.6

-3.4

-3.2

-3

-2.8

-2.6

-2.4

0 1 2 3 4 5

Entropy - Maxwellian Molecules

(3) (4)

Fig. 3. 3D homogeneous case II: evolution of the temperature and the entropy forhard sphere molecules (1)-(2) and for Maxwellian molecules (3)-(4).


6

0

−6

Z

6

0

−6X 6

0

−6

Y

Sum of two Gaussian distributions : t=0.00

6

0

−6

Z

6

0

−6X 6

0

−6

Y


6

0

−6

Z

6

0

−6X 6

0

−6

Y


6

0

−6

Z

6

0

−6X 6

0

−6

Y


Fig. 4. 3D homogeneous case II: evolution of the distribution function at time t=0,0.1, 0.4 and 2 for hard sphere molecules (level set f(t, v) = 3.10−3)

method. The TRMC method is used with 100 cells in x containing 100 par-ticles whereas the Spectral-PFC scheme is used with 64 points in x and thesize of the velocity grid is 64 × 64 points for the transport and the totalnumber of modes 32 × 32. We observe that the two solutions are in this casevery comparable even if small oscillations, due to the statistical noise, per-sist. Concerning the computational time on one processor, the Spectral-PFCscheme is more efficient than Monte Carlo in this situation because the av-eraging highly increases the computational time (see Table 1). Let us notethat in the two cases (Monte-Carlo and Spectral methods), the Time Relaxedscheme allows to use a large variety of Knudsen number (kn = 10−1, 10−2)without increasing the computational cost. Finally, the computational time ofthe Spectral-PFC scheme can be highly reduced using the parallel algorithmpresented before.


We also give the result of the computations close to the Euler limit (kn =10−4) using 128 space cells for the Spectral-PFC method. In this case, asmaller time step (∆t = 0.001) is needed to keep good accuracy3, whichincreases the computational time, while a small time step for the TRMCmethod does not influence the numerical solution due to the low order of theMonte-Carlo scheme (see Table 1). For this reason a large time step is used,which explains the lower computational cost of the TRMC scheme.

Finally, the profiles obtained with TRMC and Spectral-PFC methods arereported in Fig. 4. The use of first order scheme for the transport for theTRMC scheme is clearly not sufficient to give accurate results. On the oppo-site, using a small time step (∆t = 0.001), an accurate solution is obtainedby the Spectral-PFC method, which is much less diffusive.

TRMC S-PFC

kn = 10−1 17 mn 25 sec 10 mn 50 sec

kn =10−2 17 mn 25 sec 10 mn 50 sec

kn =10−4 17 mn 25 sec 44 mn 20 sec

Table 1. Riemann problem: the first column represents the value of Knudsen num-bers kn, the second one is the computational time obtained for the TRMC schemeand the third one is the computational time for the third order PFC scheme coupledwith the spectral method for the collision operator.

5.4 Shock profile: stationary solutions.

This test deals with the numerical solution of the non homogeneous 1D× 3DBoltzmann equation for hard sphere molecules (α = 1). We present numericalresults for one dimensional stationary shock-profiles for different Knudsennumber and compare the solution with one obtained by the Monte-Carlomethod.

The gas is initially at the upstream equilibrium state in the left half-space and in the downstream equilibrium state in the right-half space. Theupstream state are determined from downstream state using the Rankine-Hugoniot relations [28]. In the present calculations, the downstream state ischaracterized by

ρr = 1, T r = 1, M = 2,

where M is the Mach number of the shock. The downstream mean velocity isthen given by

(urx, ury) = (−M

√γ T , 0),

3 The degradation of accuracy is typical of Strang splitting when one of the term isstiff (see [13]. Such degradation can be cured using a different approach for timediscretization


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

p(t,x)

x

S-PFCTRMC

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

u(t,x)

x

S-PFCTRMC

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 0.2 0.4 0.6 0.8 1

T(t,x)

x

S-PFCTRMC

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

p(t,x)

x

S-PFCTRMC

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

u(t,x)

x

S-PFCTRMC

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 0.2 0.4 0.6 0.8 1T

(t,x)

x

S-PFCTRMC

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

-4 -2 0 2 4

p(t,x)

x

S-FB

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-4 -2 0 2 4

u(t,x)

x

S-FB

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

-4 -2 0 2 4

T(t,x)

x

S-FB

(1) (2) (3)

Fig. 5. Riemann problem (kn = 10−2): evolution of (1) the density ρ, (2) meanvelocity u and (3) temperature T at t = 0.05, 0.15, 0.20.


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

p(t,x)

x

p(t,x)

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

u(t,x)

x

U(t,x)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 0.2 0.4 0.6 0.8 1

T(t,x)

x

T(t,x)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

p(t,x)

x

S-PFCTRMC

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

u(t,x)

x

S-PFCTRMC

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 0.2 0.4 0.6 0.8 1T

(t,x)

x

S-PFCTRMC

(1) (2) (3)

Fig. 6. Riemann problem (kn = 10−4): (1) the density ρ, (2) mean velocity u and(3) temperature T at t = 0.20 obtained by the central scheme for Euler equations(up) and by Spectral-PFC and TRMC methods for Boltzmann equations.

with γ = 5/3 since we have considered a 3D monoatomic gas in velocity space.The results of the computation are shown in Fig. 5 and 6. On the one

hand, we compute a solution using the Spectral-PFC scheme (128 cells inspace and only 32 × 32 × 32 modes in velocity) up to t = 1.0, so that theprofile is practically stationary. On the other hand, Monte Carlo calculations(TRMC) are performed by time-averaging the numerical solution after timelarge enough (t = 2.0). We observe that there is a good agreement betweenthe TRMC and Spectral-PFC method. However, the TRMC scheme is usedwith nx = 250 cells in space in order to avoid a too large numerical diffusion.Indeed, with nx = 128, the shock is not well described with this method. Inany case, the TRMC is much cheaper in term of computational cost since weare only interested in the stationary solution.


0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Monte-Carlo MethodSpectral Method

Initial Data

-2.6

-2.4

-2.2

-2

-1.8

-1.6

-1.4

-1.2

-1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2


Initial Data

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2


Initial Data

(1) (2) (3)

Fig. 7. Shock profiles (ε = 1. 10−1): (1) the density ρ, (2) mean velocity u and (3)temperature T obtained by the Spectral-PFC method and by the TRMC method.

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2


Initial Data

-2.6

-2.4

-2.2

-2

-1.8

-1.6

-1.4

-1.2

-1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2


Initial Data

1

1.2

1.4

1.6

1.8

2

2.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2


Initial Data

(1) (2) (3)

Fig. 8. Shock profiles (ε = 5. 10−2): (1) the density ρ, (2) mean velocity u and (3)temperature T obtained by the Spectral-PFC method and by the TRMC method.

6 Conclusion.

In this chapter we present an accurate deterministic method for the numericalapproximation of the space non homogeneous, time dependent Boltzmannequation. The method, based on a fractional step approach, couples a Positiveand Flux Conservative scheme for the treatment of the transport step with aFourier spectral method for the collision step.

It possesses a high order of accuracy for this kind of problems. In fact itis second order accurate in time, third order accurate in space, and spectrallyaccurate in velocity. The high accuracy is evident from the quality of the


numerical results that can be obtained with a relatively small number of gridpoints in velocity domain.

An effective time discretization allows the treatment of problems with aconsiderable range of mean free path, and the decoupling between the trans-port and the collision step makes it possible the use of parallel algorithms,which become competitive with state-of-the-art numerical methods for theBoltzmann equation.

The numerical results, and the comparison with other techniques, showthe effectiveness of the present method for a wide class of problems.

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2. Bird, G.A.: Molecular gas dynamics. Clarendon Press, Oxford (1994)3. Bobylev, A.V. and Rjasanow, S.: Difference scheme for the Boltzmann equation

based on the Fast Fourier Transform. Eur. J. Mech. B/Fluids 16, 293-306 (1997)4. Buet, C.: A discrete velocity scheme for the Boltzmann operator of rarefied gas

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