PHYSICAL REVIEW E 94, 023313 (2016)

Simplification of the unified gas kinetic scheme

Songze Chen,1,* Zhaoli Guo,1,† and Kun Xu2,‡1State Key Laboratory of Coal Combustion, School of Energy and Power Engineering, Huazhong University of Science and Technology,

Wuhan 430074, China2The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

(Received 9 February 2016; published 22 August 2016)

The unified gas kinetic scheme (UGKS) is an asymptotic preserving (AP) scheme for kinetic equations. Itis superior for transition flow simulation and has been validated in the past years. However, compared to thewell-known discrete ordinate method (DOM), which is a classical numerical method solving the kinetic equations,the UGKS needs more computational resources. In this study, we propose a simplification of the unified gaskinetic scheme. It allows almost identical numerical cost as the DOM, but predicts numerical results as accurateas the UGKS. In the simplified scheme, the numerical flux for the velocity distribution function and the numericalflux for the macroscopic conservative quantities are evaluated separately. The equilibrium part of the UGKS fluxis calculated by analytical solution instead of the numerical quadrature in velocity space. The simplificationis equivalent to a flux hybridization of the gas kinetic scheme for the Navier-Stokes (NS) equations and theconventional discrete ordinate method. Several simplification strategies are tested, through which we can identifythe key ingredient of the Navier-Stokes asymptotic preserving property. Numerical tests show that, as long as thecollision effect is built into the macroscopic numerical flux, the numerical scheme is Navier-Stokes asymptoticpreserving, regardless the accuracy of the microscopic numerical flux for the velocity distribution function.

DOI: 10.1103/PhysRevE.94.023313

I. INTRODUCTION

In recent years, multiscale computation is recognized as apowerful tool for studying the interaction on different scalesand/or different hierarchies. It has become an active researchfield and has been applied in many areas, for instance, rarefiedgas dynamics, radioactive, plasma, and phonon transfer.

In the rarefied gas dynamics, the physical scales arecharacterized by the typical geometric length (L) and meanfree path (λ). The ratio of these two characteristic lengthsis known as the Knudsen number (Kn = λ/L). When theKnudsen number is much smaller than 1, it is well knownthat the Navier-Stokes equations are established and governthe fluid behavior. However, when the Knudsen number is nottoo small, the Navier-Stokes equations are no longer accurateenough to describe the flow motion, and the kinetic equationshould be adopted as the governing equation. The simplestkinetic equation for monatomic gas is the BGK equation [1],which takes the form

∂f

∂t+ u · ∇f = g − f

τ, (1)

where f represents the particle velocity distribution functiondepending on the location (x), the time (t), and particle velocity(u), and g denotes the corresponding local equilibrium stateshown as

g = M(W ) = ρ

{1

2RT π

}3/2

exp

[− 1

2RT(u − U)2

], (2)

W = (ρ,ρU,ρE)T . (3)

*jacksongze@hust.edu.cn†zlguo@hust.edu.cn‡makxu@ust.hk

where ρ is the gas density, U is the gas velocity, T is the gastemperature, R is the gas constant, and ρE denotes the totalenergy. Since the collision process is conserved, g and f shareidentical conservative quantities,

〈ψg〉 = 〈ψf 〉, ψ = (1,u, 1

2 u2)T

. (4)

The symbol 〈f 〉 is defined as 〈f 〉 = ∫ +∞−∞ f du.

Typically, the flow regimes can be categorized into fourregimes: continuum flow (Kn < 0.001), slip flow (0.001 <

Kn < 0.1), transition flow (0.1 < Kn < 10), and free molec-ular flow (Kn > 10). The Navier-Stokes equations are onlyvalidated in the continuum flow regime and can be furtherextended to solve a small portion of slip flow problemsby considering slip boundary conditions. For the other flowregimes, the kinetic theory, including the Boltzmann equationand other kinetic equations, must be adopted to account for thedelicate molecular motion. For example, when a vehicle travelsthrough the atmosphere, the density of ambient gas changesdramatically [2]. In another scenario, the gas is driven by thetemperature gradient as it goes through different chambersin a multistage Knudsen pump [3,4]. The Knudsen numbervaries widely in the above two applications and leads to thefailure of the Navier-Stokes equations. Thus, particle-basedmethods [5,6] and the kinetic equation are necessary to takeover in the nonequilibrium domain where NS equations breakdown. An intuitive idea is the domain decomposition method,in which the flow field is solved on different subdomainsby appropriate numerical solvers, the Navier-Stokes solvers,or the kinetic solvers. However, the major difficulty of thedomain decomposition method is the information exchange inthe buffer zone or the overlap region between two numericalmethods on different scales. Moreover, in many multiscaleproblems, the Knudsen number varies both in space andtime. The domain decomposition method is incapable for suchproblems.

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SONGZE CHEN, ZHAOLI GUO, AND KUN XU PHYSICAL REVIEW E 94, 023313 (2016)

Another promising multiscale approach is the asymptoticpreserving (AP) scheme that can recover large-scale systemsfrom small-scale kinetic simulation uniformly [7]. When theKnudsen number goes to zero, the numerical scheme forthe kinetic equation should be an analog of the analyticalasymptotic analysis of the kinetic equation. In 1991, Coronand Perthame [8] proposed a scheme which is asymptoticpreserving in terms of Euler equations. After this study,variant AP schemes for the rarefied gas system were proposedin the past two decades, including implicit schemes forthe collision terms [9,10], penalization methods [11–13],exponential relaxation methods [8,14], unified gas kineticschemes [15–18], discrete unified gas kinetic schemes [19–21],etc.

From the previous literature, two key ingredients of theasymptotic preserving scheme can be seen. The first keyingredient is the special treatment of the collision term [right-hand side of Eq. (1)]. The traditional discrete ordinate method(DOM) solves the collision term explicitly. The time step isalways restricted by the Knudsen number and cannot obtaina physical solution in near-continuum and continuum flowregimes unless using infinite computation resources. Actually,the stiffness of the collision term due to the small parametermakes the explicit schemes for the kinetic equation uselessin the continuum flow regime. Therefore, the exponentialcollision solver [8,14,22] and implicit treatment of the collisionterm [11,12,23] were proposed to remove the stiffness of thecollision term.

The other ingredient of the AP scheme is that the completekinetic equation must be employed to solve the numericalflux at a cell interface and the volume force inside a cell inorder to attain the correct Navier-Stokes limit [24]. In the earlystage, the operator splitting method is employed to simplifythe numerical scheme. The governing equation is modifiedfor different purposes. For solving the interfacial numericalflux, the convection term is reserved, but the source term isdiscarded. The governing equation becomes

∂f

∂t+ u · ∇f = 0. (5)

For solving the volume force, only the source term is reserved,while the convection term is abandoned:

∂f

∂t= g − f

τ. (6)

Bennoune et al. [25] investigated the influence of the implicitschemes for the collision term and found that, if the operatorsplitting method is employed to evaluate the collision term,the resulting distribution function will be too close to theequilibrium state; thus, the schemes cannot attain the physicalviscosity. Chen and Xu [24] studied the Navier-Stokes asymp-totic preserving property and concluded that not only does thevolume force need both convection and collision terms, but thenumerical fluxes also need these two terms in order to obtainthe correct Navier-Stokes limit.

In 2010, Xu et al. proposed the unified gas kinetic scheme,which couples the collision and convection terms by a localintegral solution of the complete BGK equation (1). Whenthe Knudsen number goes to zero, the numerical flux of theunified gas kinetic scheme (UGKS) turns to the Chapman-

Enskog expansion gradually. Theoretically, UGKS can recoverthe NS limit and the Euler limit. With the same spirit, Guoet al. proposed a discrete unified gas kinetic scheme (DUGKS)which replaces the local integral solution by a discrete timeintegral. Although the discrete approximation is adopted, theDUGKS still possesses the NS AP property.

However, quadrature, which accounts for the numericalintegral in discrete velocity space, is an obstacle for attainingcorrect asymptotic limit in the continuum flow regime. As weknow, in the free molecular flow regime, the Newton-Cotesquadrature is more suitable compared to the Gauss-Hermitequadrature because the distribution function deviates largelyfrom the equilibrium state. However, in the continuum flow,the Gauss-Hermite quadrature is always used due to its highaccuracy for the integral of exponential function. If differentquadratures are employed, massive interpolations will beneeded to exchange data on different velocity points. Also,it will introduce additional numerical errors. In practice, aunified scheme is still a burdensome way to reproduce thecontinuum flow limit.

From another perspective, the Navier-Stokes equations canbe derived from the kinetic equation analytically, and thetraditional numerical schemes for the Navier-Stokes equationsare highly efficient. The lower-dimensional NS equationsremind us that there must be something that can be simplified.Why do we just use the analytical results instead of numericallyrecovering the asymptotic limit from a high-dimensionaldistribution function? In this study, we revisit the unified gaskinetic scheme and estimate the contribution of each termin asymptotic limit. For the part which can be calculatedby traditional Navier-Stokes solver, we use analytical resultsinstead of the discrete velocity representation and proposeseveral simplifications of the UGKS.

The article is organized as follows. In Sec. II, the unifiedgas kinetic scheme is introduced briefly; in Sec. III, weanalyze the behavior of the UGKS in different flow regime andpropose three different simplification strategies; in Sec. IV,the numerical discretization and the boundary condition areintroduced; in Sec. V, numerical comparisons are provided,from which the key ingredients of the unified scheme andthe best simplification strategy are identified for industrialapplications. Finally, we conclude this study in Sec. VI.

II. UNIFIED GAS KINETIC SCHEME

A. Volume force

In this paper, we consider only the finite-volume schemes.As a standard finite-volume method, the quantities inside acell are updated by considering both the numerical flux andthe volume force. Because of the conservation constraint onthe collision term, the source terms for conservative variablesare zero:

〈ψ(f − g)〉 = 0 or 〈ψ(f − g)〉k = 0. (7)

The symbol 〈f 〉k denotes the quadrature of f in discretevelocity space, namely, the summation 〈f 〉k = ∑

k ωkfk ,where ωk is the weight function at velocity point uk . Please notethe difference between 〈f 〉 and 〈f 〉k . With the conservationconstraint, the conservative variables can be updated by takingaccount of only the numerical macroscopic flux during the

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time interval t ,

Wn+1 = Wn − ∇ · FW . (8)

After obtaining Wn+1, the equilibrium state gn+1 is knownthrough the formula Eq. (2). The time discretization of thekinetic equation Eq. (1) can be written as

f n+1k − f n

k

t+ 1

t∇ · F k = gn+1

k − f n+1k

τ. (9)

Then the distribution function at the n + 1 step is solved:

f n+1k = τ

τ + t

(f n

k − ∇ · F k

) + t

τ + tgn+1

k . (10)

As shown above, the convection term ∇F is also consideredwhen evaluating the volume force. Equation (10) is adopted tocalculate the volume force in the numerical schemes hereafter.

B. DOM flux and UGKS flux

Before discussing the AP property of the UGKS, webriefly recall the numerical flux of the conventional DOMfor the kinetic equation. As mentioned in the Introduction, thecollisionless kinetic equation [Eq. (5)] is taken as the governingequation to evaluate the numerical flux. The solution at theinterface (x = 0) is then

f (0,t,u) = f (−tu,0,u). (11)

Considering first-order spatial expansion, we have

fdom(0,t,u) = f (0,0,u) − tu · ∇f. (12)

The numerical flux for the distribution function is then

Fdom =∫ t

0ukfdom(0,t,uk)dt

= uk

(tfk − 1

2t2uk · ∇fk

). (13)

For simplicity, we ignore the arguments of the distributionfunction f and assume that u is aligned with the normaldirection of the cell interface. The numerical flux of the DOMis very simple; only the numerical fluxes for the distributionfunction are considered in the DOM. In order to compare withthe UGKS, the equivalent numerical fluxes for macroscopicvariables are derived by taking the moments of the numericalmicroscopic flux,

FWdom = 〈Fdom〉k =

∑k

ukψk

(tfk − 1

2t2uk · ∇fk

), (14)

where the superscript W denotes the macroscopic flux. Themechanism of the above formulations for the macroscopicfluxes is equivalent to the kinetic flux vector splitting (KFVS)method for the Euler equations. In this study, we use “DOM” todenote the numerical scheme which couples the collisionlessflux [Eqs. (13) and (14)] and the implicit time discretization[Eqs. (8) and (10)] for the volume force.

The unified gas kinetic scheme is an asymptotic preservingscheme benefiting from the local analytical solution of kineticequation. Integrating along the characteristic of the BGK

equation [Eq. (1)], a local analytical solution can be derived:

f (0,t,u) = e−t/τ f (−ut,0,u)

+ 1

τ

∫ t

0g( − u(t − t ′),t ′,u)e−(t−t ′)/τ dt ′. (15)

The forepart is the nonequilibrium part. When the system ap-proaches equilibrium, e−t/τ will become zero asymptotically;i.e., the nonequilibrium contribution will vanish. Meanwhile,the second term on the right-hand side, which represents theequilibrium part, will dominate.

Suppose, after the numerical reconstruction, the physicalquantities are linearly distributed around the cell and areexpressed as follows:

f (x,0,u) = f (0,0,u) + x · ∇f, (16)

g(x,t,u) = g(0,0,u) + x · ∇g + gt t. (17)

Substitute these formulas into the analytical solution:

fugks(0,t,u) = e−t/τ [f (0,0,u) − tu · ∇f ]

+ (1 − e−t/τ )g(0,0,u)

+[−τ + (τ + t)e−t/τ ]u · ∇g

+ (t − τ + τe−t/τ )gt .

This is the distribution function at the cell interface. Thenumerical microscopic flux is

Fugks =∫ t

0ukfugks(0,t,uk)dt. (18)

Then, taking moments of above solution, we get the numericalmacroscopic flux at the cell interface:

FWugks = 〈ψFugks〉k. (19)

C. The numerical flux stemming from the equilibriumand nonequilibrium parts

The numerical fluxes of the unified gas kinetic schemeare composed of the equilibrium and nonequilibrium terms.The competition of all these terms determines the asymptoticbehavior of the numerical schemes in different flow regimes.This issue has been discussed by Mieussens [26] for the UGKSof radiative transfer equation. We investigate every term indetail and deduce the asymptotic coefficient of each term. Thenumerical flux [Eq. (18)] can be further unfolded as follows:

Fugks = uk

{γ

ugks

0 fk + γugks

1 uk · ∇fk + γugks

2 gk

+ γugks

3

(uk · ∇gk + ∂gk

∂t

)+ γ

ugks

4

∂gk

∂t

}, (20)

FWugks = γ

ugks

0 〈uψf 〉k + γugks

1 〈uψu · ∇f 〉k+ γ

ugks

2 〈uψg〉k + γugks

3 〈uψ(u · ∇g + gt )〉k+ γ

ugks

4 〈uψgt 〉k. (21)

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SONGZE CHEN, ZHAOLI GUO, AND KUN XU PHYSICAL REVIEW E 94, 023313 (2016)

For the sake of simplicity, we define the coefficients in theUGKS flux as

γugks

0 = τ (1 − e−β),

γugks

1 = −τ (−te−β + τ − τe−β),

γugks

2 = t − τ (1 − e−β ), (22)

γugks

3 = τ (−te−β − 2τe−β + 2τ − t),

γugks

4 = {12t2 + τ [(t + τ )e−β − τ ]

},

where β is defined as the ratio of the time step t to therelaxation time τ , namely, β = t/τ . The first two terms onthe right-hand side of the Eq. (20) are the nonequilibrium partswhich are deduced from the nonequilibrium initial conditionat the beginning of the time step. The last three terms on theright-hand side stemming from the collision term represent theNavier-Stokes flux. As shown above, the nonequilibrium partf0 does not vanish directly when t

τ→ +∞. A small term

[O(τ )] still influences the numerical fluxes. For the sake ofthe implicit discretization [Eq. (10)] of the collision term,the following assumption seems natural and rational. Theinitial condition f deviates from the equilibrium by O(τ ),namely,

f = g + O(τ ). (23)

After one cycle of iteration, we can get more precise estimationfor the nonequilibrium part O(τ ) [24], that is,

f (0,0,u) = g(0,0,u) − τ (u · ∇g + gt ) + O(τt), (24)

where we choose the approximate Chapman-Enskog ex-pansion [Eq. (24)] as the initial condition for the UGKS.Substituting the estimation [Eq. (24)] into Eqs. (20) and (21),the numerical fluxes become

Fugks = uk

[tgk − tτ

(uk · ∇gk + ∂gk

∂t

)

+ t2

2

∂gk

∂t

]+ O(τ 2),

FWugks =

⟨uψ

[tg − tτ

(u · ∇g + ∂g

∂t

)

+ t2

2

∂g

∂t

]⟩k

+ O(τ 2). (25)

The Chapman-Enskog expansion for the Navier-Stokes equa-tion is exactly recovered up to O(τ ). Please note that, β is notrequired to approach zero as we derive the Chapman-Enskogexpansion. As we know, the numerical scheme must convergeas the time step goes to zero. In this sense, the asymptoticbehavior when τ → 0, t → 0, and β is finite, is moreimportant to the numerical scheme.

Under the more precise assumption [Eq. (24)], the estima-tion of the numerical fluxes in the DOM is written as

Fdom = uk

[tgk − t

(τ + t

2

)(uk · ∇gk + ∂gk

∂t

)

+ t2

2

∂gk

∂t

]+ O(τ 2),

FWdom =

⟨uψ

[tg − t

(τ + t

2

)(u · ∇g + ∂g

∂t

)

+ t2

2

∂g

∂t

]⟩k

+ O(τ 2). (26)

The equivalent viscosity in Eq. (26) is enlarged by the freestreaming. We use αdom to denote the enlarging factor, whichis

αdom = (τ + t/2)/τ = 1 + 12β.

When β varies from 0 to ∞, αdom diverges. The enlargedviscosity is an analog to the numerical viscosity in the latticeBoltzmann method [27] before the remedy of the viscosity.

III. SIMPLIFICATION OF THE UNIFIEDGAS KINETIC SCHEME

In the UGKS fluxes [Eqs. (20) and (21)], the last threeterms that stem from the collision term are also discretizedin velocity space. Therefore, it takes huge computationalresources compared to the traditional Navier-Stokes solvers.In fact, more than half of the computation is taken to evaluatethe equilibrium part. Actually, the quadratures 〈g〉k, 〈∇g〉k ,and 〈gt 〉k are only approximation of 〈g〉,〈∇g〉, and 〈gt 〉. Thequadrature of g and its derivatives can be calculated analyti-cally; for instance, 〈g〉 = (ρ,ρU,ρE)T . If the quadratures ofthe equilibrium state g and its derivatives are handled in thetraditional way in terms of analytical macroscopic flux [28],the unified scheme will be much more efficient. Therefore, wepropose the first simplification (S1), that is, using traditionalDOM to calculate the flux for distribution function and usingthe macroscopic gas kinetic scheme [28] to evaluate the lastthree terms in Eq. (21),

F s1 = uk

(tfk − 1

2t2uk · ∇fk

), (27)

FWs1 = γ s1

0 〈uψf 〉k + γ s11 〈uψu · ∇f 〉k

+ γ s12 〈uψg〉 + γ s1

3 〈uψ(u · ∇g + gt )〉 + γ s14 〈uψgt 〉,

γ s1i = γ

ugks

i , i = 0,1,2,3,4. (28)

Compared to the numerical macroscopic flux of the UGKS[Eq. (21)], the equilibrium part is solved analytically (note thedifferent symbols 〈·〉 and 〈·〉k), and the numerical microscopicflux [Eq. (20)] is replaced with the traditional DOM [Eq. (13)].With the assumption of Eq. (24), if the difference between〈·〉 and 〈·〉k is ignored, the numerical microscopic fluxbecomes

F s1 = uk

[tgk −

(τ + 1

2t

)t

(uk · ∇gk + ∂gk

∂t

)

+ 1

2t2 ∂gk

∂t

]+ O(τ 2),

FWs1 =

⟨uψ

[tg − tτ

(u · ∇g + ∂g

∂t

)

+ t2

2

∂g

∂t

]⟩+ O(τ 2). (29)

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SIMPLIFICATION OF THE UNIFIED GAS KINETIC SCHEME PHYSICAL REVIEW E 94, 023313 (2016)

TABLE I. The coefficients of the numerical flux in UGKS and S2 scheme, where β = t/τ .

Formula β → ∞ β → 0

γugks

0 τ (1 − e−β ) τ + O(e−β ) t − t2

2τ+ O(β2)

γ s20 te−β O(e−β ) t − t2

τ+ O(β2)

γugks

1 −τ (−te−β + τ − τe−β ) −τ 2 + O(e−β ) −t2

2 + t3

3τ+ O(β2)

γ s21 − 1

2 t2e−β O(e−β ) −t2

2 + t3

2τ+ O(β2)

γugks

2 t − τ (1 − e−β ) t − τ + O(e−β ) t2

2τ+ O(β2)

γ s22 (1 − e−β )t t + O(e−β ) t2

τ+ O(β2)

γugks

3 τ [2τ − t − (t + 2τ )e−β ] 2τ 2 − τt + O(e−β ) −t3

6τ+ O(β2)

γ s23 −τt(1 − e−β ) −τt + O(e−β ) t2 − t3

2τ+ O(β2)

γugks

412 t2 + τ [(t + τ )e−β − τ ] 1

2 t2 − τ 2 + O(e−β ) t3

3τ+ O(β2)

γ s24

12 t2(1 − e−β ) 1

2 t2 + O(e−β ) t3

2τ+ O(β2)

If the quadrature (〈·〉k) is accurate, the numerical macroscopicflux of the S1 scheme is identical to the macroscopic flux of theUGKS. Only the flux for the distribution function is different.We present some numerical comparisons to demonstrate thatthe inaccurate microscopic numerical flux has very littleinfluence to the NS AP property of the numerical scheme. Thissimplification only reduces the computational cost, but the for-mula and the coding are still complicated. Hence, we propose asecond simplified method (S2), which is barely a combinationof the DOM and the gas kinetic scheme for the Navier-Stokesequations. The numerical fluxes are given as follows:

F s2 = uk

(tfk − 1

2t2uk · ∇fk

), (30)

FWs2 = e−β

⟨uψ

(tf − 1

2t2u · ∇f

)⟩k

+(1 − e−β )

⟨uψ

[tg0 − tτ

(u · ∇g + ∂g

∂t

)

+ 1

2t2 ∂g

∂t

]⟩. (31)

This method is very simple. We can easily combine twoexisting flux solvers to construct a unified scheme for gaskinetic equation. Assume that the initial condition at thebeginning of the time step satisfies the near-equilibriumassumption, namely, Eq. (24) is applied. The numerical fluxof the second simplified method becomes,

F s2 = uk

[tgk −

(τ + t

2

)t

(uk · ∇gk + ∂gk

∂t

)

+ 1

2t2 ∂gk

∂t

]+ O(τ 2),

FWs2 =

⟨uψ

[tg − t

(τ − t

2e−β

)(u · ∇g + ∂g

∂t

)

+ 1

2t2 ∂g

∂t

]⟩+ O(τ 2). (32)

Obviously, in the continuum flow regime, this simplification isaccurate enough to lead to the Navier-Stokes numerical flux.The viscosity of the S2 scheme is enlarged by a factor,

αs2 = (τ − te−β/2)/τ = 1 − 12βe−β. (33)

It only varies inside the interval [1 − e−1/2,1]. The minimumis attained when β = 1. Table I compares the coefficients ofthe UGKS and the second simplified scheme. As shown in thesecond column, the formulas are apparently different when β

has a finite value. When β goes to infinity and τ goes to zero,i.e., in the continuum flow regime, the coefficients are identicalup to O(τ ). The S2 scheme approaches the equilibrium state alittle faster than the UGKS because the coefficients of thenonequilibrium part, γ s2

0 and γ s21 , approach to zero more

rapidly. Consider the free molecular flow limit, namely thatβ goes to zero and τ goes to infinity. The coefficients of theUGKS and the S2 scheme are identical up to O(β), except γ s2

3 .It deviates from γ

ugks

3 in free molecular flow regime largely.This means the simple combination cannot recover the freemolecular flow regime. We will also find a large discrepancygenerated from γ s2

3 in the numerical comparison section.Therefore, we propose a third simplified method (S3) which

modified the coefficient in front of the Navier-Stokes viscousterm. The basic idea is to construct a coefficient γ s3

3 which canpreserve the asymptotic limit of γ

ugks

3 . The third simplifiedmethod is

F s3 = uk

(tfk − 1

2t2uk · ∇fk

), (34)

FWs3 = e−β

⟨uψ

(tf − 1

2t2u · ∇f

)⟩k

+ (1 − e−β )

⟨uψ

[tg0 − trτ τ

(u · ∇g + ∂g

∂t

)+ 1

2t2 ∂g

∂t

]⟩, (35)

rτ = 1 − e−β − t2τ

(1 + e−β )e−β

1 − e−β. (36)

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SONGZE CHEN, ZHAOLI GUO, AND KUN XU PHYSICAL REVIEW E 94, 023313 (2016)

The coefficient γ s33 becomes

γ s33 = −τt

[1 − e−β − t

2τ(1 + e−β )e−β

],

γ s33 = −τt + O(e−β), when

t

τ→ ∞,

γ s33 = −t3

τ+ O(β2), when

t

τ→ 0.

The only difference from the second simplified method is that the coefficient in front of the viscous term is multiplied by a factorrτ . Therefore, the third method can also be taken as a simple combination between the DOM and the Navier-Stokes solver. Aswe can see, the coefficient γ s3

3 has the same limit in free molecular flow regime up to O(β). Then, considering the continuumflow regime, with the assumption Eq. (24), the third simplified method becomes

F s3 = uk

[tgk −

(τ + 1

2t

)t

(uk · ∇gk + ∂gk

∂t

)+ 1

2t2 ∂gk

∂t

]+ O(τ 2),

FWs3 =

⟨uψ

[tg − tτ

(u · ∇g + ∂g

∂t

)+ t2

2

∂g

∂t

]⟩+ O(τ 2)

−⟨uψ

{t

[1

2te−β + τ (1 − eβ)(rτ − 1)

](u · ∇g + ∂g

∂t

)}⟩

=⟨u[tg − t

(τ − t

2e−2β

)(u · ∇g + ∂g

∂t

)+ t2

2

∂g

∂t

]⟩+ O(τ 2). (37)

For the third simplified method, the equivalent viscosity isenlarged by

αs3 = (τ − te−2β/2)/τ = 1 − 12βe−2β . (38)

It only varies inside the interval [1 − e−1/4,1]. The minimumis attained when β = 1/2. Figure 1 shows the enlarging factorα versus β. The numerical flux of the UGKS is based onthe analytical solution. Therefore, its viscosity is unchangedin the second-order temporal discretization [Eq. (25)], i.e.,αugks = 1. The simplified schemes somehow modifies theviscous coefficient. As shown in Fig. 1, the S3 scheme ismore accurate than the S2 scheme in terms of the viscositycoefficient.

We analyze the behavior of these simplified numericalschemes. The S1 scheme replaces the quadrature related tothe equilibrium state with the analytical solution. Althoughit has correct asymptotic limits and less computational cost,

0 2 4 6 8 100.8

0.85

0.9

0.95

1

β

α

S2S3

FIG. 1. The enlarging factor of the viscosity for the simplifiedschemes (S2, S3).

the scheme is still complicated in terms of coding. TheS2 scheme is a simple combination of Navier-Stokes solverand traditional DOM. It cannot reproduce the free molecularflow regime. The S3 scheme has correct asymptotic limits infree molecular flow regime and also in the continuum flowregime. For the transition flow regime, the coefficients areapparently different from those in the analytical solution. Weuse the numerical experiment to investigate the performanceof different simplifications.

IV. NUMERICAL DISCRETIZATION

The previous section introduced the numerical flux expres-sion in terms of time. Several simplified numerical fluxes areconstructed based on the unified gas kinetic scheme. In thissection the spatial discretization and the boundary conditionare provided.

A. Spatial discretization

The value and its spatial derivative of a certain quantity areneeded in the expressions of the numerical flux [for exampleEq. (20)]. For the velocity distribution function, we adoptthe third-order weighted essentially non oscillatory (WENO)to interpolate its value at the cell interface (i + 1/2), wherei denotes the index along the interpolation direction. Theformula is

fl = w−1f−1 + w0f

0

w−1 + w0, fr = w0f

0 + w1f1

w0 + w1,

where the subscripts “l” and “r” represent left side and rightside, respectively, and w denotes the weight. Their formulasare written as

w−1 = 1

4(s2i−1 + ε

) , w0 = 3

4(s2i + ε

) , w1 = 1

4(s2i+1 + ε

) ,

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SIMPLIFICATION OF THE UNIFIED GAS KINETIC SCHEME PHYSICAL REVIEW E 94, 023313 (2016)

where ε = 1 × 10−6 is used to prevent zero denominator, andsi = fi+1 − fi ,

f −1 = 32fi − 1

2fi−1, f 0 = 12fi+1 + 1

2fi,

f 1 = 32fi+1 − 1

2fi+2.

For high-speed flow, the third-order WENO is also em-ployed to calculate the macroscopic variables at the cell inter-face, owing to the discontinuous shock wave in the flow field.For low-speed flow, the macroscopic conservative variablesare interpolated by the central difference method; that is,

Wi+1/2 = 12 (Wi + Wi+1). (39)

The derivatives of the microscopic and macroscopic vari-ables are evaluated by a second-order central differencemethod.

B. Boundary condition

The boundary condition is another crucial ingredient for APschemes. At first, we recall the diffusion boundary conditionfor the traditional DOM in free molecular flow regime. Thedistribution function of the reflecting particles is subjected tothe Maxwell distribution. Since no penetration occurs duringthe collision with the wall, the mass flux of the particle can bewritten as∫ t

0

∫ +∞

0uf indudt + ρdom

∫ t

0

∫ 0

−∞ug∗dudt = 0, (40)

where f in represents the incident molecular distributionfunction which is interpolated from the interior of the flowfield, ρdom is the density of the reflecting molecular stream, andu denotes the component of particle velocity perpendicular tothe boundary. The reflecting molecular distribution functionis assumed to be the Maxwell equilibrium on the wall, whichreads

g∗ =(

2RT ∗

π

)3/2

e− 12RT ∗ u2

, (41)

where T ∗ denote the temperature of the boundary. Accordingto Eq. (40), the density of the reflecting distribution isdetermined; that is,

ρdom = −∑

uk>0 ωkukfk∑uk�0 ωkukgk

. (42)

The velocity distribution function at the wall for the micro-scopic variables is

fdom ={f in, u > 0,

ρdomg∗, u � 0.(43)

The numerical fluxes are written as{F∗

dom = ukfdom,k,

F∗Wdom = 〈uψfdom〉k.

(44)

The diffusion boundary condition is valid in the free molecularflow regime, but cannot automatically recover the no-slipboundary condition in the continuum flow regime. Theboundary condition for the simplified method (S2, S3) shouldbe designed carefully to preserve the asymptotic limits. Fortu-nately, this task is very easy to complete, since the simplified

scheme is a simple combination of existing schemes. Herewe just combine the boundary flux of the diffusion boundarycondition and the boundary flux of the gas kinetic scheme forthe Navier-Stokes equations to develop a boundary conditionfor the simplified scheme.

We modify the nonequilibrium bounce-back boundarycondition [29] to implement the isothermal boundary conditionfor gas kinetic scheme. We adopt the extrapolation from theinterior, then construct a NS distribution at the cell interfaceas the incident distribution function,

f ingks = gin − rτ τ

(u · ∇gin + gin

t

) + gint t, for u > 0, (45)

where rτ is defined in Eq. (36). The reflecting distributionfunction is constructed as follows:

f outgks(u) = 2ρgksg

∗(u) − f ingks(−u), for u � 0. (46)

Then the complete velocity distribution function in the gaskinetic scheme is

fgks ={

f ingks, u > 0,

f outgks , u � 0.

(47)

The no-penetration condition is also employed to determinethe density at the wall boundary:

ρgks =√

2π

RT ∗

∫u>0

uf ingksdu. (48)

The numerical flux for the conservative variables and forthe distribution function are given, respectively,{

F k = ukfdom,k,

FW = e−β∑

k ukψkfdom + (1 − e−β)〈uψfgks〉.(49)

We have tested another choice of rτ , say, rτ = 1, for thesecond simplified method (S2). When applying this boundarycondition, in the free molecular flow regime, there were largeoscillations near the boundary, since the coefficient γ s2

3 isinconsistent with the analytical solution (γ ugks

3 ). Therefore,only the simplified boundary condition [Eq. (49)] is adoptedfor all the numerical simulations in the next section.

V. NUMERICAL COMPARISON

The numerical schemes compared in this section aresummarized in Table II. In the following numerical tests,the CFL number is 0.4. All the numerical setting are exactlyidentical.

TABLE II. The numerical schemes compared in Sec. V.

Scheme Volume force Numerical flux

DOM Eqs. (8) and (10) Eqs. (13) and (14)UGKS Eqs. (8) and (10) Eqs. (20) and (21)S1 Eqs. (8) and (10) Eqs. (27) and (28)S2 Eqs. (8) and (10) Eqs. (30) and (31)S3 Eqs. (8) and (10) Eqs. (34) and (35)

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SONGZE CHEN, ZHAOLI GUO, AND KUN XU PHYSICAL REVIEW E 94, 023313 (2016)

x

ρ

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1DOMUGKSS1S2S3

(a)

x

U

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

DOMUGKSS1S2S3

(b)

x0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1DOMUGKSS1S2S3

(c)

x

U

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

DOMUGKSS1S2S3

(d)

x

ρ

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1DOMUGKSS1S2S3

(e)

x

U

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

DOMUGKSS1S2S3

(f)

ρ

FIG. 2. The density and velocity profile of the shock-tube problem at different Knudsen numbers: (a),(b) Kn = 0.0001; (c),(d) Kn = 0.01;(e),(f) Kn = 1.

A. Sod shock tube

At first, the one-dimensional shock-tube problem is testedunder different Knudsen numbers:

Kn = μref√

RTref

prefL. (50)

The computational domain is [0,1] in the x direction and isdiscretized into 200 cells. The initial condition is given asfollows:

ρl = 1.0, Ul = 0.0, pl = 1.0, for x � 0.5,

ρr = 0.125, Ur = 0.0, pr = 0.1, for x > 0.5. (51)

The quantities on the right-half domain are selected to definethe reference variables. We use a 150-point uniform grid

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SIMPLIFICATION OF THE UNIFIED GAS KINETIC SCHEME PHYSICAL REVIEW E 94, 023313 (2016)

x

ρ

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1DOMUGKSS1S2S3

(a)

x

ρU

0.2 0.4 0.6 0.80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

DOMUGKSS1S2S3

(b)

FIG. 3. The density and momentum profile of the shock-tube problem at Kn = 10.

in the velocity space [−6,6]. The computation stops at t =0.15. Figures 2 and 3 show the numerical results for Kn =0.0001, 0.01, 1, 10. Five different flux solvers are employedto simulate this problem. As expected, all the methods providevery good results.

In the free molecular flow regime, say, the Knudsen numberis 10; we find that, except for the S2 scheme, all the numericalschemes predict the same density and momentum profile inFig. 3. This is because the leading-order terms are identical forthese schemes [Table I, Eqs. (13), (20), and (27)]. Inaccurateresults from the S2 scheme verify that a simple combinationof the DOM and a Navier-Stokes solver cannot lead to thecorrect asymptotic limit. The quantities plotted in Fig. 3 arethe macroscopic variables updated by Eq. (8). In the transitionflow regime, the results derived from different schemes arestill indistinguishable. In the continuum flow regime, the S1,S2, S3, DOM, and UGKS provide almost identical solutions(Fig. 2). It testified that the inaccuracy of the initial distributionfunction affects little in the numerical performance in thecontinuum flow regime.

These numerical observation are consist with our analysisin the previous section. The discrepancy is hardly noticed inall the flow regimes. All the numerical methods (except theS2) converge to the Euler solution in the continuum regimeand converge to a collisionless solution in the free molecularflow regime.

B. Lid-driven cavity flow

The one-dimensional numerical results show that all thenumerical schemes converge to the Euler solution at Kn → 0.However, as mentioned in Ref. [24], the one-dimensional nu-merical experiment cannot distinguish the NS AP scheme fromthe Euler AP scheme. Thus, we simulate a two-dimensionallid-driven cavity flow which is characterized by a strongviscous effect. The gas flow is confined in a square domainwhose extent is [0,1] × [0,1]. Each edge of the computationaldomain is uniformly discretized by 61 nodes. The top boundarymoves from left to right with a constant velocity of 0.2. Thegas pressure is 1 and the density is also 1. The Mach number

0.994

0.994

0.99

6

0.996

0.99

8

0.998

11

1

1.002

1.002

1.002

1.004

1.004

1.004

1.006

1.006

1.008

1.01

X

Y

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8Level RT11 1.01210 1.019 1.0088 1.0067 1.0046 1.0025 14 0.9983 0.9962 0.9941 0.992

(a)

0.992 0.994

0.99

6

0.996

0.99

8

0.99

8

0.998

1

1

1

1.002

1.002

1.002

1.004

1.004

1.004

1.006

1.006

1.008

1.0081.0

11.012

X

Y

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8Level RT11 1.01210 1.019 1.0088 1.0067 1.0046 1.0025 14 0.9983 0.9962 0.9941 0.992

(b)

FIG. 4. The temperature (RT ) contours in the lid-driven cavity flow at Kn = 2, Re = 0.1. The black dash lines represent the f -basedtemperature from the DOM. (a) The red dash-double-dotted lines represent the W -based temperature from the S2 scheme. (b) The reddash-double-dotted lines represent the f -based temperature from the S2 scheme.

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SONGZE CHEN, ZHAOLI GUO, AND KUN XU PHYSICAL REVIEW E 94, 023313 (2016)

0.994

0.99

6

0.996

0.99

8

0.998

0.998

1

1

1

1.002

1.002

1.002

1.002

1.004

1.004

1.004

1.006

1.006

1.00

8

1.012

X

Y

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8Level RT11 1.01210 1.019 1.0088 1.0067 1.0046 1.0025 14 0.9983 0.9962 0.9941 0.992

(a)

0.99

2

0.994

0.99

6

0.996

0.99

8

0.998

998

11

1

1.002

1.002

1.002

1.004

1.004

1.004

1.006

1.006 1.008

1.01

X

Y

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8Level RT11 1.01210 1.019 1.0088 1.0067 1.0046 1.0025 14 0.9983 0.9962 0.9941 0.992

(b)

FIG. 5. The temperature (RT ) contours in the lid-driven cavity flow at Kn = 2, Re = 0.1. The black dashed lines represent the f -basedtemperature from the DOM. (a) The red dash-double-dotted lines represent the W -based temperature from the S3 scheme. (b) The reddash-double-dotted lines represent the f -based temperature from the S3 scheme.

based on the velocity of the top wall is about 0.15. The Knudsennumber is defined as Eq. (50).

When the relaxation time τ goes to infinity, the evolutionof the distribution function [Eq. (10)] is totally independentto the evolution of the macroscopic variables [Eq. (8)] sincethe collision term vanishes. As a result, the conservativevariables W and the distribution function f are evolvingseparately in the free molecular flow regime. Therefore, wecan get two sets of macroscopic variables. We use the W -basedvariable to denote the macroscopic variable deduced fromthe conservative variables W and use f -based variable todenote the macroscopic variable deduced from the distributionfunction f . Figure 4 shows the W -based temperature andthe f -based temperature derived from the S2 scheme. Theflow condition is Kn = 2 and Re = 0.1, and the velocityspace [−5,5] × [−5,5] is discretized into 100 × 100. Sincethe DOM is accurate at high Knudsen number, we choose thef -based temperature derived from the DOM as a benchmarksolution and plot it on the background. As shown in Fig. 4,the f -based temperature [Fig. 4(b)] is identical to the DOMsolution, while the W -based temperature [Fig. 4(a)] deviatesfrom the DOM solution. This is because the asymptotic limitof the coefficient γ s2

3 (Table I) is incorrect when β goes tozero and leads to the incorrect macroscopic flux FW

s2 , whichis used to update the W -based variables. After remedying thecoefficient, the S3 scheme has the same asymptotic limit as theanalytical solution. As we can see in Fig. 5, the results obtainedfrom the S3 scheme, both f -based and W -based temperaturescoincide with the results derived from the DOM. The f -basedtemperature and heat flux from all the considered numericalmethods collapse to the DOM results in Fig. 6.

However, remarkable discrepancies are observed when theReynolds number increases to 1000. In this case, we use onlyeight velocity points in one direction to discretize the velocityspace ranging from −5 to 5, and the rectangular quadraturein velocity space is adopted. All these numerical settings arefor the purpose of illustrating the influence of the inaccuratequadrature in velocity space. Central difference interpolation

is adopted for both microscopic and macroscopic variables.As shown in Figs. 7(b) and 7(d), the DOM cannot simulatethe continuum flow properly; therefore, the DOM’s results arenot shown in Figs. 7(a) and 7(c). The UGKS and S1 schemesobtained much better numerical results, which are closer tothe reference data [30]. However, due to the inaccuracy ofthe quadrature in the velocity space, the numerical resultsare not as good as the numerical results in the previousliterature [17,21,31]. The numerical contour lines oscillatenear the boundaries [Figs. 7(a) and 7(c)]. On the other hand,the simplified schemes (S2, S3) perform best in this test case.

The asymptotic limits of the numerical schemes coincidewith our analysis in the previous section. For the transition flowregime, the numerical results are shown in Fig. 8. The Reynoldsnumber is 10, and the corresponding Knudsen number is 0.02.

X

Y

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8RT

1.0121.011.0081.0061.0041.00210.9980.9960.9940.992

FIG. 6. The f -based temperature (RT ) contour and the f -basedheat flux in the lid-driven cavity flow at Kn = 2, Re = 0.1 for allthe numerical schemes (include DOM, UGKS, S1, S2, and S3). Thenumerical results derived from different numerical schemes collapseto the DOM results.

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SIMPLIFICATION OF THE UNIFIED GAS KINETIC SCHEME PHYSICAL REVIEW E 94, 023313 (2016)

-0.06

-0.06-0.04

-0.04

-0.04

-0.04

-0.02

-0.02

-0.02

-0.0

2

-0.02

00

0

0

0

0

0

0

0

0

0

0

0.02

0.020.

02

0.04

0.04

0.04 0.06

0.060.08 0.08

0.10.10.12

0.14 0.140.16

X

Y

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8Level U12 0.1611 0.1410 0.129 0.18 0.087 0.066 0.045 0.024 03 -0.022 -0.041 -0.06

(a)

DOMUGKSS1S2S3

U/Utop

Y

-0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Ghia

(b)

-0.1

2 -0.1

-0.1

-0.1

-0.0

8

-0.08

-0.0

8

-0.0

6

-0.0

6

-0.0

6

-0.0

4

-0.04

-0.0

4

-0.0

4-0

.02

-0.02

-0.0

2

-0.02

00

0 0

00

00

0

0

0.02

0.02

0.02

0.02

0.02

0.04

0.04

0.04

0.06

0.06

0.06

X

Y

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8Level V10 0.069 0.048 0.027 06 -0.025 -0.044 -0.063 -0.082 -0.11 -0.12

(c)

DOMUGKSS1S2S3

X

V/U

top

0 0.2 0.4 0.6 0.8 1

-0.4

-0.2

0

0.2

0.4

Ghia

(d)

FIG. 7. The velocity contours (a),(c) and velocity profiles (b),(d) at Kn = 0.0002, Re = 1000. DOM, black solid line; UGKS, green dashedline; S1, blue dash-dotted line; S2, pink long dashed line; S3, red dash-double-dotted line.

We use 61 points in each direction in physical space and use60 points in each direction in velocity space. The velocitycontours are almost identical for all the schemes. Only minordifferences can be noticed in density contour and temperaturecontour.

From the above results, we demonstrate that the simplifiedschemes proposed in the paper possess correct asymptotic limitin free molecular flow regime and the continuum flow regimeand provide enough accurate numerical results in the transitionflow regime.

C. The high efficiency of the simplified methods

In the Eq. (27), three of the five terms are evaluated by an-alytical formulas. These computational costs are infinitesimalcompared to the quadrature in velocity space. We also observethat the S1 reduces about half computation time compared tothe UGKS, and the DOM, S1, S2, and S3 schemes have almostidentical computational efficiency.

On the other hand, the numerical results derived fromthe simplified methods are closer to the results from theNS solver in the continuum flow regime. It is worth notingthat the coefficient for S2, say, γ s2

0 , deviates from 0 by a

exponential truncation error, while γugks

0 and γ s10 preserve

τ as the leading-order term. As illustrated in Eqs. (21),(24), and (25), the physical asymptotic process is not simplyattained by vanishing the nonequilibrium terms f0 and u · ∇f .The nonequilibrium terms still contribute a little [O(τ )] tothe total distribution function, and the remaining terms ofnonequilibrium part are canceled by the equilibrium part, thenresult in the Chapman-Enskog expansion. Such a balance isvery delicate and sophisticated. It is definitely computationallyburdensome or clumsy to simulate this subtle asymptoticprocess in velocity space. The simplified methods proposedin this study circumvent the delicate balance; instead, usemore rapid decaying coefficients in front of the nonequilibriumterms. As we can see in the numerical comparisons, the S2and S3 schemes provide more accurate numerical resultsin the continuum flow regime, since the delicate balancebetween the nonequilibrium part and the equilibrium part isreplace with a prior knowledge and circumvents the numericalsimulation of the asymptotic process. The quadrature ofthe distribution function is totally replaced with the ana-lytical expression. Hence, the simplified schemes lead tomore accurate results and less discrete points in velocityspace.

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SONGZE CHEN, ZHAOLI GUO, AND KUN XU PHYSICAL REVIEW E 94, 023313 (2016)

0.966

.98 0.99

0.997

0.997

0.997

1

1

1

1.001

1.001

1.001

.001

1.002

1.002

1.003

1.003

1.003

1.008

1.008

1.02

X

Y

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8Level ρ12 1.0411 1.0210 1.0089 1.0038 1.0027 1.0016 15 0.9974 0.993 0.982 0.9661 0.942

(a) 0.9960.9980.998

1

1

11

1.0004

1.00041.0004

1.0004

1.00041.0004

1.0007

1.0007

1.0007

1.0007

1.0007

1.0007

1.002

1.002

1.0021.002

1.004

1.004

1.006

1.008

X

Y

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8Level RT9 1.0088 1.0067 1.0046 1.0025 1.00074 1.00043 12 0.9981 0.996

(b)

-0.02

-0.02

-0.02

-0.02

0

0

0.02

0.02

0.02

.04

0.04

0.04

0.060.06

0.08

0.080.1

0.10.12 0.120.14 0.140.16

X

Y

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8Level U10 0.169 0.148 0.127 0.16 0.085 0.064 0.043 0.022 01 -0.02

(c)-0.05

-0.05

-0.04

-0.04

-0.04

-0.03-0.03

-0.03

-0.02

-0.02

-0.02

-0.01

-0.01

-0.01

0

0

0

0.01

0.01

0.01

02

0.02

0.02

0.02

0.03

0.030.04

0.04

0.050.05

X

Y

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8Level V12 0.0511 0.0410 0.039 0.028 0.017 06 -0.015 -0.024 -0.033 -0.042 -0.051 -0.06

(d)

FIG. 8. The flow field at Kn = 0.02, Re = 10. (a) The density contour; (b) the temperature contour; (c),(d) the velocity contour. DOM,black solid line; UGKS, green dashed line; S1, blue dash-dotted line; S2, pink long dashed line; S3, red dash-double-dotted line.

VI. CONCLUSION

In this study, we analyzed the asymptotic behavior ofthe unified gas kinetic scheme and reduced the unnecessaryquadrature in the UGKS numerical flux for the equilib-rium part. In the first simplified scheme, the quadraturein velocity space for the equilibrium part is replaced withthe analytical results. The numerical comparisons showedthat this replacement reduced about half the computationload and did not affect numerical results. Based on theasymptotic expression of the UGKS flux, several othersimplification strategies have been proposed. The numericalcomparisons demonstrated that simple combination (S2)of a kinetic flux and the macroscopic flux cannot obtainthe correct asymptotic limit in the free molecular flowregime. With a rescaled viscosity coefficient, the simplifiedscheme (S3) possesses the correct asymptotic limit in both

the free molecular flow regime and the continuum flow regime.Moreover, it can be constructed by combining two existingflux solvers which handle the kinetic equation and the Navier-Stokes equations, respectively. The simplified scheme (S3)is efficient in terms of coding and computing; hence, it is apromising approach for engineering application. Its accuracyis also acceptable and controllable. The flux hybrid strategyproposed in this study can be further extended to the othermultiscale problems.

ACKNOWLEDGMENTS

This work was supported by NSF Grant No.91530319, Hong Kong Research Grant Council (Grants No.620813, No. 16211014, and No. 16207715), and HKUST(Grants No. PROVOST13SC01, No. IRS15SC29, and No.SBI14SC11).

[1] P. L. Bhatnagar, E. P. Gross, and M. Krook, Phys. Rev. 94, 511(1954).

[2] G. Bird, Comput. Math. Appl. 35, 1(1998).

[3] S. McNamara and Y. B. Gianchandani, J. Microelectromech.Syst. 14, 741 (2005).

[4] J. Zhang, J. Fan, and F. Fei, Phys. Fluids (1994-present) 22,122005 (2010).

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[5] G. A. Bird, Molecular Gas Dynamics and the Direct Simulationof Gas Flows (Clarendon Press, Oxford, UK, 1994).

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