Journal of Computational Physics · 2014-01-08 · Affordable robust moment closures for CFD based...

Journal of Computational Physics 251 (2013) 500–523

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Journal of Computational Physics

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Affordable robust moment closures for CFD based on themaximum-entropy hierarchy

0021-9991/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jcp.2013.05.046

⇑ Corresponding author. Tel.: +49 241 80 98 664.E-mail addresses: [email protected] (J. McDonald), [email protected] (M. Torrilhon).

James McDonald ⇑, Manuel TorrilhonCenter for Computational Engineering Science, RWTH Aachen University, Schinkelstr. 2, 52062 Aachen, Germany

a r t i c l e i n f o

Article history:Received 8 October 2012Received in revised form 22 May 2013Accepted 27 May 2013Available online 13 June 2013

Keywords:Non-equilibrium gas dynamicsHyperbolic moment closuresKinetic theory

a b s t r a c t

The use of moment closures for the prediction of continuum and moderately non-equilib-rium flows offers modelling and numerical advantages over other methods. The maximum-entropy hierarchy of moment closures holds the promise of robustly hyperbolic stablemoment equations, however their are two issues that limit their practical implementation.Firstly, for closures that have a treatment for heat transfer, fluxes cannot be written inclosed form and a very expensive iterative procedure is required at every flux evaluation.Secondly, for these same closures, there are physically possible moment states for whichthe entropy-maximization problem has no solution and the entire framework breaksdown. This paper demonstrates that affordable closed-form moment closures that areinspired by the maximum-entropy framework can be proposed. It is known that closingfluxes in the maximum-entropy hierarchy approach a singularity as the region of non-solv-ability is approached. This paper shows that, far from a disadvantage, this singularityallows for smooth and accurate prediction of shock-wave structure, even for high Machnumbers. The presence of unphysical ‘‘sub-shocks’’ within shock-profile predictions of tra-ditional closures has long been regarded as an unfortunate limitation of the entiremoment-closure technique. The realization that smooth shock profiles are, in fact, possiblefor moment methods with a moderate number of moments greatly increases the method’sapplicability to high-speed flows. In this paper, a 5-moment system for a simple one-dimensional gas and a 14-moment system for realistic gases are developed and examined.Numerical solution for shock-waves at a variety of incoming flow Mach numbers demon-strate both the robustness and the accuracy of the closures.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

The development of models for moderately rarefied gas-flow prediction has proven difficult. In this transition regime, be-tween the particle-collision-dominated continuum regime and the free-molecular regimes, continuum methods are inaccu-rate and particle-based methods, such as direct-simulation Monte Carlo (DSMC) [1], become expensive. The expenseassociated with stochastic particle methods becomes even more extreme for low-speed flows or for time-accurate calcula-tions. Techniques based on the addition of higher-order terms to the Navier–Stokes equations through expansion techniqueshave failed to provide stable sets of partial differential equations (PDEs) [2]. Moment closures offer the promise of manyadvantages as compared to traditional methods for both continuum and rarefied gas-flow prediction [3,4]. The resultingfirst-order balance laws provide a natural stable hyperbolic treatment for effects of local thermal non-equilibrium, suchas anisotropic pressures, through their inclusion in an expanded solution vector.

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Moment closures also have the potential for more convenient numerical solution than traditional fluid-dynamic equa-tions, such as the Navier–Stokes equations, which require the evaluation of second derivatives and are of mixed hyper-bolic-parabolic type. The requirement to compute only first derivatives means that an extra order of spatial accuracy canpotentially be obtained for a given stencil. Also, the requirement for only first derivatives makes numerical solution of sys-tems of moment equations much less sensitive to grid irregularities [5]. Grid irregularities are typical for practical situationsin which complex geometries make the generation of high-quality meshes inconvenient. They are also commonly encoun-tered when adaptive mesh refinement or embedded-boundary treatments are used. Recently, moment closures have alsobeen coupled to DSMC techniques in order to reduce the statistical fluctuations of the latter approach [6].

Traditional moment closures do, unfortunately, suffer from their own issues. The original hierarchy proposed by Grad [3]suffers from a restrictive region of hyperbolicity. This is due to the fact that the flux Jacobian can develop complex eigen-values for moderate departures from equilibrium. Alternatively, the closures of the maximum-entropy hierarchy are globallyhyperbolic whenever the underlying constrained entropy-maximization problem can be solved [4,7–9]. Theoretical andpractical studies related to maximum-entropy moment closures have been carried out by many authors, including: Brownet al. [10,11], Le Tallec and Perlat [12], McDonald and Groth [13–17], Lam and Groth [18], Suzuki et al. [19,20], Barth [21],Coulombel et al. [22,23], and Carrillo et al. [24]. Unfortunately, the maximum-entropy hierarchy suffers from two apparentproblems. Firstly, for closures of sufficiently high order to include a treatment for heat transfer, it is not possible to write theclosing fluxes in closed-form. The result is that a very expensive and often ill-conditioned iterative procedure must be usedat every flux evaluation. This expense cannot be justified for practical calculations. Also, it has been shown that, again for allclosures that include heat transfer, there are physically realistic moment states for which the entropy maximization problemdoes not have a solution and the whole framework of the closure breaks down [25–27]. If moment closures are to come intopractical use, new closures must be developed that are both affordable and robust.

Another issue that is considered a limitation to the usefulness of all hyperbolic moment closures is the presence of arti-ficial discontinuities in flow solutions. These are especially apparent in shock-structure predictions as they cause un-physical‘‘sub-shocks’’ to form [4]. Many practical applications involving thermal non-equilibrium also involve super-sonic or hyper-sonic flows and moment closures that predict artificial discontinuities inside shock-wave profiles are obviously unsuitable ifthe the internal structure of shock waves are of interest [20].

Recently proposed regularized variants of the Grad moment closures allow for the prediction of smooth shock structuresthrough the inclusion of second-order terms [9,28–30]. These closures do not remain first-order or purely hyperbolic due tothe parabolic nature the regularizing terms. All numerical advantages associated with a first-order balance-law form aretherefore lost. The limited hyperbolicity of the base Grad closure also remains and closure breakdown can still be expectedfor significant deviations from equilibrium.

1.1. Scope of current study

This paper begins with a review of the theory of moment closures with emphasis on the family of maximum-entropy clo-sures. The two major issues related with this hierarchy, the lack of a closed-form flux and limited moment realizability, arereviewed. Using a simple one-dimensional closure, it is then shown that these issues can be handled in practice through aninterpolation technique. It is demonstrated that the nature of the non-realizability of some moment states leads to a singu-larity in the closing flux that is actually beneficial, as it opens the door to smooth shock profiles for high Mach numbers.Numerical solutions demonstrate the amazing accuracy of the model for shock-structure calculations. This technique is thenextended to the fully realistic three-dimensional setting. A closed-form set of moment equations for realistic gases is pro-posed that has a vastly expanded region of hyperbolicity and is robust enough for the numerical prediction of strong shockwaves. Comparisons to Navier–Stokes solutions and direct discretization of the kinetic equation are made.

2. Moment closures

Moment closures follow from the field of gaskinetic theory. In this theory, a monatomic gas is described by a probabilitydensity function, Fðxi;v i; tÞ, that describes the probability of finding a gas particle at a given position, xi, with a velocity, v i, ata time, t. This distribution function gives a huge amount of information about the microscopic state of a gas. However, formost practical problems, it is only a small number of macroscopic properties that are of interest. These properties are relatedto the distribution function through moment relations, taking monomials of the particle velocity as weights, /ðv iÞ. Forexample,

/ðv iÞ ¼ 1 : q ¼RRR1mF dv i ¼ mFh i;

/ðv iÞ ¼ v i : qui ¼ mv iFh i;/ðciÞ ¼ cicj : Pij ¼ mcicjF

� �;

/ðciÞ ¼ cicjck : Q ijk ¼ mcicjckF� �

;

Here, the notation h�i denotes integration over all of velocity space, m is the mass of one gas particle, q is the mass density, ui

is the bulk velocity, ci ¼ v i � ui is the peculiar component of the gas-particle velocity, Pij is an anisotropic pressure tensor,and Q ijk is the generalized heat-flux tensor. The thermodynamic pressure is found as p ¼ 1

3 Pii and the traditional heat-flux

502 J. McDonald, M. Torrilhon / Journal of Computational Physics 251 (2013) 500–523

vector is given by hi ¼ 12 Q ijj. Higher-order moments can be similarly defined, but do not have an obvious macroscopic phys-

ical interpretation.In this paper, third-, fourth-, and fifth-order moments of the peculiar particle velocity are assigned the letters ‘‘Q’’, ‘‘R’’, and

‘‘S’’ respectively. All higher-order tensors of the peculiar velocity are given the symbol ‘‘V’’ and the order of the moment canbe determined by counting the number of indices. For example,

Qijj ¼ mcicjcjF� �

; Rijkk ¼ mcicjckckF� �

;

Sijjkk ¼ mcicjcjckckF� �

; Vijklm...p ¼ mcicjckclcm . . . cpF� �

:

Moments of the full particle velocity are called ‘‘full’’ or ‘‘convective’’ moments and are denoted by the symbol ‘‘U’’, as

Uijk...n ¼ mv iv jvk . . . vnF� �

: ð1Þ

The Einstein summation convention is always used.The evolution of a distribution function is governed by the Boltzmann equation,

@F@tþ v i

@F@xi¼ dF

dt; ð2Þ

shown here in the absence of external acceleration fields. The left-hand side of Eq. (2) describes simple particle streamingwhile the right-hand side is responsible for the effects of inter-particle collisions on the distribution function. This collisionoperator is a high-dimensional integral that, in general, cannot be evaluated in closed form. Nevertheless, some informationabout the behaviour of this operator is readily available. For one thing, it conserves mass, momentum, and energy. It was alsoshown by Boltzmann that it always increases the entropy monotonically until the distribution function has the form of aMaxwellian (or Maxwell–Boltzmann) distribution,

M¼ nq

2pp

� �32

expq2p

cici

� �; ð3Þ

where n is the local particle number density (q ¼ nm). At this point the collision operator has no further effect. Therefore, Eq.(3) is the distribution function with the maximum entropy for a fixed pressure and density.

Equations governing the evolution of macroscopic moments can be found by taking moments of Eq. (2). This leads toMaxwell’s equation of change,

m/@F@t

� �þ mv i/

@F@xi

� �¼ @

@tm/Fh i þ @

@ximv i/Fh i ¼ m/

dFdt

� �: ð4Þ

Usually one is interested in a set of macroscopic moments, thus is it convenient to define a vector of generating weights,Uðv iÞ ¼ ½/1ðv iÞ;/2ðv iÞ; . . . ;/nðv iÞ�T. This vector can then be used in Eq. (4),

@

@tmUFh i þ @

@ximv iUFh i ¼ @U

@tþ @F i

@xi¼ C; ð5Þ

where U ¼ hmUFi is the solution vector containing the full moments of interest, F i ¼ hmv iUFi is the corresponding fluxdyad, and C ¼ hmU dF

dt i is the source vector representing the effect of the collision operator on the moments contained inU. These equations are of first-order balance-law form. Unfortunately, this system of equations is not closed. There are al-ways moments present in the flux dyad that are of higher order than any moment in the solution vector. The effect ofthe collision operator on the moments is also not known in general.

2.1. The BGK collision operator

As mentioned above, the collision operator in Eq. (2) is very difficult to use in practice. As the goal of the current researchis in the development of effective moment closures for the modelling of the left-hand side of the Boltzmann equation, thesimple BGK relaxation-time collision operator is used [31].

dFdt¼ �F �M

s: ð6Þ

In this model, any non-equilibrium distribution, F , simply relaxes to the equilibrium Maxwellian,M, that has the same den-sity, momentum, and energy. Once the distribution function is a Maxwellian, the operator has no more effect. Though themodel has limited physical accuracy, this simple collision operator is convenient for the evaluation of moment equationsas it allows an easy automatic closure for the right-hand side of Eq. (4).

2.2. Moment closure

In order to obtain a useful model, the system of equations resulting from Eq. (5) must be closed. One way to do this is torestrict the distribution function, F , to some assumed form. This form must have the same number of free parameters,


a ¼ ½a1;a2; . . . ;a3�T , as there are entries in U. The value of these free parameters is chosen so that the moment relations, givenin Eq. (1), are satisfied. This ensures that all entries in both the flux dyad, F i, and the source vector, C, are functions of theknown moments of the solution vector. Thus, the system is closed.

The earliest moment closures were those proposed by Grad [3], in which the distribution function was assumed to be apolynomial expansion around the equilibrium Maxwellian, Eq. (3), with the same density, momentum, and energy,

FGrad ¼MaTU: ð7Þ

Unfortunately, moment closures based on such an assumed form suffer from several problems. Firstly, the distribution func-tion is not always positive, as the polynomial, aTU, can be negative, and is thus not a properly defined probability densityfunction. More devastatingly, for modest departures from equilibrium, the hyperbolicity of the resulting system of equationscan break down [32–34]. This is caused by the fact that the flux Jacobian can develop complex eigenvalues. As a result, thesystem of moment equations may not be well posed for initial-value problems.

3. Maximum-entropy moment closures

A more mathematically elegant hierarchy of closures can be obtained by assuming the distribution function is that whichmaximizes the entropy while remaining consistent with the moments in the solution vector [4,7–9]. For classical gases, forwhich the entropy density is know to be hF lnFi, this assumption leads to distribution functions of the form

FMaxEnt ¼ eaTU: ð8Þ

This distribution function is positive valued, and, through careful selection of the generating velocity weights in U, it can beassured that the distribution remains finite [8].

Aside from the mathematical beauty of the maximum-entropy theory, there are strong physical arguments as to why itshould provide accurate predictions. Not only does the collision operator always acts to increase entropy, as was stated inSection 2, but the maximum-entropy distribution is also statistically the most likely distribution that is consistent with theknown moments [9,35,36]. It is therefore, in some sense, the best possible choice given the limited information in the solu-tion vector.

3.1. Properties of maximum-entropy moment closures

In this section, useful mathematical properties of the maximum-entropy hierarchy are reviewed. Following Levermore,[8], this study begins by defining density and flux potentials,

hðaÞ ¼ meaTUD E

; f iðaÞ ¼ mv ieaTUD E

: ð9Þ

It is clear that the conserved moments and fluxes of the system can be expressed as

@h@a¼ h;a ¼ mUeaTU

D E¼ U;

@fi

@a¼ fi;a ¼ mv iUeaTU

D E¼ F i: ð10Þ

Eq. (5) can therefore be written as

@

@th;a þ

@

@xifi;a ¼ C; ð11Þ

The terms h;a and fi;a can be differentiated again to give

@U@a¼ h;a;a ¼ mUUTeaTU

D E;

@F i

@a¼ fi;a;a ¼ mv i UUTeaTU

D E: ð12Þ

The flux Jacobian is therefore

@F i

@U¼ @F i

@a@U@a

� ��1

¼ fi;a;a h;a;að Þ�1: ð13Þ

In fact, the global hyperbolicity of the system of moment equations can be demonstrated by re-expressing Eq. (11) as

h;a;a@a@tþ fi;a;a

@a@xi¼ C: ð14Þ

This relation describes the time evolution of the closure coefficients for a maximum-entropy distribution. Hyperbolicity ofthis system is assured by the symmetry of fi;a;a and symmetric positive definiteness of h;a;a [8,37,38]. The positive-definite-ness of h;a;a is known because for any vector w,

wTh;a;aw ¼ mwTUUTweaTUD E

¼ m wTU� 2

eaTUD E

P 0: ð15Þ

and hence h;a;a is both symmetric and positive definite.


3.2. Issues related to the maximum-entropy hierarchy

The two lowest-order member of the maximum-entropy family are 5-moment and 10-moment closures. The 5-momentclosure leads to the familiar Euler equations, while the 10-moment system is known as the Gaussian closure. The Gaussianclosure gives a hyperbolic treatment for viscous gas flows, but does not have a treatment for heat flux. Numerical Compu-tations have demonstrated that this model can be very successful for continuum and transition-regime flows for which heat-transfer is not important [10,11,15,19,21].

Given the mathematical elegance and physical arguments presented above, it may initially seem surprising that the max-imum-entropy hierarchy has not gained more popularity. This is due to several issues related to the higher-order members ofthis family.

Firstly, if the vector of generating weights, U, contains super-quadratic terms, moments of the distribution function can-not be given in closed form. That is, the relationship

UðaÞ ¼ mUFðaÞh i ð16Þ

cannot be evaluated for general a. The inversion of this expression, giving the coefficients as a function of the knownmoments, aðUÞ, is also impossible. The system’s fluxes, FiðaÞ, therefore cannot be expressed as functions of the knownmoments, F iðUÞ. Practically, this means that, when the closure coefficients are needed for flux evaluation during the numer-ical solution of the moment equations, they must be found iteratively. This iterative process requires the accurate integrationof the distribution function over all velocity space be evaluated numerically many times [8,12]. This expense cannot be jus-tified for practical computations. As the heat flux is a third-order moment, any closure with a treatment for heat flux willsuffer from this problem.

In what has been regarded as a more devastating issue related to the maximum-entropy hierarchy, Junk has shown that,again for all maximum-entropy distributions with super-quadratic terms in U, there are physically realistic states for whichclosure coefficients cannot be found [25]. In these regions, the entropy-maximization problem on which the theory isfounded does not have a solution, the arguments of Section 3.1 do not hold, and the entire theory of maximum-entropy mo-ment closures breaks down. In the following, approximations to maximum-entropy closures are presented that maintain theattractive features of the true maximum-entropy hierarchy while addressing these two deficiencies.

4. Closed-form approximation to a one-dimensional moment closure

The two issues relating to maximum-entropy moment closures may seem devastating at first. However, a recent studyhas found that both issues can be handled in practice for a simple one-dimensional system [17]. The procedure has now beenrefined and is shown here. A new simpler and superior moment system is the result.

4.1. Moment closures for one-dimensional physics

For one space dimension and one velocity dimension, the velocity-distribution function reduces to Fðx;v ; tÞ, whereposition and velocity are now scalars. The Boltzmann equation with the BGK collision operator shown in Section 2.1becomes

@F@tþ v @F

@x¼ �F �M

s: ð17Þ

Here, the equilibrium state given by the one-dimensional Maxwell–Boltzmann distribution is

M¼ nffiffiffiffiffiffiffiffiffiq

2pp

rexp � q

2pc2

� �: ð18Þ

The simplest member of the one-dimensional maximum entropy hierarchy with super-quadratic generating weights is a fivemoment system. It has a distribution function given by

F 5 ¼ exp a0 þ a1v þ a2v2 þ a3v3 þ a4v4� : ð19Þ

The corresponding system of moment equations is

@q@tþ @

@xquð Þ ¼ 0; ð20Þ

@

@tquð Þ þ @

@xqu2 þ p�

¼ 0; ð21Þ

@

@tqu2 þ p�

þ @

@xqu3 þ 3upþ q�

¼ 0; ð22Þ


@

@tqu3 þ 3upþ q�

þ @

@xqu4 þ 6u2pþ 4uqþ r�

¼ � qs; ð23Þ

@

@tqu4 þ 6u2pþ 4uqþ r�

þ @

@xqu5 þ 10u3pþ 10u2qþ 5ur þ s�

¼ �1s

4uqþ r � 3p2

q

� �: ð24Þ

The moments present in the system are

q ¼ mF 5h i; qu ¼ mvF 5h i;p ¼ mc2F 5

� �; q ¼ mc3F 5

� �;

r ¼ mc4F 5� �

; s ¼ mc5F 5� �

:

ð25Þ

Lower-case symbols are used to denote one-dimensional physics. It is the fifth-order moment, s, which is present in the flux,but not the solution vector, that must be found through the assumed form of the distribution to close the system. As statedearlier, a closed-form expression for this moment as a function of the lower-order moments in not possible for the maxi-mum-entropy hierarchy. Nevertheless, an investigation into the behaviour of this moment reveals that closed-form approx-imations which maintain many of the attractive features of the true maximum-entropy flux can be found.

4.2. Non-dimensionalization of the five-moment closure

In order to investigate the properties of the resulting system, it is convenient to apply a shift of the probability densityfunction in velocity space such that u ¼ 0, followed by a non-dimensionalization such that q ¼ 1 and p ¼ 1. This leads tonon-dimensionalized third, fourth, and fifth moments given by

qH¼ 1

qqp

� �32

q; rH ¼1q

qp

� �2

r; sH ¼1q

qp

� �52

s: ð26Þ

4.3. Realizability of a five-moment closure

When analyzing the realizability of moment closures, there are two types of realizability that must be considered. First,there is the question of physical realizability, which asks which moment states can be reached by an arbitrary positive dis-tribution function. Second, one must consider which moment states can be reached by distribution functions of the assumedform used to close the system.

Physical realizability can be calculated through the solution of the Hamburger moment problem [39,40]. This is done byfirst defining a vector of velocity weights, M ¼ ½1;v ;v2; . . . ;vn�T. This is then used to define a matrix

Y ¼ mMMTFD E

: ð27Þ

For a positive distribution, F , it is obviously necessary that this matrix be be positive definite, as

WTYW ¼WT mMMTFD E

W ¼ mWTMMTWFD E

¼ mðWTMÞ2FD E

> 0; ð28Þ

for any vector W. For one-dimensional distribution functions it can also be shown that this is a sufficient condition for phys-ical realizability [41].

For the one-dimensional five-moment closure, one should take M ¼ ½1;v ;v2�T. This leads to the realizability condition

rH P q2Hþ 1: ð29Þ

One must now consider which physically possible states can be realized by a distribution of the form given in Eq. (19). Junkhas shown, [25], that this distribution function cannot realize states on the line

qH¼ 0 and rH > 3: ð30Þ

In other words, if there is no heat flux, the distribution function cannot have a fourth moment that is higher than it is for anequilibrium Maxwellian, rH ¼ 3. Fig. 1 shows the region of realizability for this 5-moment system.

4.4. Parabolic mapping of phase space

In order to make interpolation of the closing flux more convenient, a parabolic mapping is employed. This is done bydefining a new variable, r, that replaces rH, through the relation

rH ¼1r

q2Hþ 3� 2r for 0 6 r 6 1: ð31Þ

q*

r *

-3 -2 -1 0 1 2 30

2

4

6

8

10

PhysicallyRealizable

Not PhysicallyRealizable

No Maximum-EntropyDistribution Exists

Local Equilibrium(0,3)

*

Fig. 1. Physical realizability and realizability of maximum-entropy distribution function for the one-dimensional 5-moment system.


On the physical-realizability boundary, r ¼ 1. As r decreases, lines of constant sigma are parabolas of increasing curvatureand increasing y intercept. At r ¼ 0, the parabola collapses to a line coinciding with the Junk subspace of non-realizability forthis maximum-entropy distribution function. The expression for r as a function of q

Hand rH can easily be found to be

rðqH; rHÞ ¼

3� rH þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið3� rHÞ2 þ 8q2

H

q4

: ð32Þ

In this new mapped description, both realizability limits described in Section 4.3 represent domain boundaries. This makes aclosed-form interpolation of the closing flux more convenient to propose.

4.5. Exact flux expressions on boundaries

If the goal is the development of closed-form approximations to the closing flux, one sensible strategy is to find the exactvalue wherever possible and interpolate. An investigation into the behaviour of the closing moment, sH, is therefore of inter-est. The non-dimensional closing flux for the five-moment maximum-entropy system is shown in Fig. 2. It can readily beseen that, as the Junk subspace is approached, the flux becomes singular.

400

800

1600

2000

1200

-400-800

-1200-1600

-2000

q*

r *

-10 -5 0 5 100

20

40

60

80

100

s*2000160012008004000-400-800-1200-1600-2000

Fig. 2. Non-dimensionalized closing flux, sH , for the one-dimensional 5-moment maximum-entropy closure.


Numerical investigation has also revealed that, on the physical-realizability boundary, the non-dimensionalized distribu-tion function degenerates into two delta functions. This investigation simply involved numerically solving the entropy-max-imization problem for many states that are very close to the boundary and examination of the resulting distributionfunctions. The weights and positions of the deltas can be found by ensuring agreement with the zeroth through third mo-ment (the fourth moment agrees automatically). The distribution function on this boundary is found to be

r *

1

Fig. 3.equilibr

FHðvHÞ ¼2

q2Hþ 4� q

H

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq2

Hþ 4

p d vH �q

H�


Hþ 4

p2

!þ 2

q2Hþ 4þ q

H


Hþ 4

p d vH �q

Hþ


Hþ 4

p2

!: ð33Þ

This can be integrated analytically to give a closing flux on the boundary of

sH ¼ q3Hþ 2q

H: ð34Þ

Using Eq. (13), it is also possible to find partial derivative of this moment with respect to the other non-dimensionalized mo-ments at the equilibrium state. The moments in the Hessians of the density and flux potentials can be taken analytically atequilibrium and the derivatives can then be found as

@sH

@qH

��ðqH¼0;rH¼3Þ

¼ 10;@sH

@rH

��ðqH¼0;rH¼3Þ

¼ 0: ð35Þ

It must be remembered that there is a singularity intersecting this state and derivatives are therefore not properly defined inall directions. Nevertheless, these values, computed here using the theories of Section 3.1, are used for the followingderivations.

4.6. Postulated interpolated closure

It now comes time to propose a form for the closing flux. The goal, of course, is a postulated flux that correctly agrees withall the information that could be analytically determined about the maximum-entropy closing flux and transitions in closeagreement with the true flux.

It has been found that a simple interpolation function that preserves the singularity is a very accurate approximation tothe true maximum-entropy closing flux. This function is given by

sH ¼q3

H

r2 þ 2r35q

Hþ 10 1� r3

5

� q

H¼ q3

H

r2 þ 10� 8r35

� q

H: ð36Þ

The exponents 2 and 35 were determined through numerical experimentation in an attempt to closely match the maximum-

entropy closure with the simplest possible expression. This was done by numerically solving the entropy-maximizationproblem for many moment states and searching for the exponents which allowed the simple expression in Eq. (36) to mostclosely match the resulting closing flux. Special attention was payed to the area near the physical-realizability boundary, on

400

800

1200

1600

2000

-400-800

-1200

-1600

-2000

q*

-10 -5 0 5 100

20

40

60

80

00

s*2000160012008004000-400-800-1200-1600-2000

(a)

0.2

0.4

0.6

0.8

1-1

-0.8

-0.2

-0.4

-0.6

q*

r *

-0.04 -0.02 0 0.02 0.042.95

3

3.05

3.1

s*10.60.2-0.2-0.6-1

(b)

Fitted non-dimensionalized closing flux, sH , for the one-dimensional 5-moment closure, Eq. (36): (a) large-scale view, (b) zoom-in aroundium.


which the maximum-entropy closure itself loses hyperbolicity, as eigenvalues and corresponding eigenvectors of the fluxJacobian become equal on this boundary.1

Plots of sH as a function of qH

and rH are shown in Fig. 3. Comparison with Fig. 2 shows that this simple expression agreesvery well with the true maximum-entropy closing flux throughout realizable moment space. Fig. 3(b) shows a zoom-in of thearea surrounding local equilibrium (q

H¼ 0, rH ¼ 3). It can clearly be seen that the singularity touches equilibrium from

above.

4.7. Hyperbolicity of closed-form one-dimensional closure

One goal of this research is moment closures with hyperbolicity that is expanded as compared to traditional moment clo-sures. The original, and still the most popular, family of moment closures is that proposed by Grad [3], shown above in Sec-tion 2.2. It is well know that this family suffers from a restrictive region of hyperbolicity [32–34]. This is caused by the fluxJacobian developing complex eigenvalues.

The five-moment one-dimensional Grad moment closure gives the simple closing relation

1 In aexpressundesirresults,

sH ¼ 10qH: ð37Þ

The small region of hyperbolicity of this closure is contained within a circle around local equilibrium with a radius of approx-imately three in the q

H—rH plane. By contrast, the new closure given in Eq. (36) has a vastly expanded region of hyperbolicity.

Numerical investigation suggests that this new closure is hyperbolic for 0:001 < r < 0:999 and for rH all the way up to avalue of 50,000—truly extreme non-equilibrium.

4.8. Numerical results

In order to investigate the behaviour of the new moment equations, a numerical investigation of the internal structure ofstationary shock waves is shown. Comparisons are made between the five-moment model, shown above; the numericalsolution of the BGK equation, using the discrete-velocity method of Mieussens [42]; and the Navier–Stokes-like equationsthat result from a Chapman–Enskog expansion of Eq. (17) [17]. The system resulting from this expansion is

@q@tþ @

@xquð Þ ¼ 0; ð38Þ

@

@tquð Þ þ @

@xqu2 þ p�

¼ 0; ð39Þ

@

@tqu2 þ p�

þ @

@xqu3 þ 3up�

� @

@x3ps @

@xpq

� �� ¼ 0; ð40Þ

and can be regarded as the continuum (Navier–Stokes) model for one-dimensional physics.An upwind Godunov-type finite-volume scheme with piece-wise limited-linear reconstruction [43] and HLL flux function

[44] is used for the numerical solution of the moment equations. Wave speeds are determined by numerically computing theeigenvalues of the analytic flux Jacobian. A second-order-accurate time-marching scheme is used that treats the hyperbolicleft-hand side of the moment equations explicitly while employing a point-implicit treatment for the stiff, but local, right-hand side [17]. It is given by

�Upi ¼ �Un

i þDtDx

Fni�1

2� Fn

iþ12

� þ Dt �Cp

i ; ð41Þ

�Unþ1i ¼ �Un

i þDt

2DxFn

i�12� Fn

iþ12þ Fp

i�12� Fp

iþ12

� þ Dt

2�Cn

i þ �Cnþ1i

� : ð42Þ

Here, the subscript denotes the cell index and the superscript denotes the time-step index, with a ‘‘p’’ denoting the predictor-step value. An overbar denotes the cell-average value and Fn

i�12

denoted the numerical flux between cell i� 1 and cell ievaluated at the nth time step. A spatial discretization of five thousand equal-sized volumes is used. As Eq. (36) approachesa singularity and the Junk line is approached, special care must be taken during numerical calculations. For this study, thevalue of sigma used in the flux calculation is simply assigned a lower limit of ~r ¼ 2:0� 10�4, if r is less than this value, ~r isused in Eq. (36). This is obviously a crude fix, but it is sufficient for the current purposes.

The relaxation time for the BGK collision operator is held fixed at a value of s ¼ 10�7 s for all computations. The computedresults are therefore not physically accurate, however they do allow an evaluation of the behaviour of the closure for

previous study, [17], fitting software was used to find an approximation to the closing flux in Eq. (24). The resulting function is a complicatedion that does not preserve the singularity present in the maximum-entropy system. At the time it was thought that the presence of this singularity isable and therefore a function that transitioned smoothly across this line was sought. The expression given in Eq. (36) is much simpler, produces superiorand is recommended.


non-equilibrium flows. The upstream values of pressure and density are taken to be standard atmospheric values,101;325 Pa and 1:225 kg=m3, respectively. The mean free path, k, is taken to be

k ¼ 16s5

ffiffiffiffiffiffiffiffiffiffip

2pq

r¼ 3:67� 10�5 m: ð43Þ

This follows from the expression for hard-sphere collisional processes given by Bird [1],

k ¼ 16l5

ffiffiffiffiffiffiffiffiffiffiffiffi2pqp

p ; ð44Þ

and the relationship between the relaxation time and fluid viscosity for the BGK collision operator,
l ¼ sp: ð45Þ
This approximation for the mean free path is not perfectly correct for this one-dimensional situation, however, as this rela-tion is only used to non-dimensionalize the x axis in the results that follow, it provides an acceptable approximation.

Fig. 4 shows the numerical solutions of the 5-moment system as compared to the solution to the one-dimensional Na-vier–Stokes-like equation and the direct numerical solution of the BGK equation with the BGK collision operator usingthe conservative discrete-velocity method of Mieussens [42]. For the solution of the BGK equation, velocity space is discret-ized using 500 velocities spanning �5000 m/s to 5000 m/s for the cases with an upstream Mach number of 2 and 4. For thecase with an upstream Mach number of 8, 1000 velocities spanning �10,000 m/s to 10,000 m/s is needed. Comparison ismade to the BGK equation rather than the Boltzmann equation with the true collision operator so as more accurately makea comparison to the moment closure. Since both solutions have the same collision operator, the only source of deviation isthe moment approximation, which is the interest of this study.

It can be seen that the moment solutions are in far better agreement with direct solution of the kinetic equation than theNavier–Stokes is for all Mach numbers. It can also be observed that, as Mach number increases and non-equilibrium effectsbecome more strong, the Navier–Stokes solutions become worse and worse, whereas the moment solutions maintain a goodagreement with the BGK-equation solutions. Another striking observation is that the hyperbolic moment closure solutionsare smooth—even at high Mach numbers.

4.9. Remarks regarding the smooth shock-structure predictions

For hyperbolic balance laws, it is well known that information travels at finite velcoities that are described by the eigen-values of the flux Jacobian. For shock-structure predictions, if the incoming flow velocity is larger than the fastest equilib-rium wavespeed in the system, then all characteristics will point in the downstream direction and there is no possibilityof upstream information propogation. The only way the flow can transition to the downstream state is through an initialdiscontinuity, or ‘‘sub-shock’’. All previous hyperbolic closures have suffered from this mathematical artifact [4,45].

If the theory of Section 3.1 is followed, the maximum wave speed for a gas at rest and local equilibrium predictied by afive-moment maximum-entropy closure appears to be less than Mach 2. Therefore, why is there no discontinuity in any ofthe results? It is because of the singularity in the closing flux. As the Junk subspace is approached and the closing flux be-comes singular, the maximum wavespeeds also become arbitrarily large. Because this singularity touches equilibrium, thereare states arbitrarily close to equilibrium where the wavespeeds are arbitrarily large. If the solution leaves equilibrium with atrajectory that takes it into a region of very high wavespeeds, discontinuities can travel upstream at very high velocity.

For the shock-structure predictions presented in this paper, the gas always leaves a state of equilibrium with a trajectoryin moment space that takes it into a region where the wavespeeds are high enough that discontinuities generated by thebreakup of the initial conditions always travel upstream and out of the computational domain. Due to the singular natureof the closing flux, the details regarding the behaviour of the moment system near equilibrium are difficult to assess andmuch of the theory for hyperbolic systems is not applicable. This is the subject of ongoing research and will be more deeplyexamined in future work.

5. Closed-form approximation to a three-dimensional moment closure

The above analysis demonstrates the potential for closed-form robustly hyperbolic moment closures to provide afford-able and highly-accurate predictions for both equilibrium and non-equilibrium flow predictions. The resulting system, witha singularity touching local equilibrium, is also a very interesting from a mathematical standpoint. Nevertheless, to be usefulfor practical applications, an extension to realistic, three-dimensional gases is required. The technique used to find the abovefive-moment system followed three steps:

1. First, the region of realizability was determined and a suitable remapping of moment space was employed.2. Next, information about the behaviour of the closing flux at equilibrium and on the realizability boundaries was

determined.3. Finally, a form for the closing flux that transitions between the known boundary states was postulated.

This analysis has been extended to the physically realistic three-dimensional case and is shown here.

x/

0

-20 -15 -10 -5 0 5 10 15 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

5-momentBGKNavier-Stokes

(a) Normalized density, Ma = 2.

x/

q *

-20 -15 -10 -5 0 5 10 15 20-0.8

-0.6

-0.4

-0.2

0


(b) Non-dimensionalized heat flux, Ma = 2.

x/

0

-40 -30 -20 -10 0 10 20 30 40

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


(c) Normalized density, Ma = 4.

x/

q *

-40 -30 -20 -10 0 10 20 30 40

-2

-1

0


(d) Non-dimensionalized heat flux, Ma = 4.

x/

0

-80 -60 -40 -20 0 20 40 60

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


(e) Normalized density, Ma = 8.

x/

q *

-80 -60 -40 -20 0 20 40 60-6

-5

-4

-3

-2

-1

0


(f) Non-dimensionalized heat flux, Ma = 8.

Fig. 4. Predicted normalized density and non-dimensionalized heat-transfer through a stationary shock wave for a one-dimensional gas as determinedusing the 5-moment closure, direct numerical solution of the BGK kinetic equation, and Navier–Stokes-like equations: (a)–(b) Ma ¼ 2, (c)–(d) Ma ¼ 4, and(e)–(f) Ma ¼ 8.



5.1. Fourteen-moment closure

The simplest member of the maximum-entropy hierarchy which has a treatment for heat-transfer for a physically real-istic gas is a 14-moment closure. This is the closure that will be investigated. The vector of generating weights isU ¼ ½1;v i;v iv j;v iv2;v4�T . This leads to a solution vector containing the following moments: q ¼ mFh i, ui ¼ mv iFh i

q ,Pij ¼ mcicjF

� �, Qijj ¼ mcic2F

� �, and Riijj ¼ mc4F

� �. The resulting system of moment equations is

@

@tqþ @

@xiqui½ � ¼ 0; ð46Þ

@

@tqui½ � þ @

@xjquiuj þ Pij� �

¼ 0; ð47Þ

@

@tquiuj þ Pij� �

þ @

@xkquiujuk þ uiPjk þ ujPik þ ukPij þ Q ijk

� �¼ Cij; ð48Þ

@

@tquiujuj þ uiPjj þ 2ujPij þ Q ijj

� �þ @

@xk½quiukujuj þ uiukPjj þ 2uiujPjk þ 2ujukPij þ ujujPik þ uiQ kjj þ ukQ ijj þ 2ujQijk þ Rikjj� ¼ Cijj; ð49Þ

@

@tquiuiujuj þ 2uiuiPjj þ 4uiujPij þ 4uiQ ijj þ Riijj� �

þ @

@xk½qukuiuiujuj þ 2ukuiuiPjj þ 4uiuiujPjk þ 4uiujukPij

þ 2uiuiQ kjj þ 4uiukQ ijj þ 4uiujQ ijk þ 4uiRikjj þ ukRiijj þ Skiijj� ¼ Ciijj; ð50Þ

In order to close the system, expressions for the following moments, which appear only in the flux dyad, are needed:Q ijk ¼ mcicjckF

� �, Rijkk ¼ mcicjc2F

� �, and Sijjkk ¼ mcic4F

� �.

5.2. Realizability of a fourteen-moment three-dimensional closure

In order to determine the region of physical realizability for the moment in the solution vector of the 14-moment closure,an analysis similar to that of Section 4.3 can be undertaken. For this case, the vector, M, is taken as M ¼ ½1;vx;vy;vz;v2�.Choosing the bulk velocity to be zero, this leads to a matrix

Y ¼ mMMTFD E

¼

q 0 0 0 Pii

0 Pxx Pxy Pxz Q xii

0 Pxy Pyy Pyz Q yii

0 Pxz Pyz Pzz Q zii

Pii Q xii Q yii Q zii Riijj

26666664

37777775: ð51Þ

This matrix is positive definite whenever

Riijj P Q kiiðP�1ÞklQ ljj þPiiPjj

q: ð52Þ

The theory of physical realizability for one-dimensional distribution functions in Section 4.3 gives a sufficient condition onrealizability, however, for the multi-dimensional case, it provides only a necessary condition. Nevertheless, for the choice offourteen moments considered here, experience and experimentation involving the numerical solution of many entropy-maximization problems suggests that Eq. (52) does, in fact, describe all physically realizable states.

For this 14-moment closure, the Junk subspace of physically realizable states that cannot be realized by a maximum-en-tropy distribution function is given by

Q ijj ¼ 0 Riijj >2PjiPij þ PiiPjj

q: ð53Þ

This is very similar to the one-dimensional case. If there is any heat flux, there is no problem and if the heat flux is zero, thereis some maximum value of the fourth moment. For the one-dimensional case, this value was that of local equilibrium. Here,when there is no heat flux, the maximum value of Riijj is the value this moment takes when the distribution function is aGaussian, 10-moment distribution [8,45].


5.3. Parabolic mapping of phase space

The similarity of the realizable region for the three-dimensional gas to that of the one-dimensional gas makes a similarparabolic mapping possible. The following expression is used to define the variable, r, which transitions from the Junk sub-space for r ¼ 0 to the physical-realizability boundary for r ¼ 1,

Fig. 5.

Riijj ¼1r

Q kiiðP�1ÞklQ ljj þ2PjiPij þ PiiPjj

q� r 2PjiPij

q¼ 1

rQ kiiðP�1ÞklQ ljj þ

2ð1� rÞPjiPij þ PiiPjj

q: ð54Þ

Given a state, one can therefore compute sigma as

r ¼½2PijPji þ PiiPjj � qRiijj� þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½2PijPji þ PiiPjj � qRiijj�2 þ 8qPmnPnmQ kiiðP�1ÞklQ ljj

q4PijPji

: ð55Þ

5.4. Exact flux expressions on boundaries

A numerical investigation into the maximum-entropy distribution functions corresponding to moment states on thephysical-realizability boundary reveals that, on this boundary, all particles exist on a sphere in velocity space. Fig. 5 showsfour such distribution functions (non-dimensionalized axi-symmetric cases, with v2

r ¼ v2y þ v2

z , are chosen for ease ofillustration).

The fact that the distribution function is non-zero only on a sphere is advantageous. To realize this advantage, one mustfirst define a shifted particle velocity, such that

v i ¼ v i þ ~v i; ð56Þ

vx

v r

-4 -2 0 2 4

-4

-2

0

2

4

(a)ρ = 1, ux = 0, Pxx = 2,Pyy = Pzz = 2, and Qxii = 0

vx

v r

-4 -2 0 2 4

-4

-2

0

2

4

(b)ρ = 1, ux = 0, Pxx = 2,Pyy = Pzz = 7, and Qxii = 0

vx

v r

-4 -2 0 2 4

-4

-2

0

2

4

(c)ρ = 1, ux = 0, Pxx = 2,Pyy = Pzz = 2, and Qxii = 5

vx

v r

-4 -2 0 2 4

-4

-2

0

2

4

(d)ρ = 1, ux = 0, Pxx = 1,Pyy = Pzz = 4, and Qxii = −7

Non-dimensionalized axi-symmetric probability-density function corresponding to several moment states on the physical realizability boundary.


where ~v i is the shift and v i is the particle velocity in the new frame of reference. This is obviously a simple Galilean trans-formation. As the form of the distribution function is invariant under Galilean transformation [8,26], the effect is simply tolook at the same distribution function from a different inertial frame of reference. Moments of the peculiar component ofparticle velocity are obviously unaffected by this translation. The only effect on macroscopic moments is a shift of the appar-ent bulk velocity,

ui ¼ ui þ ~v i: ð57Þ

It is obvious that one such shift results in the sphere of particles being centred on the origin of the new v i space. For theremainder of this work, ~v i is taken to be the particular shift that accomplished this centring for a given moment state. There-fore, ui is the apparent bulk velocity in the frame of reference that centres the sphere for the state of interest.

The fact that, on the realizability boundary, particles exist only on a sphere can now be used to relate some higher-ordermoments to known lower-order moments. This is because the magnitude of the particle velocity is constant on the sphereand any v2 terms in higher-order contracted moment relation can then be taken out of the integral of the moment definition.For example,

bUii ¼ mv2F� �

¼ v2 mFh i ¼ v2q: ð58Þ

The radius of this spherical distribution function is therefore

! ¼

ffiffiffiffiffiffiffibUii

q

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPii þ quiui

q

s: ð59Þ

The bulk velocity of the gas, as viewed from the frame of reference that centres the spherical distribution function on theorigin, can also easily be found, as

mv iv2F� �

¼ !2 mv iFh i;

quiujuj þ uiPjj þ 2ujPij þ Qijj ¼ ujuj þPjj

q

� �quið Þ;

þ þ 2ujPij þ Qijj ¼ þ :

This leads to a bulk velocity of

uj ¼ �12ðP�1ÞijQ ikk: ð60Þ

This technique can also be used to find expressions for Rijkk and Sijjkk on this boundary. These longer calculations are shown inAppendices B and C.

5.5. Postulated interpolated closure

The task now comes to postulating closing relations for the tree tensors mentioned in Section 5.1. These tensors are inves-tigated individually in the following subsections. Long calculations are contained in the appendix to this paper.

5.5.1. Postulated Qijk

As Qijk is not a contracted tensor, the knowledge of the shape of the distribution function on the realizability boundary isof no use. It is possible to determine the derivatives of this moment when the distribution function is a Gaussian. This der-ivation is shown in Appendix A, the result is

@Qijk

@Q mnn¼ 2PilðP2Þjk þ 2PklðP2Þij þ 2PjlðP2Þikh i

B�1lm ¼ Kijkm; ð61Þ

with

Blm ¼ 2PlmðP2Þaa þ 4ðP3Þlm: ð62Þ

Given the small amount of information available about this tensor, the assumed expression for Qijk is taken to be

Q ijk ¼ KijkmQmnn; ð63Þ

with Kijkm defined in Eq. (61).


5.5.2. Postulated Rijkk

Next an expression for Rijkk is developed. At a Gaussian distribution, this can be integrated analytically and has the value

Rijkk ¼1q

PijPkk þ 2PikPkj�

: ð64Þ

On the realizability boundary, this moment can be determined by expanding the moments in the equationbUijkk ¼ hmv iv jv2Fi ¼ !2hmv iv jFi ¼bU kkqbUij and solving for Rijkk. On this boundary it has the expression

Rijkk ¼ Q ijl P�1�

lmQmkk þ

PijPkk

q: ð65Þ

The derivation of this is shown in Appendix B.Remembering the definition of r in Eq. (54), the interpolation that seems most appropriate is

Rijkk ¼1r

Qijl P�1�

lmQmkk þ

2ð1� rÞPikPkj þ PijPkk

q: ð66Þ

5.5.3. Postulated Sijjkk

Finally, an expression for Sijjkk is sought. When the distribution function is Gaussian, the following derivative can be found

@Sijjkk

@Qmnn¼ 1

q2PilðPaaÞ3 þ 12PilðP3Þaa þ 14ðP2ÞaaðP

2Þil þ 20PaaðP3Þilh

þ20ðP4Þil � 2ðP2ÞaaPbbPil � 6ðPaaÞ2ðP2ÞiliB�1

lm

¼Wim: ð67Þ

This derivation is also shown in Appendix A.On the physical realizability boundary, one can use bUijjkk ¼ hmv iv4Fi ¼ !4hmv iFi ¼

bU jj

q

� �2 bUi to find Sijjkk on this bound-ary. This derivation is shown in Appendix C and leads to the expression

Sijjkk ¼ P�1kn P�1

lm Q nppQ mjjQikl þ 2PjjQ ikk

q: ð68Þ

Using this knowledge, the following expression for Sijjkk is proposed

Sijjkk ¼1r2 P�1

kn P�1lm QnppQ mjjQ ikl þ 2r3

5PjjQ ikk

qþ ð1� r3

5ÞWimQ mnn: ð69Þ

The exponents 2 and 35 are chosen for consistency with the one-dimensional study above.

5.6. Numerical results

With the aim of assessing the behaviour of this closure for one-dimensional flows, stationary shock-structure solutionsare again investigated, now for a realistic three-dimensional gas. When restricted to only spacial variations in the x direction,the distribution function remains axi-symmetric about the vx axis. Therefore, the radial particle velocities can be defined asv2

r ¼ v2y þ v2

z and c2r ¼ c2

y þ c2z . This leads to moment relations: uy ¼ uz ¼ 0, Prr ¼ Pyy þ Pzz, Pxy ¼ Pxz ¼ Pyz ¼ 0, and

Qyii ¼ Qzii ¼ 0. The fourteen-moment equations with BGK collision operator then reduce to six moment equations,

@q@tþ @

@xqux½ � ¼ 0; ð70Þ

@

@tqux½ � þ @

@xqu2

x þ Pxx� �

¼ 0; ð71Þ

@

@tqu2

x þ Pxx� �

þ @

@xqu3

x þ 3uxPxx þ Q xxx

� �¼ Prr � 2Pxx

3s; ð72Þ

@

@tPrr½ � þ

@

@xuxPrr þ Q xrr½ � ¼ 2Pxx � Prr

3s; ð73Þ

@

@tqu3

x þ uxð3Pxx þ PrrÞ þ Qxii

� �þ @

@xqu4

x þ u2x ð6Pxx þ PrrÞ þ 2uxðQ xii þ Q xxxÞ þ Rxxii

� �¼ 2uxðPrr � 2PxxÞ � 3Q xii

3s; ð74Þ

Table 1Discreti

MiniMaxMaxnx

nr

TotaDvx

Dvr


@

@tqu4

x þ u2x ð6Pxx þ 2PrrÞ þ 4uxQ xii þ Riijj

� �þ @

@xqu5

x þ u3x ð10Pxx þ 2PrrÞ þ u2

x ð6Q xii þ 4QxxxÞ þ uxðRiijj þ 4RxxiiÞ þ Sxiijj� �

¼ 13s

4u2x ðPrr � 2PxxÞ � 12uxQ xii þ 5

ðPxx þ PrrÞ2

q� 3Riijj

" #: ð75Þ

The closing relations postulated above reduce to

Q xxx ¼ AQxii; Q xrr ¼ ð1� AÞQ xii; ð76Þ

Rxxii ¼Ar

Q 2xii

Pxxþ 2ð1� rÞP2

xx þ ðPxx þ PrrÞPxx

q; ð77Þ

Sxiijj ¼Ar2

Q 3xii

P2xx

þ 2q

Pxx þ Prr þ 1� r35

� P3rr þ 2P2

rrPxx þ 24P3xx

P2rr þ 6P2

xx

!Q xii; ð78Þ

with

A ¼ 6P2xx

P2rr þ 6P2

xx

; ð79Þ

and

r ¼ð2P2

xx þ P2rrÞ þ ðPxx þ PrrÞ2 � qRiijj þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2P2

xx þ P2rrÞ þ ðPxx þ PrrÞ2 � qRiijj

h i2þ 4qð2P2

xx þ P2rrÞ

Q2xii

Pxx

r2ð2P2

xx þ P2rrÞ

ð80Þ

Comparisons will again be made to the Navier–Stokes equations that result from a Chapman–Enskog expansion of the BGKkinetic equation [46]. For a realistic three-dimensional monatomic gas, this results in the following equations for one-dimen-sional flows,

@q@tþ @

@xqux½ � ¼ 0; ð81Þ

@

@tqux½ � þ @

@xqu2

x þ p� �

� @

@x43sp@ux

@x

� �¼ 0; ð82Þ

@

@t12qu2

x þ32

p� �

þ @

@x12qu3

x þ52

uxp� �

� @

@x43

uxsp@ux

@xþ 5

2sp

@

@xpq

� �� ¼ 0: ð83Þ

These are the classical one-dimensional Navier–Stokes equations with viscosity, l ¼ sp, and thermal conductivity,j ¼ 5ksp=ð2mÞ (here, k is Boltzmann’s constant).

The same numerical scheme presented in Section 4.8 is used with the upstream density and pressure given by standardatmospheric values and a relaxation time of s ¼ 10�7s. The spatial domain is now discretized into one thousand cells. Onceagain, upstream Mach numbers of 2;4, and 8 are considered. The speed of sound for a monatomic gas at these conditions is

vsound ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi53

101;325 Pa1:225 kg=m3

s� 371 m=s: ð84Þ

zation of velocity space for three-dimensional conservative BGK numerical solutions.

Mach 2 Mach 4 Mach8

mum vx �2000 m/s �3000 m/s �5000 m/simum vx 3000 m/s 4000 m/s 7000 m/simum vr 2000 m/s 3000 m/s 5000 m/s

250 350 600100 150 250

l number of discrete velocities 25,000 52,000 150,00020 m/s 20 m/s 20 m/s20 m/s 20 m/s 20 m/s

x/

0

-20 -10 0 10 20

0

0.2

0.4

0.6

0.8

1

BGKNavier-Stokes14-Moment


x/

P 0

-20 -10 0 10 20

0

0.2

0.4

0.6

0.8

1Pxx - BGKPyy - BGKPxx - 14-MomentPyy - 14-Moment

(b) Normalized directional pressures, Ma = 2.

x/

P 0

-20 -10 0 10 20

0

0.2

0.4

0.6

0.8

1BGKNavier-Stokes14-Moment

(c) Normalized thermodynamic pressure, Ma = 2.

x/

0

-20 -10 0 10 20

0

0.2

0.4

0.6

0.8

1

1.2

xx - BGKyy - BGKxx - 14-Momentyy - 14-Moment

(d) Normalized directional temperatures, Ma = 2.

x/

0

-20 -10 0 10 20

0

0.2

0.4

0.6

0.8


(e) Normalized thermodynamic temperature, Ma = 2.

x/

Qxi

i *

-20 -10 0 10 20

-1

-0.8

-0.6

-0.4

-0.2

0



Fig. 6. Predicted normalized density, pressure, and temperature profiles as well as non-dimensionalized heat-transfer through a stationary shock wavewith an inflow Mach number of 2 for a realistic three-dimensional gas as predicted by the closed-form 14-moment equations, Navier–Stokes equations, andBGK kinetic equation. Symbol spacing is not representative of grid resolution.


x/

0

-30 -20 -10 0 10 20

0

0.2

0.4

0.6

0.8

1



x/

P 0

-30 -20 -10 0 10 20

0

0.2

0.4

0.6

0.8



x/

P 0

-30 -20 -10 0 10 20

0

0.2

0.4

0.6

0.8



x/

0

-30 -20 -10 0 10 20

0

0.2

0.4

0.6

0.8

1

1.2xx - BGKyy - BGKxx - 14-Momentyy - 14-Moment


x/

0

-30 -20 -10 0 10 20

0

0.2

0.4

0.6

0.8



x/

Qxi

i *

-30 -20 -10 0 10 20-5

-4

-3

-2

-1

0





x/

0

-60 -40 -20 0 20 40

0

0.2

0.4

0.6

0.8

1



x/

P 0

-60 -40 -20 0 20 40

0

0.2

0.4

0.6

0.8



x/

P 0

-60 -40 -20 0 20 40

0

0.2

0.4

0.6

0.8



x/

0

-60 -40 -20 0 20 400

0.2

0.4

0.6

0.8

1

1.2

1.4

xx - BGKyy - BGKxx - 14-Momentyy - 14-Moment


x/

0

-60 -40 -20 0 20 40

0

0.2

0.4

0.6

0.8



x/

Qxi

i *

-60 -40 -20 0 20 40-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0






For the BGK computations, the velocity distribution function will remain axi-symmetric and a two-dimensional discretiza-tion of velocity space is sufficient. Table 1 shows the maximum and minimum velocities; number of discrete velocities, nx

and nr; and the spacing of discrete velocities, Dvx and Dv r , used for the three cases considered.Results of this study are shown in Fig. 6–8. Again, the moment-closure solutions are smooth and are in much better agree-

ment with those of the kinetic equation than the Navier–Stokes solutions. The predicted non-dimensionalized heat flux is infar better agreement, especially at higher Mach numbers. The moment-closure solution also predicts pressure and temper-ature anisotropies that are not possible within the Navier–Stokes treatment. The overshoot of the temperature normal to theshock wave is in amazing agreement with the solution to the kinetic equation at all Mach numbers. This overshoot is ex-pected in shock waves, as energy is added more quickly to this mode and is then later redistributed by inter-particle colli-sions as local equilibrium is recovered.

5.7. Hyperbolicity of three-dimensional closure

Though the closure for a one dimensional gas shown in Section 4 is thought to be globally hyperbolic for all physicallyrealizable states. This closure for a three-dimensional gas is not. For the numerical results shown above, the eigenvaluesof the analytic flux Jacobian in each cell were computed at each time step for use in the HLL flux function and CFL time-steprestriction. During the computation of the shock profiles for flows with Mach numbers of 2 and 4, these eigenvalues werealways real and the system deemed hyperbolic. For the case with a flow Mach number of 8, some complex eigenvalues withsmall imaginary components were observed. The region of hyperbolicity does, however, seem to be large enough for the reli-able computation of flow solutions involving large deviation from local thermal equilibrium. It is hoped that further refine-ment of this model will bring true global hyperbolicity and may lead to even better solution agreement with the kineticequation.

6. Conclusions

It has been demonstrated that affordable closed-form moment closures can be postulated if one takes the maximum-entropy hierarchy as a guide. The widespread adoption of moment-closure-based techniques has always been hampered bythe lack of such affordable and robustly hyperbolic closures. For the one-dimensional gas examined in Section 4, a simplesystem with an extremely, possibly global, region of hyperbolicity is demonstrated. Though the closed-form closure for arealistic gas developed in Section 5 is not globally hyperbolic, it does appear to have a region of hyperbolicity that is largeenough to allow the reliable prediction of flows with much larger non-equilibrium effects than traditional closures haveallowed.

The presence of the Junk subspace of non-realizability for the maximum-entropy framework has been considered a some-what devastating characteristic of these closures. This paper shows that, for the closures considered, the nature of this regionand the behaviour of the closing fluxes as it is approached actually gives the possibility of smooth shock profiles, even forflows with very high Mach numbers. The undesirable ‘‘sub-shocks’’, present in other moment-closure solutions, have longbeen considered an inevitable characteristic of all hyperbolic moment closures. This works shows that this is not the case.Smooth shock profiles are possible in a hyperbolic setting.

Acknowledgement

This work was partially funded by a post-doctoral-fellowship grant from the Natural Science and Engineering ResearchCouncil of Canada. The authors are grateful for this support.

Appendix A. Determination of ›Q ijk›Q lmm

and ›Sijjkk›Q lmm

at a Gaussian distribution

As stated in Section 3.1, the flux Jacobian in the n direction for a maximum-entropy moment closure can be computedusing Eq. (13). When the distribution function is a Gaussian [8,45] with zero bulk velocity, all odd moments are zero. By sym-metry, the derivatives of all even moments with respect to odd moments as well as the derivatives of odd moments withrespect to even moments must be zero. This causes Eq. (13) to decompose into two separate systems. One of these systemsis given by

g@Fn

@U¼ gðfn;a;aÞ gðh;a;aÞ�1: ðA:1Þ

Using the fact that, with zero bulk velocity, @Unij

@Uxkk¼ @Qnij

@Qxkkand @Uniijj

@Uxkk¼ @Sniijj

@Qxkk, these matrices are given by the following

expressions:


g@Fn

@U¼

@Uf

@Ux

@Uf

@Uy

@Uf

@Uz0 0 0

@Unxx@Ux

@Unxx@Uy

@Unxx@Uz

@Qnxx@Qxii

@Qnxx@Qyii

@Qnxx@Qzii

@Unxy

@Ux

@Unxy

@Uy

@Unxy

@Uz

@Qnxy

@Qxii

@Qnxy

@Qyii

@Qnxy

@Qzii

@Unxz@Ux

@Unxz@Uy

@Unxz@Uz

@Qnxz@Qxii

@Qnxz@Qyii

@Qnxz@Qzii

@Unyy

@Ux

@Unyy

@Uy

@Unyy

@Uz

@Qnyy

@Qxii

@Qnyy

@Qyii

@Qnyy

@Qzii

@Unyz

@Ux

@Unyz

@Uy

@Unyz

@Uz

@Qnyz

@Qxii

@Qnyz

@Qyii

@Qnyz

@Qzii

@Unzz@Ux

@Unzz@Uy

@Unzz@Uz

@Qnzz@Qxii

@Qnzz@Qyii

@Qnzz@Qzii

@Uniijj

@Ux

@Uniijj

@Uy

@Uniijj

@Uz

@Sniijj

@Qxkk

@Sniijj

@Qykk

@Sniijj

@Qzkk

2666666666666666666664

3777777777777777777775

; ðA:2Þ

gðfn;a;aÞ ¼

Pnx Pny Pnz Rnxjj Rnyjj Rnzjj

Rnxxx Rnxxy Rnxxz Vnxxxii Vnxxyii Vnxxzii

Rnxxy Rnxyy Rnxyz Vnxxyii Vnxyyii Vnxyzii

Rnxxz Rnxyz Rnxzz Vnxxzii Vnxyzii Vnxzzii

Rnxyy Rnyyy Rnyyz Vnxyyii Vnyyyii Vnyyzii

Rnxyz Rnyyz Rnyzz Vnxyzii Vnyyzii Vnyzzii

Rnxzz Rnyzz Rnzzz Vnxzzii Vnyzzii Vnzzzii

Vnxiijj Vnyiijj Vnziijj Vnxiijjkk Vnyiijjkk Vnziijjkk

266666666666664

377777777777775; ðA:3Þ

gðh;a;aÞ�1 ¼

Pxx Pxy Pxz Rxxii Rxyii Rxzii

Pxy Pyy Pyz Rxyii Ryyii Ryzii

Pxz Pyz Pzz Rxzii Ryzii Rzzii

Rxxii Rxyii Rxzii Vxxiijj Vxyiijj Vxziijj

Rxyii Ryyii Ryzii Vxyiijj Vyyiijj Vyziijj

Rxzii Ryzii Rzzii Vxziijj Vyziijj Vzziijj

2666666664

3777777775

�1

¼Pij Rijkk

Rijkk Vijkkll

� ��1

¼A BB C

� �; ðA:4Þ

with

A ¼ Pij � RikllðV�1ÞkmnnooRjmpp

� �1; ðA:5Þ

B ¼ �ðP�1ÞinRnopp Vojkkll � RokllðP�1ÞkmRjmnn

� �1; ðA:6Þ

C ¼ Vijkkll � RikllðP�1ÞkmRjmnn

� �1: ðA:7Þ

Multiplication of these matrices reveals the following expressions:

@Qnqr

@Q jll¼ Vnoqrpp � RniqrðP�1ÞinRnopp

h iVojkkll � RokllðP�1ÞkmRjmnn

h i�1; ðA:8Þ

and

@Snqqrr

@Q jll¼ Vniqqrrpp � VniqrssðP�1ÞinRnopp

h iVojkkll � RokllðP�1ÞkmRjmnn

h i�1: ðA:9Þ

These higher-order moments can be integrated exactly when the distribution function is a Gaussian and have the values

Rijkl ¼PijPkl þ PikPjl þ PilPjk

q; ðA:10Þ

Rijkk ¼PijPkk þ 2ðP2Þij

q; ðA:11Þ

Vijklmm ¼Pmm PijPkl þ PikPjl þ PilPjk

� �q2 þ 2

PijðP2Þkl þ PikðP2Þjl þ PilðP2Þjk þ PklðP2Þij þ PjkðP2Þil þ PjlðP2Þikh i

q2 ; ðA:12Þ


Vijkkll ¼ðPkkÞ2Pij þ 4PkkðP2Þij þ 2ðP2ÞkkPij þ 8ðP3Þij

q2 ; ðA:13Þ

Vijkkllmm ¼3ðPkkÞ3Pij þ 12ðP3ÞkkPij þ 18ðP2ÞkkðP

2Þij þ 36PkkðP3Þij þ 36ðP4Þijq3 : ðA:14Þ

Substituting these expressions into Eq. (A.8) and (A.9) yields

@Q nqr

@Q jll¼ 2PnqðP2Þro þ 2PqoðP2Þnr þ 2PqrðP2Þno

h i2PjoðP2Þkk þ 4ðP3Þjoh i�1

; ðA:15Þ

and

@Snqqrr

@Qjll¼ 2PnoðPkkÞ3þ12PnoðP3Þkkþ14ðP2ÞkkðP

2Þnoþ20PkkðP3Þno

hþ20ðP4Þno�2ðP2ÞkkPllPno�6ðPkkÞ2ðP2Þno

i2PjoðP2Þkkþ4ðP3Þjoh i�1

: ðA:16Þ

Appendix B. Determination of Rijkk on physical-realizability boundary

Following the technique shown in Section 5.4 the moment Rijkk on the physical-realizability boundary can be found usingthe relation,

bUijkk ¼ mv iv jvkvkF� �

¼ !2 mv iv jF� �

¼bUkk

qbUij: ðB:1Þ

This is expanded, then simplified using Eq. (60) as follows,

bUijkk ¼bUkk

qbUij;

quiujukuk þ uiujPkk þ 2uiukPjk þ 2ujukPik þ ukukPij þ uiQ jkk þ ujQ ikk þ 2ukQ ijk þ Rijkk ¼qukuk þ Pkk

q

� �quiuj þ Pij�

;

þ þ 2uiukPjk þ 2ujukPik þ þ uiQ jkk þ ujQ ikk þ 2ukQ ijk þ Rijkk

¼ þ þ þ PijPkk

q;

2uiukPjk þ 2ujukPik þ uiQ jkk þ ujQ ikk|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}Q jkk ¼ �2ukPjk

Q ikk ¼ �2ukPik

þ2ukQ ijk þ Rijkk ¼PijPkk

q;

þ � � þ 2ukQ ijk þ Rijkk ¼PijPkk

q;

Rijkk ¼ �2ukQ ijk|fflfflfflfflfflffl{zfflfflfflfflfflffl}uk¼�1

2ðP�1ÞklQ lmm

þ PijPkk

q;

Rijkk ¼ Qijl P�1�

lmQ mkk þ

PijPkk

q: ðB:2Þ

Appendix C. Determination of Sijjkk on physical-realizability boundary

The derivation of Sijjkk on the realizability boundary is similar to that for Rijkk given in Appendix B. It begins with therelation

bUijjkk ¼ mv iv jv jvkvkF� �

¼ !4 mv iFh i ¼bUjj

q

bUkk

qbUi: ðC:1Þ

Again, this is expanded and reduced, making repeated use of Eq. (60), as follows


bUijjkk ¼bUjj

q

bUkk

qbUi;

quiujujukuk þ 2uiujujPkk þ 4ukujujPik þ 4uiujukPjk

þ2ujujQ ikk þ 4uiujQ jkk þ 4ujukQ ijk þ 4ujRijkk

þuiRjjkk þ Sijjkk

¼ qujuj þ Pjj

q

� �qukuk þ Pkk

q

� �qui;

þ þ 4ukujujPik þ 4uiujukPjk þ 2ujujQ ikk þ 4uiujQ jkk þ 4ujukQ ijk þ 4ujRijkk þ uiRjjkk

þ Sijjkk

¼ þ þ uiPjjPkk

q

4ukujujPik þ 4uiujukPjk þ 2ujujQ ikk

zfflfflfflfflfflffl}|fflfflfflfflfflffl{Qikk¼�2ukPik

þ 4uiujQ jkk

zfflfflfflfflfflffl}|fflfflfflfflfflffl{Qjkk¼�2ukPjk

þ4ujukQ ijk þ 4ujRijkk þ uiRjjkk þ Sijjkk ¼uiPjjPkk

q;

þ � � þ 4ujukQijk þ 4ujRijkk þ uiRjjkk þ Sijjkk ¼uiPjjPkk

q;

4ujukQ ijk þ 4ujRijkk þ uiRjjkk þ Sijjkk ¼ 4uiujukPjk þuiPjjPkk

q;

4ujukQ ijk þ 4ujRijkk þ ui Q ljjðP�1ÞlmQ mkk þPjjPkk

q

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

¼Rjjkk

þSijjkk ¼ 4uiujukPjk þuiPjjPkk

q;

4ujukQ ijk þ 4ujRijkk þ ui Q ljjðP�1ÞlmQ mkk|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}Q ljj ¼ �2ujPjl

Q mkk ¼ �2ukPkm

þ þ Sijjkk ¼ 4uiujukPjk þ ;

References

[1] G.A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon Press, Oxford, 1994.[2] A.V. Bobylev, The Chapman–Enskog and grad methods for solving the Boltzmann equation, Sov. Phys. Dokl. 27 (1) (1982) 29–31.[3] H. Grad, On the kinetic theory of rarefied gases, Commun. Pure Appl. Math. 2 (1949) 331–407.[4] I. Müller, T. Ruggeri, Rational Extended Thermodynamics, Springer-Verlag, New York, 1998.[5] J.G. McDonald, J.S. Sachdev, C.P.T. Groth, Gaussian moment closure for the modelling of continuum and micron-scale flows with moving boundaries, in:

H. Deconinck, E. Dick (Eds.), Proceedings of the Fourth International Conference on Computational Fluid Dynamics, ICCFD4, Ghent, Belgium, July 10–14,2006, Springer-Verlag, Heidelberg, 2009, pp. 783–788.

[6] P. Degond, G. Dimarco, L. Pareschi, The moment guided Monte Carlo method, Int. J. Numer. Meth. Fluids 67 (2) (2011) 189–213.[7] W. Dreyer, Maximization of the entropy in non-equilibrium, J. Phys. A: Math. Gen.[8] C.D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys. 83 (1996) 1021–1065.[9] H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows, Springer-Verlag, Berlin, 2005.

http://refhub.elsevier.com/S0021-9991(13)00413-0/h0005











[10] S.L. Brown, P.L. Roe, C.P.T. Groth, Numerical solution of a 10-moment model for nonequilibrium gasdynamics, Paper 95–1677, AIAA (June 1995).[11] S.L. Brown, Approximate Riemann solvers for moment models of dilute gases, Ph.D. thesis, University of Michigan (1996).[12] P.L. Tallec, J.P. Perlat, Numerical analysis of Levermore’s moment system, Tech. Rep. 3124, INRIA Rocquencourt (1997).[13] J.G. McDonald, C.P.T. Groth, Numerical modeling of micron-scale flows using the Gaussian moment closure, Paper 2005–5035, AIAA (June 2005).[14] J.G. McDonald, C.P.T. Groth, Extended fluid-dynamic model for micron-scale flows based on Gaussian moment closure, Paper 2008–0691, AIAA

(January 2008).[15] C.P.T. Groth, J.G. McDonald, Towards physically-realizable and hyperbolic moment closures for kinetic theory, Continuum Mech. Thermodyn. 21

(2009) 467–493.[16] J.G. McDonald, Extended fluid-dynamic modelling for numerical solution of micro-scale, Ph.D. thesis, University of Toronto (2011).[17] J.G. McDonald, C.P.T. Groth, Towards realizable hyperbolic moment closures for viscous heat-conducting gas flows based on a maximum-entropy

distribution, Continuum Mech. Thermodyn.DOI:10.1007/s00161-012-0252-y.[18] C.K.S. Lam, C.P.T. Groth, Numerical prediction of three-dimensional non-equilibrium gaseous flows using the Gaussian moment closure, Paper 2011–

3401, AIAA (June 2011).[19] Y. Suzuki, B. van Leer, Application of the 10-moment model to MEMS flows, Paper 2005–1398, AIAA (January 2005).[20] Y. Suzuki, L. Khieu, B. van Leer, CFD by first order PDEs, Continuum Mech. Thermodyn. 21 (2009) 445–465.[21] T.J. Barth, On discontinuous Galerkin approximations of Boltzmann moment systems with Levermore closure, Comput. Meth. Appl. Mech. Engrg. 195

(2006) 3311–3330.[22] J. Coulombel, F. Golse, T. Goudon, Diffusion approximation and entropy-based moment closure for kinetic equations, Asymptot. Anal. 45 (1–2) (2005)

1–39.[23] J. Coulombel, T. Goudon, Entropy-based moment closure for kinetic equations: Riemann problem and invariant regions, J. Hyperbolic Differ. Equat. 3

(4) (2006) 649–671.[24] J.A. Carrillo, T. Goudon, P. Lafitte, F. Vecil, Numerical schemes of diffusion asymptotics and moment closures for kinetic equations, J. Sci. Comput. 36 (1)

(2008) 113–149.[25] M. Junk, Domain of definition of Levermore’s five-moment system, J. Stat. Phys. 93 (5/6) (1998) 1143–1167.[26] M. Junk, A. Unterreiter, Maximum entropy moment systems and Galilean invariance, Continuum Mech. Thermodyn. 14 (2002) 563–576.[27] M. Junk, Maximum entropy moment problems and extended Euler equations, in: proceedings of IMA Workshop Simulation of Transport in Transition

Regimes, Vol. 135 of IMA Vol. Math. Appl, Springer, 2004, pp. 189–198.[28] H. Struchtrup, M. Torrilhon, Regularization of grad’s 13 moment equations: derivation and linear analysis, Phys. Fluids 15 (2003) 2668–2680.[29] H. Struchtrup, M. Torrilhon, H theorem, regularization, and boundary conditions for the linearized 13 moment equations, Phys. Rev. Lett. 99 (2007)

014502.[30] M. Torrilhon, H. Struchtrup, Boundary conditions for regularized 13 moment equations for micro-channel-flows, J. Comput. Phys. 227 (2008) 1982–

2011.[31] P.L. Bhatnagar, E.P. Gross, M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component

systems, Phys. Rev. 94 (3) (1954) 511–525.[32] M. Torrilhon, Characteristic waves and dissipation in the 13-moment-case, Continuum Mech. Thermodyn. 12 (2000) 289–301.[33] F. Brini, Hyperbolicity region in extended thermodynamics with 14 moments, Continuum Mech. Thermodyn. 13 (2001) 1–8.[34] M. Torrilhon, Hyperbolic moment equations in kinetic gas theory based on multi-variate Pearson-IV-distributions, Commun. Comput. Phys. 7 (4)

(2010) 639–673.[35] L. Boltzmann, Weitere studien über das wärmegleichgewicht unter gasmolekülen, Sitz. Math.-Naturwiss. Cl. Akad. Wiss. Wien 66 (1872) 275–370.[36] I. Müller, Thermodynamics, Pitman Publishing, Boston, 1985.[37] S.K. Godunov, An interesting class of quasilinear systems, Sov. Math. Dokl. 2 (1961) 947–949.[38] K.O. Friedrichs, P.D. Lax, Systems of conservation laws with a convex extension, ProcNAS 68 (1971) 1686–1688.[39] H.L. Hamburger, Hermitian transformations of deficiency-index (1,1), Jacobian matrices, and undetermined moment problems, Amer. J. Math. 66

(1944) 489–552.[40] M. Reed, B. Simon, Methods of Modern Mathematical Physics, II: Fourier Analysis, Academic Press, New York, Self-Adjointness, 1975.[41] N.I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Oliver & Boyd, Edinburgh, 1965.[42] L. Mieussens, Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics, Math. Models Methods Appl. Sci. 10 (8)

(2000) 1121–1149.[43] V. Venkatakrishnan, On the accuracy of limiters and convergence to steady state solutions, Paper 93-0880, AIAA (January 1993).[44] A. Harten, P.D. Lax, B. van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev. 25 (1) (1983) 35–61.[45] C.D. Levermore, W.J. Morokoff, The gaussian moment closure for gas dynamics, SIAM J. Appl. Math. 59 (1) (1996) 72–96.[46] T.I. Gombosi, Gaskinetic Theory, Cambridge University Press, Cambridge, 1994.









































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