Acta Mathematicae Applicatae Sinica, English SeriesVol.18, No.1 (2002) 37–62
A Relaxation Scheme for Solving the Boltzmann EquationBased on the ChapmanEnskog Expansion
Shi Jin
1
, Lorenzo Pareschi
2
, Marshall Slemrod
3
1
Department of Mathematics, University of Wisconsin, Madison, Van Vleck Hall, WI 53706, USA.(Email: jin@math.wisc.edu)
2
Department of Mathematics, University of Ferrara, Via Machiavelli 35, I44100, Italy & Department of Mathematics, University of WisconsinMadison, Van Vleck Hall, WI 53706, USA. (Email: pareschi@dm.unife.it)
3
Department of Mathematics, University of WisconsinMadison, Madison, WI 537151149, USA.(Email: slemrod@math.wisc.edu)
Abstract
In [16] a viscoelastic relaxation system, called the relaxed Burnett system, was proposed by Jinand Slemrod as a moment approximation to the Boltzmann equation. The relaxed Burnett system is weaklyparabolic, has a linearly hyperbolic convection part, and is endowed with a generalized entropy inequality. Itagrees with the solution of the Boltzmann equation up to the Burnett order via the ChapmanEnskog expansion.We develop a onedimensional nonoscillatory numerical scheme based on the relaxed Burnett system forthe Boltzmann equation. We compare numerical results for stationary shocks based on this relaxation scheme,and those obtained by the DSMC (Direct Simulation Monte Carlo), by the NavierStokes equations and bythe extended thermodynamics with thirteen moments (the Grad equations). Our numerical experiments showthat the relaxed Burnett gives more accurate approximations to the shock proﬁles of the Boltzmann equationobtained by the DSMC, for a range of Mach numbers for hypersonic ﬂows, than those obtained by the otherhydrodynamic systems.
Keywords
Boltzmann equation, ChapmanEnskog expansion, Burnett equations, relaxation, central schemes
2000 MR Subject Classiﬁcation
35L99, 65M06, 76P05, 82C40
1 Introduction
Dynamics of a moderately rareﬁed gas of monatomic molecules is often represented by theBoltzmann equation. Observable quantities such as density, velocity, temperature, etc., arederived as expectations of a probability density function
f
(
xxx,ξξξ,t
) satisfying the Boltzmannequation (see [7,36])
f
+
ξξξ
·∇
f
=1
εQ
(
f,f
)
,
where
xxx
denotes the position of a particle at time
t
moving with velocity
ξξξ
,
Q
is the integral collision operator, and
is the Knudsen number which is proportional to the mean free path of thegas. The main numerical diﬃculty to solve the Boltzmann equation is its high dimensionality.There are two practical methods being used in applications. One is the DSMC (Direct Simulation MonteCarlo
[3
,
30]
and the others are moment methods that provide continuum equationsfor the observable macroscopic equations. DSMC oﬀers much less computational cost than a
Manuscript received October 8, 2001.
1
Supported by NSF grant DMS0196106;
3
Supported by NSF grant DMS9803223 and DMS00711463.
38
Shi Jin, et al.
deterministic method, but on the other hand it yields low accuracy and statistically ﬂuctuating results and the convergence in general is very slow. Moment methods, among them theGrad’s thirteen moment equations
[13]
and the extended Thermodynamics equations
[26]
, deﬁnedin physical space, are generally faster than DSMC but the results deviate from that of theBoltzmann at high Mach numbers.In this paper we propose a new numerical scheme based on the ChapmanEnskog expansion(see [7,10,36]) for the Boltzmann equation. This scheme is a numerical discretization of therelaxation approximation proposed by Jin and Slemrod
[16
,
17]
and its conceptual basis is indeedthe ChapmanEnskog expansion.The classical ChapmanEnskog procedure for the Boltzmann equation is a well known toolfor bridging the gap between kinetic theory as described by the Boltzmann equation for theevolution of a monatomic gas and continuum mechanics. The ChapmanEnskog expansion is aformal power series ordered by the viscosity
µ
which is itself proportional to the nondimensionalKnudsen number, i.e.,
T T T
=
−
pI I I
−
P P P , p
=
Rρθ,P P P
=
µ
ΠΠΠ
(1)
+
µ
2
ΠΠΠ
(2)
+
µ
3
ΠΠΠ
(3)
+
···
, qqq
=
µ
ΞΞΞ
(1)
+
µ
2
ΞΞΞ
(2)
+
µ
3
ΞΞΞ
(3)
+
···
.
The coeﬃcients ΠΠΠ
(
j
)
, ΞΞΞ
(
j
)
,
j
= 1
,
2
,
···
are obtained from the Boltzmann equation and havebeen determined up to
j
= 2 (Burnett order) (cf. [10,36]) and in one space dimension up to
j
= 3 (superBurnett order) (cf. [12]). (We remind the readers that all physical quantities inthis paper and their mathematical deﬁnitions are given in the Nomenclature at the end of thepaper.)In practice however the ChapmanEnskog expansion as a tool for solving the Boltzmannequation has had limited practical value. Truncation at ﬁrst order yields the NavierStokesequations which as
µ
ceases to be small becomes a poor approximation to solutions of theBoltzmann equation (cf. [21,26]). Truncation at order
µ
2
yields the Burnett equations whichpossesses the unphysical property of yielding linearly unstable rest states (cf. [1,5,23,24,25]).Simply by expanding to the higher order will not remove this instability (cf. [31]).In addition, the ChapmanEnskog expansion destroys the material frame indiﬀerence at theBurnett order (cf. [4]).Despite the linear instability of the Burnett equations, numerical solutions on augmentedBurnett equations (cf. [1,11,36]) suggest that they provide more accurate solutions in theshock layer than those of the NavierStokes equations when compared with the direct simulationMonteCarlo method of the Boltzmann equation. In [1,11,36] the augmented Burnett equationswere obtained either by removing the unstable term from or by adding linearly stabilizingterms of the super Burnett order to the stress and heat ﬂux. Unfortunately the augmentedBurnett equations possess two drawbacks. First, numerically they require resolution of thesuperBurnett stabilizing terms which practically means numerical resolution of derivatives upto fourth order. This is rather a cumbersome approach in several space dimensions. Secondly,the augmented Burnett equations have not been shown to have a globally deﬁned entropypossessing the usual property of satisfying an entropy inequality.In [16], a viscoelastic relaxation approximation was introduced as an approximation to theBoltzmann equation. This relaxation system has the following properties.(i) It requires at most resolution of second derivatives in spatial variables;(ii) it possesses a globally deﬁned “entropy” like function;(iii) when expanded via the ChapmanEnskog expansion, it matches the classical ChapmanEnskog expansion for the Boltzmann equation to the Burnett order.Speciﬁcally the pressure deviator and heat ﬂux were relaxed by rate equations to obtain a systemof local equations that can recover the Burnett equations via the ChapmanEnskog expansion
A Relaxation Scheme for Solving the Boltzmann Equation Based on the ChapmanEnskog Expansion
39
with a correction at the superBurnett order. By doing this, a system of thirteen local equationswere obtained that is linearly stable. This system is weakly parabolic with a linearly hyperbolicconvection part. Moreover, it is endowed with a generalized entropy inequality. The nonlinearentropy inequality guarantees the irreversibility of the relaxation process. The localness of thissystem is attractive for a robust numerical approximation to the gas dynamics valid to theBurnett order.In recent years, relaxation approximations have been used as an eﬀective tool to designnumerical methods — known as the relaxation schemes. In [18] a generic way to relax a generalsystem of hyperbolic conservation laws was introduced by Jin and Xin, which induced a classof relaxation schemes free of Riemann solver and local characteristic decomposition for inviscidgas dynamics. A physically natural pressure relaxation method was developed by Coquel andPerthame for an inviscid general gas
[9]
.In this paper, for onedimensional problem, we propose a class of relaxation schemes forthe Boltzmann equation based on the relaxed Burnett system by Jin and Slemrod. There aretwo main diﬃculties when discretizing this system. First, the equations for the stress deviatorand heat ﬂux are not in conservative form, thus canonical shock capturing methods, developedby hyperbolic systems of conservation laws, cannot be applied directly. Secondly, the stiﬀ relaxation terms need to be discretized properly so the scheme is eﬃcient even for small meanfree paths.Our relaxation schemes combine a conservative solver for the conserved part of the system(balance laws for density, momentum and energy), while for equations of
P P P
and
qqq
we discretizethe spatial derivatives using slope limiters and central diﬀerences. These discretizations arecarried out conveniently using a staggered grid, as in a staggered nonoscillatory central scheme.We compare the numerical results obtained by this relaxation scheme with those obtained byDSMC, the NavierStokes equations and the extended thermodynamics (with thirteen moments)for onedimensional stationary shocks with various Mach numbers. Our results show that therelaxed Burnett system oﬀers more accurate shock proﬁles compared to the DSMC than otherhydrodynamic theories.The paper is divided into ﬁve sections after this Introduction. Section 2 reviews the relaxedBurnett system introduced by Jin and Slemrod. We also derive boundary conditions for thissystem using the moment deﬁnition from the probability density distribution. Section 3 reviewsseveral main properties of the relaxed Burnett system, and computes the linear dispersionrelation. In Section 4 we introduce the numerical discretization for the onedimensional relaxedBurnett system. In section 5 we solve a onedimensional stationary shock problem by thenewly introduced relaxation scheme, and compare it with DSMC, NavierStokes and extendedthermodynamics. We end the paper with a few concluding remarks in Section 6.
2 The Relaxed Burnett System
2.1 The Field Equations of Balance
The ﬁeld equations of balance for continuum ﬂuid dynamics in the absence of heat sources areas follows˙
ρ
+
ρ
div
uuu
= 0 (mass conservation)
,
(1)
ρ
˙
uuu
+ grad
p
+ div
P P P
=
ρbbb
(linear momentum conservation)
,
(2)
P P P
=
P P P
T
(rotational momentum conservation)
,
(3)
ρ
˙e +
p
div
uuu
+
P P P
·
S S S
+ div
qqq
= 0 (energy conservation)
,
(4)
40
Shi Jin, et al.
wheree =
ψ
−
θ∂ψ∂θ, η
=
−
∂ψ∂θ, p
=
ρ
2
∂ψ∂ρ.
(5)Diﬀerentiation of the expression for the Helmholtz free energy
ψ
=
ε
−
θη
yields
ρθ
˙
η
=
ρ
˙
e
−
ρ
˙
ρ∂ψ∂ρ,
which when combined with (1) and (4), yields the entropy production equation
ρθ
˙
η
=
−
P P P
·
S S S
−
div
qqq.
(6)Division by
θ
yields the total entropy product rate of a ﬂuid occupying domain
B⊂
IR
3
ddt
B
ρηdV
=
−
B
P P P
·
S S S θ
+
qqq
·
grad
θθ
2
dV
−
∂
B
qqq
·
nnnθdA.
(7)The ClausiusDuhem inequality is a common albeit not universally accepted form of thesecond law of thermodynamics. It asserts
ddt
B
ρηdV
+
∂
B
qqq
·
nnnθdA
≥
0
,
which in turn from (7) requires
P P P
,
qqq
to satisfy
B
P P P
·
S S S θ
+
qqq
·
grad
θθ
2
dV
≤
0for all ﬂuid domains
B
. However the classical ClausiusDuhem inequality is inconsistent with
P P P
,
qqq
delivered by the ChapmanEnskog expansion beyond NavierStokes order.
2.2 The ChapmanEnskog Expansion
The ChapmanEnskog expansion for a monatomic gas of spherical molecules yields the constitutive relationse =32
Rθ, p
=
Rρθ, µ
=
µ
(
θ
)
,
(8)
ψ
=
Rθ
log
ρ
−
32
Rθ
log
θ
+32
Rθ
−
aθ
+
b,
(9)
η
=
−
R
log
ρ
+32
R
log
θ
+
a.
(10)where
a,b
are constants of integration.In addition the expansion provides representations for the pressure deviator tensor
P P P
andheat ﬂux vector
qqq
in terms of a series which may be ordered via powers of the viscosity
µ
interms of the total number of space plus time derivatives. Following the notation of Ferzigerand Kaper
[10]
we record
P P P
=
µP P P
(1)
+
µ
2
P P P
(2)
+
···
,
(11)
qqq
=
µqqq
(1)
+
µ
2
qqq
(2)
+
···
,
(12)where the expressions for
P P P
(1)
,
P P P
(2)
,
qqq
(1)
,
qqq
(2)
are as follows
P P P
(1)
=
−
2
S S S ,
(13)
A Relaxation Scheme for Solving the Boltzmann Equation Based on the ChapmanEnskog Expansion
41
qqq
(1)
=
−
32
M
R
grad
θ,
(14)
P P P
(2)
=
ω
1
1
p
(div
uuu
)
S S S
+
ω
2
1
p
˙
S S S
−
LLLS S S
−
S S S L
T
+23tr(
S S S L
T
)
I I I
+
ω
3
1
ρθ
grad
2
θ
−
13∆
θI I I
+
ω
4
1
ρpθ
12grad
p
⊗
grad
θ
+12grad
θ
⊗
grad
p
−
13grad
p
·
grad
θI I I
+
ω
5
1
ρθ
2
grad
θ
⊗
grad
θ
−
13

grad
θ

2
I I I
+
ω
6
1
p
S S S
2
−
13tr(
S S S
2
)
I I I
,
(15)
qqq
(2)
=
θ
1
1
ρθ
(div
uuu
)grad
θ
+
θ
2
1
ρθ
(grad
θ
)
•
−
LLL
T
grad
θ
+
θ
3
1
pρ
(
S S S
grad
p
) +
θ
4
1
ρ
div
S S S
+
θ
5
1
ρθS S S
grad
θ.
(16)One drawback of the ChapmanEnskog expansion is that, if truncated at the Burnett orhigher order, it destroys the property of material frame indiﬀerence. In particular, in (15) and(16), the
ω
2
term in
P P P
(2)
and the
θ
2
term in
qqq
(2)
are both material frame dependent. It cannotbe recovered by replacing the material derivative with the space derivative using the Euler orNavierStokes equations
[4]
.The coeﬃcients
ω
1
,
···
,ω
6
,θ
1
,
···
,θ
5
are functions of
θ
and are not independent. For agas of spherical molecules the following universal relations have been derived by Truesdell andMuncaster
[36]
generalizing more specialized relations:
ω
3
=
θ
4
,θ
1
=23
72
−
µ
(
θ
)
µ
(
θ
)
θ
θ
2
−
13
θ∂θ
2
∂θ,ω
1
=23
72
−
µ
(
θ
)
µ
(
θ
)
θ
ω
2
−
13
θ∂ω
2
∂θ.
(17)Furthermore for gases of ideal spheres in which the collisions are purely elastic or satisfy aninverse
k
th
power attraction between molecules, the coeﬃcients
ω
1
,ω
2
,
···
,θ
5
are independentof
θ
. In addition the relations
θ
1
θ
2
=
ω
1
ω
2
=
23
3
k
−
5
k
−
1
for inverse
k
th
power molecules
,
2 for ideal sphereshold.Exact determination of
ω
1
,ω
2
,
···
,θ
5
has only been accomplished for a gas of Maxwellian(
k
= 5) molecules. For the more general case only approximations to
ω
1
,ω
2
,
···
,θ
5
have beenobtained. The classical approximation result (say as found in [10, p.149]) is
ω
2
2
, ω
3
3
, ω
4
0
, ω
5
µ
(
θ
)
θω
3
µ
(
θ
)
, ω
6
8
,θ
2
458
, θ
3
−
3
, θ
4
3
, θ
5
3
354+
θµµ
(
θ
)
,
M
52
.
(18)For Maxwell molecules the relations (18) are exact
θ
1
θ
2
=
ω
1
ω
2
=53
,θµ
(
θ
)
µ
(
θ
)= 1
,
and
µ
is linear in
θ
.