Computers & Fluids 56 (2012) 39–48

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Computers & Fluids

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Progress in gas-kinetic upwind schemes for the solutionof Euler/Navier–Stokes equations – I: Overview

Lei Tang ⇑D&P, LLC, Phoenix, AZ 85016, USA

a r t i c l e i n f o

Article history:Received 2 September 2010Received in revised form 19 October 2011Accepted 22 November 2011Available online 1 December 2011

Keywords:Gas-kinetic upwind schemesKFVS schemeKWPS schemeBGK scheme

0045-7930/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compfluid.2011.11.012

⇑ Tel./fax: +1 602 957 2868.E-mail address: tanglei@d-p-llc.com

a b s t r a c t

Three gas-kinetic upwind schemes for the solution of the Euler/Navier–Stokes equations are reviewed.They are Kinetic Flux-Vector Splitting (KFVS), Kinetic Wave/Particle Splitting (KWPS), and Bhatnagar–Gross–Krook (BGK) methods. For the Euler equations, the most sophisticated BGK scheme can be inter-preted as a relaxation scheme between the two KFVS schemes with different moments, KFVS andKFVS_u0. It improves the accuracy over the KFVS scheme and the robustness over the KFVS_u0 scheme.The direct generalization of this relaxation approach to the Navier–Stokes equations leads to a much sim-pler BGK scheme than the one in the literature. In this simplified BGK scheme, there exist two types ofparticle collision time. The one in the BGK model acts as a relaxation parameter. Its role is to add somenumerical dissipation from the KFVS scheme to the KFVS_u0 scheme. On the other hand, the one in theChapman–Enskog expansion of the gas distribution function is related to physical dissipation. Followingthe same approach, another type of BGK schemes is further developed, which is a relaxation schemebetween KWPS and KWPS_u0. In spite of the fact that the KWPS scheme is more diffusive than the KFVSscheme, a BGK scheme based on KWPS and KWPS_u0 is found not only computationally more efficientbut also less diffusive than a BGK scheme based on KFVS and KFVS_u0. However, this issue needs furtherand more rigorous investigation by performing the numerical analysis of a model 1-D convection–diffusion equation.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

According to [1], the upwind shock-capturing schemes for theEuler equations adopt two different approaches to estimate thenumerical fluxes at the interfaces. The first is to follow the methodof characteristics and solve a Riemann problem locally, and thesecond is to follow the gas-kinetic theory and integrate the gas dis-tribution function over the velocity space according to the direc-tion of a specific particle velocity. While much more popular inthe CFD community, the first approach is not straightforward forrigorous extension beyond the Euler equations. For numerical solu-tion of the Navier–Stokes equations, most extensions of the firstapproach use a Riemann solver to estimate the inviscid fluxesbut a central discretization to estimate the viscous fluxes. On theother hand, the second approach, the gas-kinetic CFD approach,treats the inviscid and viscous fluxes as a single entity. This is moreconsistent and also more appealing from the algorithmic point ofview. For example, with the gas-kinetic CFD approach, the exten-sion of the discontinuous Galerkin method from the Euler equa-tions to the Navier–Stokes equations becomes straightforward.

ll rights reserved.

This is also true for the combination of CFD algorithm around thecontinuum regime and DSMC algorithm around the rarefied re-gime. Other benefits of using a gas-kinetic CFD approach includeno need for the macroscopic governing equations before numericaldiscretization and the implementation of the gas-kinetic boundaryconditions. It is also found that the gas-kinetic CFD approach hassome advantages for turbulent flow simulation [2,3]. Therefore,the approach receives more and more attention.

While the development of the gas-kinetic CFD schemes startsfrom as early as the 1960s, the first gas-kinetic upwind schemeis probably the Equilibrium Flux method (EFM) in [4] or the KineticFlux-Vector Splitting (KFVS) method in [5,6]. Although it has someunique features, a KFVS scheme is found computationally morecostly due to the use of the error and exponential functions andmore diffusive than Roe’s approximate Riemann solver in [7]. Toimprove the computational efficiency over a KFVS scheme, theso-called Kinetic Wave/Particle Splitting (KWPS) method is furtherdeveloped in [8,9], which avoids the use of error and exponentialfunctions. However, the approach is found even more diffusivethan the KFVS approach. To improve the accuracy over a KFVSscheme, on the other hand, the so-called Bhatnagar–Gross–Krook(BGK) method is developed in [10,11]. While more accurate thana KFVS scheme, the BGK scheme in the literature suffers from thecomplexity and low computational efficiency. Until today no

40 L. Tang / Computers & Fluids 56 (2012) 39–48

explicit expressions of the numerical fluxes have been given interms of the macroscopic variables. So, many moments of termsassociated with the spatial and temporal expansions are requiredto compute at each interface and at each time step. This makesthe BGK scheme in [11] extremely expensive in terms of floatingpoint operations per flux computation [12].

The objective of this series of papers is to develop an accurateand yet computationally efficient gas-kinetic upwind scheme fornumerical solution of the Euler and Navier–Stokes equations. Westart from the overview of all three gas-kinetic upwind approachesin this paper. First, the KFVS and KWPS schemes for the Euler andNavier–Stokes equations are briefly summarized in Sections 2 and3, respectively. Then Section 4 interprets the most sophisticatedBGK scheme as a relaxation scheme between the two KFVSschemes with different moments. This leads to a much simplerBGK scheme for the Navier–Stokes equations than the one in theliterature. Based on the findings in Section 4, Section 5 furtherdevelops another type of BGK schemes based on the KWPS ap-proach. Finally, several test cases are presented in Section 6 tocompare the performances of these BGK schemes.

2. Kinetic flux-vector splitting method

For the sake of clarity, let us confine our discussion to one spa-tial dimension. The methods can be easily extended to multidi-mensional cases by using the directional splitting technique.

It is well known that the Euler and Navier–Stokes equations canbe derived from the Boltzmann equations:

@f@tþ u

@f@x¼ Iðf ; f Þ ð2:1Þ

where f is the gas distribution function, u is the particle velocity,and I(f, f) is the collision integral. The Euler equations correspondto the Boltzmann equations with the Chapman–Enskog expansionof the gas distribution function truncated at the first term

f ¼ g ¼ q1

2pRT

� �Nþ32

e�1

2RT ðu�UÞ2þn2½ � ð2:2Þ

where g is the Maxwellian distribution function for the equilibriumstate, q is the density, U is the macroscopic velocity in the x direc-tion, T is the temperature, R is the gas constant, N is the number ofthe internal degrees of freedom, and n represents the remaining de-grees of freedom besides u such as the translational degrees of free-dom in y and z directions and the internal degrees of freedom. TheEuler equations can be obtained by taking the moments of (2.1)with f given by (2.2):

@

@t

ZwgdN

� �þ @

@x

ZwugdN

� �¼Z

wIðf ; f ÞdN ¼ 0 ð2:3Þ

where the moment w ¼ ð1;u; u2þn2

2 ÞT and dN ¼ dudn. These equationscan be further recast into a more familiar form:

@Q@tþ @F@x¼ 0 ð2:4Þ

with

Q ¼Z

wgdN ¼qqU

E

0B@1CA; F ¼ Z wugdN ¼

qU

qU2 þ p

UðEþ pÞ

0B@1CA ð2:5Þ

Here E ¼ q2 ½U

2 þ ðN þ 3ÞRT� and p = qRT.The Kinetic Flux-Vector Splitting (KFVS) method in [5] splits the

flux vector in (2.5) according to the sign of the particle velocity u:

F ¼ Fþ þ F� ¼Z

u>0wugLdNþ

Zu<0

wugRdN

¼ 12

qLeUL

qLULeUL þ 1þ erf ULffiffiffiffiffiffiffi

2RTL

p� �� �

pL

eULðEL þ pLÞ �ffiffiffiffiffiffiffi2RTLp

qe�

U2L

2RTLpL2

0BBBBBBB@

1CCCCCCCA

0BBBBBBB@

þ

qReUR

qRUReUR þ 1� erf URffiffiffiffiffiffiffiffi

2RTR

p� �� �

pR

eURðER þ pRÞ �ffiffiffiffiffiffiffiffi2RTRp

qe�

U2R

2RTRpR2

0BBBBBBB@

1CCCCCCCA

1CCCCCCCA ð2:6Þ

with eUL ¼ 1þ erf ULffiffiffiffiffiffiffi2RTL

p� �� �

UL þffiffiffiffiffiffiffi2RTLp

qe�

U2L

2RTL and eUR ¼ 1� erf½

URffiffiffiffiffiffiffiffi2RTR

p� �

�UR �ffiffiffiffiffiffiffiffi2RTRp

qe�

U2R

2RTR : The subscripts of L and R indicate that

the corresponding variables are at the left and right-hand sides ofthe interfaces, respectively.

Let us apply this KFVS scheme to the standard Sod’s shock-tube problem and compare its result with the numerical solutiongiven by Roe’s approximate Riemann solver in [7]. In order tomore clearly demonstrate the impact of each flux evaluation ap-proach on the resulting numerical solutions, all computationsare performed with the first-order reconstruction on a uniformmesh with Dx = 0.005. The Trapezoidal scheme is used for timediscretization with the CFL number of 0.8. We select this timediscretization simply because it has no numerical dissipation.The implicit operator used is the LU-SGS (Lower–Upper Symmet-ric Gauss–Seidel) scheme with the spectral radius approximation.Four Newton-type subiterations are used at each time step forreduction of the linearization and factorization errors and forimprovement of the time accuracy. A detailed description of thisalgorithm is referred to [13]. Fig. 2.1 presents the numerical solu-tions obtained by the above KFVS scheme after 100 time stepswith comparison to the exact solutions and Roe’s results. It isfound that the numerical solutions given by this KFVS schemeare more diffusive than those given by Roe’s scheme, especiallyat the contact discontinuity and around the foot of the expansionwave.

To further obtain the Navier–Stokes equations, one needs tokeep the first two terms of the Chapman–Enskog expansion ofthe gas distribution function. For simplicity, the Chapman–Enskogexpansion for the BGK model instead of the Boltzmann equation isconsidered here:

f ¼ g � sDg ¼ g � s @g@tþ u

@g@x

� �ð2:7Þ

where s is the particle collision time. The Navier–Stokes equationscan be obtained by taking the moments of (2.1) with f given by(2.7):

@

@t

ZwgdN� s

ZwDgdN

� �þ @

@x

ZwugdN� s

ZwuDgdN

� �¼Z

wIðf ; f ÞdN ¼ 0 ð2:8Þ

With the compatibility condition ofR

wDgdN ¼ 0, the vector of theconservative variables, Q, in (2.4) remains the same as the one givenin (2.5). On the other hand, the vector of the fluxes, F, in (2.4)

0.1

0.2

0.3

0.4

0.5

0.4 0.6 0.8 1

ExactRoeKFVSKWPS

x

ρ

(a) density

0

0.2

0.4

0.6

0.8

1

0.1 0.4 0.7 1

ExactRoeKFVSKWPS

x

U

(b) velocity

0

0.2

0.4

0.6

0.8

1

0.1 0.4 0.7 1

ExactRoeKFVSKWPS

x

p

(c) pressure

Fig. 2.1. Sod’s shock tube problem.

-0.09

-0.06

-0.03

0

0.5 0.7 0.9 1.1

Nor

mal

str

ess

U

Exact (Pr=2/3)KFVS (Pr=2/3)Exact (Pr=1)KFVS (Pr=1)Exact (Pr=10)KFVS (Pr=10)

(a) Normal stress

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.5 0.7 0.9 1.1

Hea

t flu

x

U

Exact (Pr=2/3)KFVS (Pr=2/3)Exact (Pr=1)KFVS (Pr=1)Exact (Pr=10)KFVS (Pr=10)

(b) Heat flux

Fig. 2.2. KFVS-NS solutions of M1 = 1.5 shock structure.

L. Tang / Computers & Fluids 56 (2012) 39–48 41

includes not only the vector of the inviscid fluxes defined in (2.5)but also the following vector of the viscous fluxes:

Fv ¼ �sZ

wuDgdN ¼ �l0

2 Nþ2Nþ3 Ux

2 Nþ2Nþ3 UUx þ Nþ5

2 RTx

0B@1CA ð2:9Þ

where l = sp is the viscosity coefficient.The Kinetic Flux-Vector Splitting (KFVS) method in [6] splits the

vector of the viscous fluxes in (2.9) according to the sign of the par-ticle velocity u:

Fv ¼ Fþv þ F�v ¼ �sZ

u>0wuDgLdN� s

Zu<0

wuDgRdN

¼ �lL

1þ erf ULffiffiffiffiffiffiffi2RTL

p� �2

02 Nþ2

Nþ3 UxL

2 Nþ2Nþ3 ULUxL þ Nþ5

2 RTxL

0B@1CA

2664

þ RTL

2pe�

U2L

2RTL

12RTL

2 Nþ2Nþ3 UxL � UL

TxLTL

� �TxLTL

Nþ24 2 Nþ6

Nþ3 UxL � ULTxLTL

� �0BBBB@

1CCCCAvuuuuuut

377775

� lR

1� erf URffiffiffiffiffiffiffiffi2RTR

p� �2

02 Nþ2

Nþ3 UxR

2 Nþ2Nþ3 URUxR þ Nþ5

2 RTxR

0B@1CA

2664

� RTR

2pe�

U2R

2RTR

12RTR

2 Nþ2Nþ3 UxR � UR

TxRTR

� �TxRTR

Nþ24 2 Nþ6

Nþ3 UxR � URTxRTR

� �0BBBB@

1CCCCAvuuuuuut

377775 ð2:10Þ

It is clear that our KFVS scheme for the Navier–Stokes equations,KFVS-NS, is much simpler than the one given in [6]. This is because

42 L. Tang / Computers & Fluids 56 (2012) 39–48

we adopt the Chapman–Enskog expansion for the BGK model insteadof the Boltzmann equation. On the other hand, the fixed unit Prandtlnumber problem of the BGK model has to be solved. This can beachieved by simply replacing the coefficient of Nþ5

2 in (2.9) and(2.10) with Nþ5

2Pr .The validity of this Prandtl number fix can be demonstrated

with the following standard viscous shock structure case in [11].We consider a stationary shock structure for a monatomic gas withthe freestream Mach number of 1.5. The freestream viscosity coef-ficient is l1 = 0.0005 and l / T0.8. The piecewise quadratic recon-struction, Q4c, in [14] is adopted for the viscous computations on auniform mesh with Dx = 0.000625. Since this is a steady problem,the first-order implicit time discretization is applied without anyNewton-type subiteration. The obtained normal stress and heatflux profiles across the shock are presented in Fig. 2.2 for the Pra-ndtl numbers of 2/3, 1, and 10, respectively, with comparison tothe analytical solutions of the one-dimensional Navier–Stokesequations given in [11]. The agreement between the numericaland analytical solutions is found excellent for all three Prandtlnumbers, indicating the effectiveness of our Prandtl number fix.

3. Kinetic wave/particle splitting method

Notice that the flux vector in (2.5) for the Euler equations can besplit into the convective and acoustic components as follows:

F ¼Z

wugdN ¼Z

wUgdNþZ

wcgdN ¼qU

qU2

UE

0B@1CAþ 0

pUp

0B@1CAð3:1Þ

where c is the thermal velocity. The Kinetic Wave/Particle Splitting(KWPS) method in [8,9] splits the convective flux vector accordingto the sign of U and the acoustic flux vector according to the signof c, respectively

F ¼ Fþ þ F� ¼Z

U>0wUgLdNþ

Zc>0

wcgLdNþZ

U<0wUgRdN

þZ

c<0wcgRdN ¼ 1

2

qLUL

qLULUL þ pL

ULEL þ ULpL

0B@1CAþ qRUR

qRURUR þ pR

URER þ URpR

0B@1CA

0B@1CA ð3:2Þ

with UL ¼ UL þ jULj þffiffiffiffiffiffiffi2RTLp

q, UL ¼ UL þ

ffiffiffiffiffiffiRTL2p

q, UR ¼ UR � jURj �

ffiffiffiffiffiffiffiffi2RTRp

q,

UR ¼ UR �ffiffiffiffiffiffiRTR2p

q. It is clear that the error and exponential functions

used in the KFVS scheme of (2.6) do not appear in (3.2). This makesthe KWPS scheme computationally 2.26 times faster than the KFVSscheme and 1.32 times faster than Roe’s scheme. On the other hand,as shown in Fig. 2.1, the density distribution predicted by the KWPSscheme for Sod’s shock tube problem is even more diffusive thanthe one given by the KFVS scheme.

Let us further consider the Navier–Stokes equations, startingwith the Chapman–Enskog expansion of the gas distribution func-tion for the BGK model in (2.7) again for simplicity. Then the vectorof the viscous fluxes in (2.9) for the Navier–Stokes equations can besplit into the convective and acoustic components as follows:

Fv ¼ �sZ

wuDgdN ¼ �sZ

wUDgdN� sZ

wcDgdN

¼ �s

�px

pUx � Upx

pUUx � U2

2 þ Nþ32 RT

� �px

0BB@1CCA

� s

pxNþ1Nþ3 pUx þ Upx

� Nþ52 ðRTÞ2qx þ Nþ1

Nþ3 pUUx þ U2

2 þ ðN þ 4ÞRTh i

px

0BB@1CCA ð3:3Þ

The KWPS approach further splits the above convective flux vectoraccording to the sign of U and the acoustic flux vector according tothe sign of c, respectively

Fv ¼ Fþv þ F�v ¼ �sZ

U>0wUDgLdN� s

Zc>0

wcDgLdN

� sZ

U<0wUDgRdN� s

Zc<0

wcDgRdN ¼ lL

2

aL

aLUL þ bL

aLU2L=2þ bLUL þ cL

0B@1CA

þ lR

2

aR

aRUR þ bR

aRU2R=2þ bRUR þ cR

0B@1CA ð3:4Þ

where

aL;R ¼ �UL;R

jUL;RjpxL;R

pL;R� 1

N þ 3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

pRTL;R

sUxL;R

bL;R ¼ � 2N þ 2N þ 3

� UL;R

jUL;Rj

� �UxL;R �

ffiffiffiffiffiffiffiffiffiffiffiffiffi2RTL;R

p

rpxL;R

pL;Rþ TxL;R

TL;R

!

cL;R ¼ �N þ 3

2UL;R

jUL;RjpxL;R

qL;R� N

N þ 3

ffiffiffiffiffiffiffiffiffiffiffiRTL;R

2p

rUxL;R �

N þ 52

RTxL;R

ð3:5Þ

Again the fixed unit Prandtl number problem of the BGK model canbe solved by simply replacing the coefficient of Nþ5

2 in (3.3) and (3.5)with Nþ5

2Pr . The resulting scheme is referred to be KWPS-NS. It is note-worthy that even though the KWPS-NS scheme in (3.4) looks morecomplicated than the KFVS-NS scheme in (2.10), the scheme is 2.33times faster than the KFVS-NS scheme. On the other hand, it is 1.5times slower than the Navier–Stokes solver with Roe’s scheme forthe inviscid fluxes and the second-order central discretization forthe viscous fluxes. This is because the second-order central discret-ization of the viscous fluxes is computationally much more efficientthan the KWPS discretization in (3.4). However, the second-ordercentral discretization of the viscous fluxes introduces the inconsis-tency between the reconstruction and evolution schemes becausethe derivatives of the solution used in the reconstruction stageare not adopted in the subsequent evolution stage.

4. Gas-kinetic BGK schemes

The gas-kinetic BGK scheme for the Navier–Stokes equations(BGK-NS) in [11] is based on the following integral solution ofthe BGK model under the assumption that the particle collisiontime s is locally constant,

f ðxjþ1=2; t;u; nÞ ¼1s

Z t

t0

gðx0; t0;u; nÞe�t�t0s dt0 þ e�

Dts f0ðxjþ1=2

� uDt; t0;u; nÞ ð4:1Þ

where x = x0 + u(t � t0) is the particle trajectory and Dt = t � t0. Thedetermination of this gas distribution function at the interfacex = xj+1/2 and at the time t requires the equilibrium state g(x0, t0, u, n)and the initial state f0. In [11], the equilibrium state takes a linearexpansion around the interface x = xj+1/2 and t = t0,

gðx; t;u; nÞ ¼ g0 þ ½ð1� H½x�Þ�gxL þ H½x��gxR�ðx� xjþ1=2Þ þ �gtDt ð4:2Þ

where g0 is the equilibrium state at x = xj+1/2 and t = t0, and H[x] isthe Heaviside function defined as follows:

H½x� ¼0; x < xjþ1=2

1; x P xjþ1=2

ð4:3Þ

On the other hand, the initial non-equilibrium state f0 in (4.1) is ob-tained from the Chapman–Enskog expansion of the gas distribution

0

0.2

0.4

0.6

0.8

1

0.1 0.4 0.7 1

ExactRoeKFVSKFVS_u0BGK

x

U

(b) velocity

0

0.2

0.4

0.6

0.8

1ExactRoeKFVSKFVS_u0BGK

x

p

(c) pressure

0.1

0.2

0.3

0.4

0.5

0.4 0.6 0.8 1

ExactRoeKFVSKFVS_u0BGK

x

ρ

(a) density

0.1 0.4 0.7 1

Fig. 4.1. Sod’s shock tube problem.

L. Tang / Computers & Fluids 56 (2012) 39–48 43

function truncated at the Navier–Stokes level. A linear expansionaround x = xj+1/2 and t = t0 gives,

f0 ¼ ð1� H½x�Þ½gL þ gxLðx� xjþ1=2Þ � sðgxLuþ gtLÞ� þ H½x�½gR

þ gxRðx� xjþ1=2Þ � sðgxRuþ gtRÞ� ð4:4Þ

As a result, the gas distribution function at the interface x = xj+1/2

and at the time t becomes:

fjþ1=2 ¼ 1� e�Dts

� �g0 þ s �1þ e�

Dts þ Dt

s e�Dts

� �u½H½u��gxL

þ ð1� H½u�Þ�gxR� þ s Dts� 1þ e�

Dts

� ��gt

þ e�Dts H½u�½gL � uðDt þ sÞgxL � sgtL�f

þ 1� H½u�Þ½gR � uðDt þ sÞgxR � sgtR�ð g ð4:5Þ

The numerical fluxes at the interface can be further obtained by tak-ing the moment of (4.5) over the velocity space.

Due to the complexity of (4.5) and in the determination of �gt , inthe literature, no explicit expressions of the BGK numerical fluxeshave been given in terms of the macroscopic variables. Many mo-ments of terms associated with the spatial and temporal expan-sions are required to compute at each interface and at each timestep. This makes the BGK scheme extremely expensive in termsof floating point operations per flux computation [12]. Only afterweeks of tedious derivations was the author able to obtain the ex-plicit expressions of the BGK numerical fluxes in terms of the mac-roscopic variables. Those expressions alone already take up morethan eight pages, making the approach still computationally veryinefficient.

In the literature, several attempts have been made to simplifythis BGK scheme, such as the one in [12]. However, a lack of rigor-ousness in those efforts adds more uncertainties to the derivationof the BGK scheme. For example, one uncertainty is the form thatthe equilibrium state in (4.1) should take. In the original versionof the BGK scheme [10], the spatial derivative of the equilibriumstate is assumed to be continuous across the interface. However,as shown in (4.2), the BGK scheme in [11] assumes the spatialderivative of the equilibrium state to be discontinuous across theinterface while the temporal derivative is assumed to be continu-ous. This makes the determination of �gt extremely complicated.

To simplify the above BGK scheme in a rigorous way, we firstconsider three gas-kinetic upwind schemes for the Euler equations,which have no physical viscosity. The first one is the KFVS schemein (2.6), which can also be derived from (4.1) by assuming both theequilibrium state g(x0, t0, u, n) and the initial state f0 to be discontin-uous across the interface,

f ðxjþ1=2; t;u; nÞ ¼ H½u�gL þ ð1� H½u�ÞgR ð4:6Þ

The second gas-kinetic upwind scheme is the totally thermalizedtransport scheme in [15], which can be obtained from (4.1) byassuming both the equilibrium state g(x0, t0, u, n) and the initial statef0 to be continuous across the interface. The resulting gas distribu-tion function at the interface is the Chapman–Enskog expansiontruncated at the Euler level,

f ðxjþ1=2; t;u; nÞ ¼ g0 ð4:7Þ

Therefore, the numerical fluxes at the interface are the inviscidfluxes with their flow variables reconstructed from those at the leftand right-hand sides of the interfaces as follows:Z

wu0g0dN ¼Z

u>0wu0gLdNþ

Zu<0

wu0gRdN ð4:8Þ

So, we refer to this gas-kinetic scheme as KFVS_u0 although this isreally not a flux-vector splitting scheme. The last gas-kinetic up-wind scheme considered is the so-called first-order BGK scheme

for the Euler equations or the partial thermalized transport schemein [15]:

f ðxjþ1=2; t;u; nÞ ¼ 1� e�Dts

� �g0 þ e�

Dts ½H½u�gL þ ð1� H½u�ÞgR� ð4:9Þ

which can be obtained from (4.1) by assuming the equilibrium stateg(x0, t0, u, n) to be continuous but the initial state f0 to be discontin-uous across the interface.

Let us apply these three gas-kinetic schemes to the previousSod’s shock-tube problem with the same reconstruction and thesame time discretization on the same mesh as before. As shownin Fig. 4.1, while the numerical solutions given by the KFVS scheme

44 L. Tang / Computers & Fluids 56 (2012) 39–48

are more diffusive than Roe’s results, those given by the KFVS_u0

scheme are less diffusive than Roe’s results at both shock and con-tact discontinuity. However, the KFVS_u0 scheme generates somenumerical oscillations at the foot of the expansion wave. As willbe discussed in [17], this is because the flux of the energy equationinvolves the third moment of the Maxwellian distribution functionwhile the highest moment of the Maxwellian distribution functionin the KFVS_u0 scheme is only second. The gas-kinetic BGK schemeof (4.9) makes a compromise between the above two KFVSschemes. It improves the robustness over the KFVS_u0 schemeand the accuracy over the KFVS scheme. Its solutions are almostthe same as Roe’s results, slightly better at both shock and contactdiscontinuity but slightly worse at the foot of the expansive wave.

It is important to point out that if s in (4.9) were related to thephysical dissipation, then the BGK scheme of (4.9) would reduce tothe KFVS_u0 scheme even in the non-smooth region wheneverDt – 0. For Dt = 0, Dt/s becomes undetermined. To keep the com-plete form of (4.9) for any Dt and s, therefore, it is reasonable tointerpret the term of Dt/s in (4.9) as a relaxation parameter. Its roleis just to add some numerical dissipation from the KFVS schemeinto the KFVS_u0 scheme. At the limiting case of Dt/s ?1 in thesmooth region, the BGK scheme of (4.9) reduces to the KFVS_u0

scheme; at the limiting case of Dt/s = 0 in the non-smooth region,it reduces to the KFVS scheme.

With the Chapman–Enskog expansion truncated at the Navier–Stokes level, one can further extend the above relaxation scheme tothe Navier–Stokes equations, resulting in a much simpler BGKscheme for the Navier–Stokes equations:

fjþ1=2 ¼ 1� e�Dts

� �½g0 � sðugx0 þ gt0Þ� þ e�

Dts fH½u�½gL

� sðugxL þ gtLÞ� þ ð1� H½u�Þ½gR � sðugxR þ gtRÞ�g ð4:10Þ

Compared with the BGK scheme of (4.5), this simplified BGKscheme is different in the following three aspects:

(1) Instead of taking (4.2), the equilibrium state in (4.1) takesthe following piecewise linear expansion around x = xj+1/2

and t = t0

gðx; t;u; nÞ ¼ g0 þ gx0ðx� xjþ1=2Þ þ gt0Dt ð4:11Þwhich is the same as the original version of the BGK scheme in[10]. This is because the spatial and temporal derivatives of thegas distribution function are coupled with each other throughthe compatibility condition. It is more reasonable to assumethe spatial derivative of the equilibrium state continuousacross the interface after the temporal derivative is assumedto be continuous. The practice of (4.2) makes the determina-tion of �gt or gt0 extremely complicated but no benefit has beenobserved based on the available numerical results.

(2) All time-accurate terms in (4.5) are not considered. This isbecause in the context of shock-capturing schemes, whatwe are interested in are the numerical fluxes at t = t0, neitherthe numerical fluxes at Dt nor the averaged numerical fluxesover Dt. Furthermore, the inclusion of the time-accurateterms not only complicates the resulting BGK scheme butalso complicates the implementation of an implicit schemewith local time stepping.

(3) Different from the BGK scheme of (4.5), where s is alwaysthe summation of both numerical and physical dissipations,there exist two types of the particle collision time in (4.10).The one in the BGK model acts as a relaxation parameter, i.e.,the numerical dissipation. On the other hand, the particlecollision time in the Chapman–Enskog expansion representsthe physical dissipation. For a well resolved viscous problem,therefore, the BGK scheme of (4.10) simply reduces to theKFVS_u0 scheme for the Navier–Stokes equations. On theother hand, the BGK scheme of (4.5) still needs the completeform, resulting in much poorer computational efficiency.

The numerical fluxes resulting from (4.10) can be presented interms of the macroscopic variables as follows:

Fjþ1=2 ¼ 1� e�Dts

� � q0U0

q0U20 þ p0 � 2 Nþ2

Nþ3l0Ux0

U0ðE0 þ p0Þ �l0 2 Nþ2Nþ3 U0Ux0 þ Nþ5

2Pr RTx0

� �0BBB@

1CCCA

þ e�Dts

1þ erf ULffiffiffiffiffiffiffi2RTL

p� �2

qLUL

qLU2L þ pL � 2 Nþ2

Nþ3lLUxL

ULðEL þ pLÞ �lL 2 Nþ2Nþ3 ULUxL þ Nþ5

2Pr RTxL

� �0BBB@

1CCCA26664

þ RTL

2pe�

U2L

2RTL

qL �lL

2RTL2 Nþ2

Nþ3 UxL �ULTxLTL

� �qLUL �lL

TxLTL

EL þ pL2 �lL

Nþ24 2 Nþ6

Nþ3 UxL �ULTxLTL

� �

0BBBBB@

1CCCCCA

vuuuuuuut

þ1� erf URffiffiffiffiffiffiffiffi

2RTR

p� �2

qRUR

qRU2R þ pR � 2 Nþ2

Nþ3lRUxR

URðER þ pRÞ �lR 2 Nþ2Nþ3 URUxR þ Nþ5

2Pr RTxR

� �0BBB@

1CCCA

� RTR

2pe�

U2R

2RTR

qR �lR

2RTR2 Nþ2

Nþ3 UxR �URTxRTR

� �qRUR �lR

TxRTR

ER þ pR2 �lR

Nþ24 2 Nþ6

Nþ3 UxR �URTxRTR

� �

0BBBBB@

1CCCCCA

vuuuuuuut

3777775 ð4:12Þ

where the simple Prandtl number fix is included. For a well resolvedviscous problem, this BGK scheme reduces to the KFVS_u0 schemefor the Navier–Stokes equations. If (Ux0, Tx0) in (4.12) are computedwith the two neighboring cell-averaged values, the resultingKFVS_u0 scheme becomes a Navier–Stokes solver with the KFVS_u0

scheme of (4.7) for the inviscid fluxes and a second-order centraldiscretization for the viscous fluxes. As mentioned before, such anevolution scheme is not consistent with the reconstruction schemealthough it is computationally more efficient. A consistent approachis to reconstruct (Ux0, Tx0) from (UxL, TxL) and (UxR, TxR) by:Z

wu0gx0dN ¼Z

u>0wu0gxLdNþ

Zu<0

wu0gxRdN ð4:13Þ

The resulting KFVS_u0 scheme is 3.43 times faster than the BGKscheme in [11] but still 3.5 times slower than the Navier–Stokes sol-ver with Roe’s scheme for the inviscid fluxes and the second-ordercentral discretization for the viscous fluxes.

5. Gas-kinetic BGK-KWPS schemes

Since a gas-kinetic BGK scheme can be considered as a relaxa-tion scheme between the two KFVS schemes with different mo-ments, one can further explore a gas-kinetic BGK scheme basedon the two KWPS schemes for improvement of the computationalefficiency. Let us start with the first-order BGK scheme for the Eu-ler equations.

It is obvious that in (4.9), the limiting case of Dt/s = 0 can be re-placed with the KWPS scheme of (3.2). Special attention should bepaid to the limiting case of Dt/s ?1. The major question is how tocompute (q0, U0, E0) from (qL, UL, EL) and (qR, UR, ER). A direct gen-eration of (4.8) gives:Z

wðU þ cÞ0g0dN ¼Z

c>0wc0gLdNþ

Zc<0

wc0gRdN ð5:1Þ

We refer to this gas-kinetic scheme as KWPS_u0.Based on the KWPS and KWPS_u0 schemes, the first-order BGK

scheme for the Euler equations can be further obtained as:

0

0.2

0.4

0.6

0.8

1

0.1 0.4 0.7 1

ExactRoeKWPSKWPS_u0BGK

x

U

(b) velocity

0

0.2

0.4

0.6

0.8

1

0.1 0.4 0.7 1

ExactRoeKWPSKWPS_u0BGK

x

p

(c) pressure

0.1

0.2

0.3

0.4

0.5

0.4 0.6 0.8 1

ExactRoeKWPSKWPS_u0BGK

x

ρ

(a) density

Fig. 5.1. Sod’s shock tube problem.

L. Tang / Computers & Fluids 56 (2012) 39–48 45

Fjþ1=2 ¼ 1� e�ts

� � q0U0

q0U20 þ p0

U0ðE0 þ p0Þ

0@ 1A

þ e�ts

qL12 ðUL þ jULj þ

ffiffiffiffiffiffiffi2RTLp

qÞ

qLUL12 UL þ jULj þ

ffiffiffiffiffiffiffi2RTLp

q� �þ pL

2

EL12 UL þ jULj þ

ffiffiffiffiffiffiffi2RTLp

q� �þ pL

2 UL þffiffiffiffiffiffiRTL2p

q� �0BBBBB@

1CCCCCA

2666664

þ

qR12 UR � jURj �

ffiffiffiffiffiffiffiffi2RTRp

q� �qRUR

12 UR � jURj �

ffiffiffiffiffiffiffiffi2RTRp

q� �þ pR

2

ER12 UR � jURj �

ffiffiffiffiffiffiffiffi2RTRp

q� �þ pR

2 UR �ffiffiffiffiffiffiRTR2p

q� �

0BBBBBB@

1CCCCCCA

37777775 ð5:2Þ

We call this scheme BGK-KWPS to distinguish it from the BGK-KFVSscheme of (4.9).

All KWPS, KWPS_u0, and BGK-KWPS schemes are applied to theprevious Sod’s shock-tube problem again with the same reconstruc-tion and the same time discretization on the same mesh as before.Similar to the results of the KFVS, KFVS_u0, and BGK-KFVS schemesshown in Fig. 4.1, Fig. 5.1 indicates that the numerical solutions gi-ven by the KWPS scheme are more diffusive than Roe’s results, actu-ally even more diffusive than the KFVS solutions. On the other hand,those given by the KWPS_u0 scheme are less diffusive than Roe’s re-sults at both shock and contact discontinuity but some numericaloscillations are produced after the contact discontinuity. Similar tothe KFVS_u0 scheme, this is because the highest moment of the Max-wellian distribution function in the KWPS_u0 scheme is only second.It is noteworthy that the KWPS_u0 solutions are even more oscilla-tory than the KFVS_u0 solutions. By making a compromise betweenthe above two, the BGK-KWPS scheme of (5.2), with the same relax-ation parameter as the BGK-KFVS scheme of (4.9), improves therobustness over the KWPS_u0 scheme and the accuracy over theKWPS scheme. Its solutions are even much less dissipative thanRoe’s results and the BGK-KFVS solutions at both shock and contactdiscontinuity while slightly worse at the foot of the expansive wave.This is because the numerical diffusion inherent in the KWPS_u0

scheme is much smaller than the KFVS_u0 scheme although theKWPS scheme is more diffusive than the KFVS scheme.

Similarly in the BGK-NS scheme of (4.12), the limiting case ofDt/s = 0 can be replaced with the KWPS-NS scheme of (3.4). Specialattention should be paid to the limiting case of Dt/s ?1. Now themajor question is how to reconstruct (Ux0, Tx0) from (UxL, TxL) and(UxR, TxR). A direct generation of the practice of (4.13) in theKFVS_u0 scheme gives:Z

wðU þ cÞ0gx0dN ¼Z

c>0wc0gxLdNþ

Zc<0

wc0gxRdN ð5:3Þ

As a result, the BGK-KWPS scheme for the Navier–Stokes equationscan be expressed as:

Fjþ1=2¼ 1�e�Dts

� � q0U0

q0U20þp0�2Nþ2

Nþ3l0Ux0

U0ðE0þp0Þ�l0 2Nþ2Nþ3U0Ux0þNþ5

2Pr RTx0

� �0B@

1CA

þe�Dts

qL12 ULþjULjþ

ffiffiffiffiffiffiffi2RTLp

q� �þlL

2 aL

qLUL12 ULþjULjþ

ffiffiffiffiffiffiffi2RTLp

q� �þ pL

2 þlL2 aLULþbLð Þ

EL12 ULþjULjþ

ffiffiffiffiffiffiffi2RTLp

q� �þ pL

2 ULþffiffiffiffiffiffiRTL2p

q� �þlL

2 aLU2

L2 þbLULþcL

� �

0BBBBBB@

1CCCCCCA

26666664

þ

qR12 UR�jURj�

ffiffiffiffiffiffiffiffi2RTRp

q� �þlR

2 aR

qRUR12 UR�jURj�

ffiffiffiffiffiffiffiffi2RTRp

q� �þ pR

2 þlR2 aRURþbRð Þ

ER12 UR�jURj�

ffiffiffiffiffiffiffiffi2RTRp

q� �þ pR

2 UR�ffiffiffiffiffiffiRTR2p

q� �þlR

2 aRU2

R2 þbRURþcR

� �

0BBBBBB@

1CCCCCCA

37777775ð5:4Þ

where the simple Prandtl number fix is included. For a well resolvedviscous problem, this BGK scheme reduces to the KWPS_u0 schemefor the Navier–Stokes equations, which is found as efficient as theNavier–Stokes solver with Roe’s scheme for the inviscid fluxesand the second-order central discretization for the viscous fluxes.

6. Numerical results and discussion

In this section, several test cases are presented to compare theperformances of the BGK-KFVS and BGK-KWPS schemes. To makea fair comparison, the first-order reconstruction is used in all

0

1

2

3

4

5

6

7

0.5 0.6 0.7 0.8 0.9

ExactRoeBGK-KFVSBGK-KWPS

x

ρ

(a) density

-1

3

7

11

15

0 0.3 0.6 0.9

ExactRoeBGK-KFVSBGK-KWPS

x

U

(b) velocity

10

70

130

190

250

310

370

430

0 0.3 0.6 0.9

ExactRoeBGK-KFVSBGK-KWPS

x

p

(c) pressure

0

1

2

3

4

5

6

7

0.5 0.6 0.7 0.8 0.9

ExactRoeBGK-KFVSBGK-KWPS

x

ρ

(a) density

-1

3

7

11

15

0 0.3 0.6 0.9

ExactRoeBGK-KFVSBGK-KWPS

x

U

(b) velocity

10

70

130

190

250

310

370

430

0 0.3 0.6 0.9

ExactRoeBGK-KFVSBGK-KWPS

x

p

(c) pressure

Fig. 6.1. Woodward–Colella’s blast wave problem (N = 1600, Dt = 0.000025,t = 0.038).

0 0.5 1 1.5 2 2.5 30

0.5

1 1.1 1.52 1.94 2.36 2.78 3.2 3.62 4.04 4.46 4.88 5.3

(a) Roe

0 0.5 1 1.5 2 2.5 30

0.5

1 1.1 1.52 1.94 2.36 2.78 3.2 3.62 4.04 4.46 4.88 5.3

(b) BGK-KFVS

0 0.5 1 1.5 2 2.5 30

0.5

1 1.1 1.52 1.94 2.36 2.78 3.2 3.62 4.04 4.46 4.88 5.3

(c) BGK-KWPS

Fig. 6.2. Density contours of Mach 3 wind tunnel with a step(Dx = Dy = 0.05, CFL = 5).

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2 2.5 3

BGK-KFVS

BGK-KWPS

x

ρ

Fig. 6.3. Density distributions at y = 0.725 for Mach 3 wind tunnel with a step(Dx = Dy = 0.05, CFL = 5).

46 L. Tang / Computers & Fluids 56 (2012) 39–48

inviscid computations and the same high-order reconstructionschemes are used in the viscous computations for all flux evalua-tion methods, including Roe’s approximate Riemann solver in [7].For all inviscid computations in this paper, the relaxation parame-ter of Dt/s takes 1=ðC1 þ C2

jpL�pR jpLþpR

Þ with C1 = C2 = 0.8. No specialattention has been paid to search for the optimal relaxation param-eter. This is partially because our ultimate goal, as shown in thenext two papers [16,17], is to avoid using the relaxation parameter.

6.1. Woodward–Colella’s two interacting blast wave case

Let us first use Woodward–Colella’s blast wave interaction casein [18] to test the robustness of the two BGK schemes for solutionof one-dimensional Euler equations. A uniform mesh withDx = 0.000625 is adopted for computations. The Trapezoidalscheme with four Newton-type subiterations and Dt = 0.000025is used for time discretization. Fig. 6.1 presents the numerical solu-tions obtained by the BGK-KFVS and BGK-KWPS schemes at

-3 -2 -1-3

-2

-1

0

1

2

3

-3 -2 -1-3

-2

-1

0

1

2

3

(a) Roe’s solution (b) Solid lines: BGK-KFVS; Dash lines: BGK-KWPS

Fig. 6.4. Hypersonic cylinder case (M1 = 6, CFL = 5, 500 iterations).

0

0.2

0.4

0.6

0.8

1

0 2 4 6

ExactRoe + CentralBGK-KFVSBGK-KWPS

x

U

(a) U-velocity

0

0.3

0.6

0.9

0 2 4 6

ExactRoe + CentralBGK-KFVSBGK-KWPS

x

V

(b) V-velocity

Fig. 6.5. Blasius boundary layer velocity profiles (M1 = 0.15, Re = 105).

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

TestRoe + CentralBGK-KFVSBGK-KWPS

Cp

x/c

Fig. 6.6. Surface Cp distribution over RAE2822 airfoil case (M1 = 0.725, a = 3.1,Re = 6.5 � 106).

L. Tang / Computers & Fluids 56 (2012) 39–48 47

t = 0.038 with comparison to the numerical solutions of Roe’sscheme. It is found that the BGK-KFVS solution is slightly less dis-sipative than Roe’s solution while the BGK-KWPS solution is muchmore accurate than the other two solutions. The computationaltime of BGK-KFVS for this case is only 1.6 times while BGK-KWPSis about 1.05 times the computational time of Roe’s scheme be-cause the remaining parts of the algorithm are the same.

6.2. Mach 3 wind tunnel with a step

Next, we use the case of Mach 3 wind tunnel with a step to fur-ther test the robustness of the two BGK schemes for solution oftwo-dimensional Euler equations with the Cartesian coordinates.A coarse Cartesian mesh with D x = Dy = 0.05 is used for computa-tions to emphasize the difference between the results from the twoBGK schemes. The first-order implicit time discretization is usedfor the CFL number of five without any Newton-type subiteration.Fig. 6.2 first presents the predicted density contours after 50 itera-tions. It is found that without any special treatment around thecorner point, Roe’s scheme fails to produce a physically correctsolution. On the other hand, both BGK schemes have no difficultyto handle the corner point because they are really multidimen-sional upwind schemes. Fig. 6.3 further compares the density dis-tributions at y = 0.725 obtained by the two BGK schemes. It is clearthat the BGK-KWPS scheme captures slightly sharper and strongershock profiles than the BGK-KFVS scheme. This indicates that thenumerical diffusion inherent in the BGK-KWPS scheme is smaller.The computational time of the BGK-KFVS scheme for this case isabout 1.84 times while the BGK-KWPS scheme is about 1.16 timesthe computational time of Roe’s scheme.

6.3. Hypersonic cylinder case

Let us continue to use a hypersonic cylinder case to test therobustness of the two BGK schemes for solution of two-dimensional Euler equations with the curvilinear coordinates. Abody-conforming mesh with 81 points along the body surfaceand 41 points in the normal direction is used for computations.Since this is a steady problem, the first-order implicit time discret-ization is used for the CFL number of five without any Newton-typesubiteration. Fig. 6.4 presents the predicted density contours after500 iterations. As expected, Roe’s scheme incurs the carbunclephenomenon. On the other hand, both BGK schemes are able toproduce the physically correct solutions. The computational time

of the BGK-KFVS scheme for this case is about 1.97 times whilethe BGK-KWPS scheme is about 1.2 times the computational timeof Roe’s scheme.

6.4. Blasius boundary layer case

After demonstrating the robustness of the two BGK schemes forinviscid computations, we use the laminar boundary layer over flat

48 L. Tang / Computers & Fluids 56 (2012) 39–48

plate case to examine the accuracy of the two BGK schemes forsolution of two-dimensional Navier–Stokes equations with theCartesian coordinates. The Mach number considered is 0.15 andthe Reynolds number is 105. A uniform Cartesian mesh withDx = 0.005 and Dy = 0.002 is used for the computations. Since thegas-kinetic evolution schemes use the same slopes in the recon-struction scheme, it is not proper to use the first-order reconstruc-tion in viscous computations. A high-order reconstruction schemeis needed. As a result, one cannot expect to see the same large dif-ference between the results given by the different evolutionschemes as in the above inviscid cases. Fortunately, our objectivehere is to demonstrate that while more robust for inviscid compu-tations, these BGK schemes are also accurate for viscous computa-tions. For the well-resolved flat plate case, the piecewise quadraticreconstruction, Q2d, in [14] is used and the BGK schemes simplyreduce to either KFVS_u0 or KWPS_u0 for the Navier–Stokes equa-tions. Since this is a steady problem, the first-order implicit timediscretization is used with the CFL number of five and withoutany Newton-type subiteration. Fig. 6.5 presents the obtained veloc-ity profiles inside the boundary layer at x = 0.4975, compared withthe Blasius solutions. Different from the KFVS and KWPS schemes,both BGK-KFVS and BGK-KWPS schemes are found able to producethe numerical solutions in good agreement with the analytical Bla-sius solutions. Although the difference between the results givenby the different evolution schemes is very small, it is still visiblethat the BGK-KWPS solutions are slightly more accurate than theother two numerical solutions. While the BGK-KFVS scheme takesabout 1.2 times the computational time of the Navier–Stokes sol-ver with Roe’s scheme for the inviscid fluxes and the second-ordercentral discretization for the viscous fluxes, the BGK-KWPS schemetakes almost the same amount of computational time as thatNavier–Stokes solver.

6.5. RAE2822 airfoil case

Finally, we use the RAE2822 airfoil case in [19] to demonstratethe accuracy of the two BGK schemes for solution of two-dimensional Navier–Stokes equations with the curvilinear coordi-nates and with the Spalart–Allmaras turbulence model. The Machnumber considered is 0.725, the angle of attack is 3.1�, and the Rey-nolds number is 6.5 millions, which are closer to the test setting thanother computations in the literature. The body-conforming C-typemesh has 253 points in the wraparound direction with 177 pointon the airfoil and 91 points in the normal direction with the first gridspacing of 10�5 chords. Since this is a steady transonic problem, thepopular MUSCL scheme is used for reconstruction and the first-orderimplicit time discretization is used with the CFL number of five andwithout any Newton-type subiteration. Fig. 6.6 presents the surfacepressure distributions over the airfoil surface obtained by the threeevolution schemes, compared with the test data in [19]. It is foundthat the numerical results given by both BGK-KFVS and BGK-KWPSschemes are in good agreement with the test data. The BGK-KFVSsolution is almost identical to the Roe’s solution while the BGK-KWPS solution has a slightly sharper shock profile. For this case,the computational times of the BGK-KFVS and BGK-KWPS schemesare about 1.4 and 1.2 times the Navier–Stokes solver with Roe’sscheme for the inviscid fluxes and the second-order central discret-ization for the viscous fluxes, respectively.

7. Conclusions

The simplest first-order BGK scheme for the Euler equations canbe interpreted as a relaxation scheme between the two KFVSschemes with different moments. It improves the robustness overthe KFVS_u0 scheme and the accuracy over the KFVS scheme. A

direct generation of this relaxation scheme to the Navier–Stokesequations leads to a much simpler BGK scheme for the Navier–Stokes equations than the one in the literature. In this simplifiedBGK scheme, there exist two types of the particle collision time.The one in the BGK model acts as a relaxation parameter. Its roleis to add some numerical dissipation from the KFVS scheme intothe KFVS_u0 scheme. On the other hand, the one in the Chap-man–Enskog expansion of the gas distribution function representsthe physical dissipation. Therefore, for a well resolved viscousproblem, the BGK scheme reduces to the KFVS_u0 scheme for theNavier–Stokes equations.

Following this relaxation approach, a new type of BGK schemeshas been further developed based on the two KWPS schemes with dif-ferent moments. For a less resolved inviscid problem, the resultingBGK-KWPS scheme is found not only computationally more efficientbut also less diffusive than the above BGK-KFVS scheme. However,this needs further and more rigorous investigation by performingthe numerical analysis of a model 1-D convection–diffusion equation.

Finally, it is noteworthy that the above relaxation approach iscomputationally less appealing. This is because it involves twoKFVS or KWPS schemes and thereby is computationally less effi-cient. More importantly, the approach has a relaxation parameter,an uncertainty similar to the artificial viscosity used in the centraldiscretization approach. Therefore, in the next two papers [16,17],we will improve the accuracy of the KFVS/KWPS schemes and therobustness of the KFVS_u0/KWPS_u0 schemes without relaxation.

Acknowledgment

This work is sponsored by DARPA SBIR Phase II contract ofW31P4Q-09-C-0090. The technical monitor is Dr. Matthew S.Goodman.

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