The Equivalent Differential Equation of the Gas-Kinetic
BGK Scheme
Wei Gao · Quanhua Sun · Kun Xu
Abstract The equivalent partial differential equation is derived for the gas-kinetic
BGK scheme (GKS) [Xu, J. Comput. Phys. vol. 171, pp. 289-335 (2001)] following
its detailed numerical procedure. The GKS scheme is a one-step scheme where the
flux reconstruction employs the accurate time-dependent solution of the BGK
equation. The derived equation confirms that the GKS scheme solves the
Navier-Stokes equations with a predetermined bulk viscosity. It shows that the
directional splitting method fails to include the contributions from velocity gradients
in tangential direction to the viscous flow flux, which can be resolved using a
multi-dimensional scheme. The GKS scheme is second-order accurate in both space
and time at the level of Euler equations. However, it is first-order accurate in time for
viscous flow simulations.
Keywords
Equivalent differential equation · Gas kinetic BGK scheme · Navier-
Stokes equations · Compressible flow
Mathematics Subject Classification (2010) 35Q30 · 65M08 · 76M25 · 76N15
· 82B40
W. Gao · Q. Sun (corresponding author)
State Key Laboratory of High-temperature Gas Dynamics, Institute of Mechanics, Chinese Academy of Sciences
No. 15 Beisihuan Xi Rd, Beijing 100190, China
e-mail: [email protected]
Tel: +86 (10) 82544023
K. Xu
Mathematics Department, Hong Kong University of Science and Technology
Clear Water Bay, Kowloon, Hong Kong
1 Introduction
In the past years, a gas-kinetic BGK scheme (GKS) has been developed for
compressible flow simulations [1-3]. The GKS scheme evaluates the interface flux by
integrating the velocity distribution function of gas molecules, which is different from
conventional finite volume method where macroscopic properties are employed
directly for flux evaluation. In the GKS scheme, the gas distribution function follows
the accurate time-dependent solution of the BGK equation thus the free transport and
collision mechanisms are physically coupled. It has been shown that the GKS scheme
is very successful for many engineering applications [4,5] and is more robust than
generalized Riemann problem (GRP) scheme for compressible flow simulations [6].
The GKS scheme is designed using the distribution function governed by the BGK
equation under the conditions where the Navier-Stokes (NS) equations are valid. The
scheme does not solve directly the gas distribution function using the finite volume
method. Instead, the distribution function is used to construct the evolution process
for interface flux. It is more like a physical model than a numerical scheme. The
validity of the scheme relies heavily on the construction of the distribution function on
the interface. In an early scheme [1], the initial gas distribution is reconstructed with a
slope term having the spatial derivation of a Maxwellian distribution. Later, Xu [2]
added a nonequilibrium term to account for the nonequilibrium states obtained from
the Chapman-Enskog expansion. Ohwada [7] analyzed the kinetic schemes using a
railroad method and showed that solutions in the GKS scheme were 2nd-order accurate
in both space and time. Recently, the GKS scheme has been extended to
multi-dimensional schemes [4,8], high-order schemes [9,10], and unified schemes for
rarefied gas flows [11,12]. Particularly, the gradients of flow variables in both normal
and tangential directions are explicitly included in the distribution function at cell
interface in multi-dimensional schemes. For high-order schemes, it is important to
have the initial distribution and its evolution reconstructed using high-order processes.
The validity of GKS schemes has been demonstrated by various test problems in
the literature. The underlying equations, however, are not as clear as those schemes
directly solving macroscopic equations such as the Navier-Stokes equations. Although
it may be more suitable to employ physical means to resolve problems such as the
flow in discontinuous regions, it is beneficial to know the governing equations of the
GKS scheme in smooth flow regions. If the equivalent differential equation can be
obtained, the numerical error will be obvious and may indicate research directions for
further improvement of the scheme.
The objective of this study is to find the equivalent partial differential equation
(EDE) of the GKS scheme when it is applied to a smooth flow region and show its
connection to the Navier-Stokes equations. For this purpose, the GKS scheme is
introduced in Sect. 2 along with its flux expressions. The EDE is then derived in Sect.
3. The connection between the derived equation and the NS equations is discussed in
Sect. 4. Some concluding remarks are given in the last section.
2 The Gas-Kinetic BGK Scheme and its Flux Evaluation
Several gas-kinetic schemes have been reported in the literature. The most important
one seems to be the one published in 2001 [2] and is termed as the GKS scheme. Thus
the GKS scheme is analyzed in this paper. As the scheme has been detailed in [2],
only related expressions are presented here.
The GKS scheme evaluates the interface flux with the help of the velocity
distribution function that follows the BGK equation,
f
f
f
f g  f
u v  w 

t
x
y
z
(1)
where f is the gas distribution function, and g is the equilibrium state approached by f.
Both f and g are functions of space (x, y, z), time t, molecular velocities (u, v, w), and
internal variable ξ. The internal variable has K degrees of freedom where K is equal to
 5  3    1 , and γ is the ratio of specific heats. The relaxation time τ is related to
the gas viscosity and pressure (    p ). The equivalent state g is a Maxwellian
distribution,

g   
 
K 3
2
e

   u U    v V    w W   2
2
2
2
,
where ρ is the density, (U, V, W) is the macroscopic velocity in the (x, y, z) directions,
and λ is equal to m 2kT . Here m is the molecular mass, k is the Boltzmann constant,
and T is the temperature.
The general solution of f in (1) at a cell interface x j 1 2 and time t is
f  x j 1 2 , t , u , v, w,   
1
t
g  x, t , u , v, w,   e 


 t t   
0
dt   e t  f 0  x j 1 2  ut 
(2)
where x  x j 1 2  u  t  t   is the molecular trajectory and f 0 is the initial state of f
at the beginning of each time step (t=0). Functions g and f 0 are unknown and are
reconstructed as,
g  g 0 1  1  H  x  a l x  H  x  a r x  At 
(3)
f 0  1  H  x  g l 1  a l x    a l u  Al    H  x  g r 1  a r x    a r u  Ar  
where
0, x  0
,
H  x  
1, x  0
g0 ,
gl ,
gr
(4)
are the corresponding Maxwellian
distributions constructed at the interface, the left hand side, and the right hand side of
the interface. The variables a and A are the spatial and temporal derivatives of related
distributions, respectively. Substituting (3) and (4) into (2), function f can be
expressed as,
f  x j 1 2 , t , u , v, w,    1  e  t   g 0    t   1  e t   Ag 0

   e  t   1  te  t 
  a H  u   a 1  H  u   ug
l
r
0


1  u  t    a  H  u  g  1  u  t    a  1  H  u   g 
 e  t   Al H  u  g l  Ar 1  H  u   g r
 et 
l
l
r
(5)
r
Expression (5) is rather complex and can be greatly simplified in smooth flow
regions where g 0  g l  g r , a l  a r  a l  a r ,
A  Al  Ar . The function f then
becomes as
f  g 0 1   au   t    A  .
(6)
With the above expression, the flux through an interface can be easily integrated.
Particularly, the mass, momentum, and energy fluxes in the x direction are calculated
as,
F    ug 0 1   au   t    A  d 
(7)
where d   dudvdwd1  d  K ,    1, u, v, w, 1  u 2  v 2  w2   2   ,
2

a  a1  a2u  a3v  a4 w  a5  u  v  w2   2  ,
2

2
A  A1  A2u  A3v  A4 w  A5  u 2  v 2  w2   2  .
The derivatives a and A are determined using the slopes of macroscopic quantities.
Namely,
1
 g a  d   b ,
(8)
A  d     g0 au  d  ,
(9)
0
g
where b1 
b5 
0
0
 j 1   j
 U   jU j
 V   jV j
 W   jW j
, b2  j 1 j 1
, b3  j 1 j 1
, b4  j 1 j 1
,
0 x
0 x
0 x
0 x
 j 1 E j 1   j E j
, and E  1  U 2  V 2  W 2  K  3  . The subscript 0 refers to the
2
2 
0 x
reconstructed value at the interface j+1/2. Notice that slopes b are approximated
using the neighboring cell values and are 2nd-order accurate. With the values of a
and A from (8) and (9), the fluxed in (7) are derived as,

2t  1

2
2
2
F  0  U 0 
   K  2  U 0  V0  W0  b1   K  2  U 0b2  V0b3  W0b4  b5   ,
K 3 2


1 0 K  2
 2

 U 0  2   K  3  b2  U 0b1 

0
0






K 5
2
2
2
,
   2 K  3 U 0  3V0  3W0 

F u   0 
 U 0b1
20 

t 




 K  3    3  K  1 U 2  K  5  b  6U V b  W b  b  
 2
0
0 0 3
0 4
5 
 


20 
 


(10a)
(10b)
0


 U 0V0  2 V0b1  b3 

0

,
F v   0 
2
2
2



  2 K  5 U 0  V0  W0 V0b1
 t 

 K  3    K  3  U 2b  2V   K  2  U b  V b  W b  b   
0 3
0
0 2
0 3
0 4
5 


(10c)
0


 U 0W0  2 W0b1  b4 

0

,
F w   0 
2
2
2



2
5
K
U
V
W
W
b






0
0
0  0 1
 t 

 K  3  2W   K  2  U b  V b  W b  b    K  3  U 2b  
0
0 2
0 3
0 4
5
0 4 


(10d)
F E ,Pr 1 

1   2
K 5
 0  U 0  U 0  V0 2  W0 2 


2  
20 


 K 1
 K  3 K  5   b  
2
2
2

3
U
V
W




 0 0 0
 1 
0

40
 
 2
  K  3  0 


   K  1 U 0b2  2V0b3  2W0b4   K  5  b5



  U 0 2  V0 2  W0 2    2 K  1 U 0 2  V0 2  W0 2  
 


 


   K  5    2 K  7 U 2  V 2  W 2 
b1  


0
0
0

 .



20

 



 

  2  K  2  U 0 2  2  K  2  V0 2  W0 2  
 


 
 t 
U 0 b2


K
3
K
5






 
 K 3

 
 

0



 

 
 

K 5
   2  5U 0 2  V0 2  W0 2  

 V0b3  W0 b4  b5   
0 
 
 


 
2

 2  K  3  U 0 b5
 

(10e)
It is well known that the Prandtl number in the BGK equation is fixed at 1. In order
to extend the GKS scheme to flows with arbitrary Prandtl number, a post-processing
step is used in [2] and the energy flux is updated as,
 1

F E  F E ,Pr 1    1 q ,
 Pr 

(11)

where q  1   u  U t   u  U t 2   v  Vt 2   w  Wt 2   2 fd  , which is calculated as,
2
q  F E ,Pr 1  U t Q5  U t 2  Vt 2  Wt 2  Q2  U t F u  Vt F v  Wt F w
(12)
and  Q1 , Q2 , Q3 , Q4 , Q5 T   f   d  , U t  Q2 , Vt  Q3 , Wt  Q4 . The expressions of
Q1
Q1
Q1
Q are derived as follows,
Q1  0 1  tb2  ,
(13a)

2t  1

2
2
2
Q2   0  U 0 
   K  2  U 0  V0  W0  b1   K  2  U 0b2  V0b3  W0b4  b5   , (13b)
K 3 2


Q5 


  W  t U W b  W b  U b   ,
Q3   0 V0  t U 0V0b1  V0b2  U 0b3  ,
(13c)
Q4
(13d)
0
0
0
0 1
0 2
0 4

K 3
1  2
0   U 0  V0 2  W0 2 


2 
20 

 

   K  1 U 2  V 2  W 2   K  5  U b   K  1 U 2  V 2  W 2  K  5  b   (13e)
 0 1 
 2
0
0
0
0
0
   K  3 0
20 
20   
 K 3
 t 

K 5 
4

 
 
   K  3 U 0  V0b3  W0b4  2 b5 


 
It should be emphasized that (10) and (13) use values of flow variables at the
interface. Thus the interface values are reconstructed as the averages of the
neighboring cell values. For instance,  0    j   j 1  2 . Numerical error of this
approximation can be identified using Taylor expansion at the interface as,
0   
x 2  2 
 O  x 4 
8 x 2
(14)
A summary of related expressions for interface values is listed in Appendix (A.1).
Notice that O(t 2 ) is introduced when U t , Vt , Wt are approximated as shown in
(A14-A18). With these expressions, the fluxes are derived with cell size x and
time t as follows,
 U 2 p 
   O  x 2 
F  t   U t 0  t  
x  t 0
 x
(15a)
 U 3 K  5 U
2 K  4 U 
p 

2

Fu  t    U 2  p 
t
p



 3U

  O  x  (15b)

3
3
K
x
x
K
x
x







 t 0

 t 0
 U 2V
V 
p 

F v  t    UV  
t


 V   O  x 2 


x  t 0
x  t 0

 x
(15c)
 U 2W
W 
p 

2
F w  t    UW  
W
  O  x 
  t 
x  t 0
x  t 0

 x
(15d)
1
  2K  4 2
K 5

F E  t    U U 2  V 2  W 2    K  5  Up   
U V 2 W 2 
RT  
x  K  3
2
Pr
  t 0
 U 2 U 2  V 2  W 2   K  4  K  5  U 2


p

x
K 3
x

t
p
      K  8  U 2  V 2  W 2   K  5  RT 
2 
x
2


Pr  1 K  2 U  U
2 p 
 V V W W
 2




 U 

K
x
x
x
x
x







x x
P
r
3







(15e)


  O  t 2 , x 2 




  
   t 0
Expressions (15) are the fluxes of the GKS scheme in smooth regions. These fluxes
are evaluated at time t using values of macroscopic flow variables at the beginning of
the time step. It should mention that the present derivation of fluxes is based on the
first order expansion of the velocity distribution function as in (3) and (4). For
higher-order gas-kinetic BGK schemes, the flux expressions may be slightly different.
With the flux expressions, conservative flow variables Q in the cell are updated
using the finite volume method. Namely,
Q n 1  Q n   
t
0
  F  dS dt
Vol
(16)
where Q    , U , V , W ,  E  , F is the flux along the surface, and Vol is the
cell volume. To derive the governing equation for Q, (16) is expressed in the integral
form,
t
 Q
   t
0

   F  dVol  dt  0

(17)
As (17) is applied to every cell and any time step, its differential form can be obtained
as
Q F G H



0
t x y z
(18)
where F, G and H are component fluxes in x, y and z directions. It should emphasize
that Q, F, G, and H are evaluated using variables with instantaneous values at local
position.
3 The Equivalent Differential Equation
In order to derive the equivalent differential equation of GKS, the fluxes in (18) are to
be evaluated locally at time t. Thus the variables at t=0 in (15) are converted to values
at time t using Taylor expansions. Taking the mass flux (15a) as example, we have,
U t 0  U t t  t
U
t
t t
 O t 2 
(19)
and
 U 2 p 
 U 2 p 
  
   O t  ,

x  t 0  x
x  t t
 x
(20)
then
F  t   U
t t
t
U
t
t t
 U 2 p 
t
   O  t 2 , x 2  ,
x  t t
 x
or
F  U  t
U  U 2 p 
t
   O  t 2 , x 2  .
x
t

x 

Similarly, the other fluxes are derived as follows,
(21a)
2 K  4 U

2 K  4 U 


 t  U 2  p 

x
t 
x 
K 3
K 3
 U 3 K  5 U
p 
2
2
t

 3U
p
  O  t , x 




x
K
3
x
x


Fu  U 2  p 
F v  UV  
V

V
 t  UV  
x
t 
x
F w  UW  
F E 
1
2
t
2
p 
  U V
2
2

V
t
  O  t , x 
 
x 
  x
W

W
 t  UW  
x
t 
x
 U U 2  V 2  W 2  
K 5
2
 1
2
2
2
  U U  V  W  
t  2
Up  
K 5
2
(21b)
2
p 
  U W
2
2

W
t

  O  t , x 

x 
  x
 K 2

x  K  3
Up  
(21c)
U2 
V 2 W 2 K  5


RT 
2 Pr
2

 K 2

x  K  3
U2 
V 2 W 2
2

K 5
2 Pr

(21d)
(21e)
RT  

 U U  V  W   K  4  K  5  U


p


x
K 3
x



t
p
2
2
2
2
2
     K  8  U  V  W   K  5  RT 
  O  t , x 
2
x

2

1   K  2 U   U
2 p 

 V V  W  W   

 2   1   

  U  x x   x x   
 x 
 Pr   K  3 x  x



2
2
2
2
2
All variables in (21) are now in time t. Then (21) presents the instantaneous
numerical flux where the difference between the exact one and the numerical one is
on the order of O  t , x 2  . For a finite volume conservative GKS, which is the same
for any other finite volume method, the error in the instantaneous flux is the only
source for the error in the differential equations of (18). Therefore the differential
equation for the GKS must have the following form,
Q F G H



 O  t , x 2 , y 2 , z 2   0 ,
t x y z
where
U




2K  4 U


U 2  p 

K  3 x



,
V
UV  


F 
x



W
UW  



x


2
2

  K  2 2 V W
K 5

U 

RT  
   E  p U   
x  K  3
2
2 Pr


(22)
V


W






U

U


UV  


UW  

y


z


,

.
V
2K  4 V


V 2  p 



VW  
 H 
z
K  3 y
G





2K  4 W
2
W



W  p 



VW  
K  3 z


y


2
2



2
2
  K  2 2 U V
K 5


K

2
U

W
K

5


2
W 

RT  
   E  p V  
   E  p W   
V 

RT  


z  K  3
2
2 Pr


y  K  3
2
2 Pr


G and H are obtained be rotating variables and coordinates in F. From (22), the time
derivative terms in (21) are derived as follows,


 U     U 2  p   F , y  F , z  F ,   O  t , x 2 , y 2 , z 2 
t
x
(23a)
 2
2K  4 U  U3
p K  5 U
 
 3U 
p  Fu, y  Fu,z  Fu,  O t, x2 , y2 , z2  (23b)
 U  p 
t 
K  3 x 
x
x K  3 x

V
 UV  
t 
x

W
 UW  
t 
x
VU 2
p



V
 F v , y  F v , z  F v ,   O  t , x 2 , y 2 , z 2 


x

x

WU 2
p



W
 F w, y  F w, z  F w,   O  t , x 2 , y 2 , z 2 

x
x

(23c)
(23d)

  K  2 2 V 2 W 2 
T 
U 
   E  p U   

 
2
t 
x  K  3
x 

 K  4  K  5  p U 2   K  7 U 2  E  RT  p (23e)
1 

U 2 U 2  V 2  W 2  


2 x
2  K  3
x  2
 x
 F E , y  F E , z  F E ,   O  t , x 2 , y 2 , z 2 
where the expressions of F , y , F , z , F , , Fu , y , Fu , z , Fu , , F v, y , F v , z , F v ,  , F w, y , F w, z ,
F w,  , F E , y , F E , z , F E ,  are given in Appendix (A.2), and  
K 5
R is the heat
2 Pr
transfer coefficient.
Substituting (23) to (21), the final expressions for fluxes are as follows,
F  U  t  F , y  F , z  F ,    O  t 2 , x 2 , y 2 , z 2 
(24a)
2K  4 U 

2
2
2
2
F u   U 2  p 

  t  F u , y  Fu , z  F u ,    O  t , x , y , z  (24b)
K  3 x 

F v  UV  
V
 t  F v , y  F v , z  F v ,    O  t 2 , x 2 , y 2 , z 2 
x
F w  UW  
W
 t  F w, y  F w, z  F w,    O  t 2 , x 2 , y 2 , z 2 
x
(24c)
(24d)
F E    E  p  U  
  K  2 2 V 2 W 2 
T
U 

 
x  K  3
x
2

(24e)
 t  F E , y  F E , z  F E ,   O  t 2 , x 2 , y 2 , z 2 
2


where F E ,  F E ,   1  1   K  2  U  2 p  U  U  V V  W W   .
 Pr   K  3  x  x  x
 x x x x  
With (24), the governing equation (18) is updated as follows,
 F G H t 
Q F G H
2
2
2
2



 t  t  t 
  O  t , x , y , z   0
t x y z

x

y

z


(25)
where, t , x , y , z are the time step and cell size, F, G, H are the same as
those in (22), and
 F , y  F , z  F ,  
 H  , x  H  , y  H  , 
 G , z  G , x  G ,  






 Fu , y  F u , z  Fu ,  
 H u , x  H u , y  H u , 
 G u , z  G u , x  G u ,  
Ft   F v , y  F v , z  F v ,   , Gt   G v , z  G v , x  G v ,   , H t   H  v , x  H  v , y  H  v ,   .






 F w, y  F w, z  F w,  
 H  w , x  H  w, y  H  w ,  
 G w, z  G w, x  G w,  
F

H

G

  E , z  G E , x  G E , 
  E , y  F E , z  F E , 
  E , x  H  E , y  H  E , 
The detailed expressions of Gt and H t are not given here, but can be easily
obtained by rotating the coordinate and velocity components in Ft .
Clearly, Eq. (25) is the equivalent partial differential equation for the GKS scheme.
It shows that the GKS scheme is 1st-order accurate in time and 2nd-order accurate in
space. The 1st-order time error comes from the reconstruction of the interface flux.
Detailed analysis is given in the next session.
4 Connection with the Navier-Stokes Equations
The GKS scheme is based on the Maxwell distribution and its gradients, which means
that the scheme is targeted at solving the continuum equations. The Navier-Stokes
equations are the well-known equations for fluid flow at this level. Thus the
equivalent differential equation of the GKS scheme is compared with the NS
equations.
In the NS equations, the fluxes in the x direction can be written as,
U








U
V
W
U
V
W






2


 B 


U 2  p    2  




3  x y z 
 x y z 


 V U 


UV     

FNS  


x
y






 W U 

UW   



 x z 




 U V W 
 U V W 
 V U 



2
W
U
T


   E  p  U  U  2






  BU 
  V 
  W 
   
x 
3
 x z 
 x y z 
 x y z 
 x y 

(26)
where  B is the bulk viscosity. In order to compare with the GKS flux, (26) is
separated into two terms,
FNS
0
U

 

 
2
  V W 
4
 U

 

U 2  p      B 

   B  

3

  y z 


3
 x



U
V

 

UV  

y

x

 
W

U



UW  

x
z

 


V
W
T    2
4
 U
 V
 W

   E  p U      B U
     B  U  V  W   V U  W U
x    3
x
x
3
 x

y
z
  y z 



 (27)










The fluxes in the EDE of the GKS scheme can be written as,
FGKS
U




4
2

U
K




U 2  p    

(28)
 F , y  F , z  F ,  


3K  9  x
3






F
F
F
u , z
u , 
V
 u , y


UV  
2
2
2
2

  t  F v , y  F v , z  F v ,    O  t , x , y , z 
x




W


F
F
F
  w, y
 w, z
 w,  


UW  
F

x


  E , y  F E , z  F E , 

2K
V
W
T 
4
 U
 U
 V
 W

   E  p U    

3K  9  x
x
x
x 
3

Comparing (27) and (28), the connection between EDE and the NS equations can be
clearly identified.
First, under the limitation of Δt, Δx, Δy, Δz0 and quasi-1D assumption (eg.
 y  0 ,  z  0 for fluxes in the x-direction), the EDE of GKS equals to the NS
equations only that the EDE fixes the bulk viscosity at
2K
 . In the NS equations,
3K  9
the bulk viscosity is a free parameter, which is usually neglected for many
computational fluid dynamics applications. In the GKS, it is zero for monatomic gases
(K=0), which is physically true. For other gases, it accounts for the difference between
the equilibrium pressure and kinetic pressure. The BGK model predicts the value of
bulk viscosity as 4/15 of the dynamic viscosity when K=2 (or γ=7/5). However, the
accurate value for the bulk viscosity is generally unknown. In [13], it is reported that
the bulk viscosity of nitrogen at moderate temperatures is around 0.8 times of the
dynamic viscosity when the absorption of sound waves was measured at frequencies
below the relaxation frequency of rotational energy.
Second, the EDE lacks the gradients of flow variables in other directions, which is
the outcome of directional splitting used in the initial reconstruction. This is the main
reason that multi-dimensional schemes [4,8] are desired for viscous flows. The EDE
of multi-dimensional schemes can be derived in a similar procedure as in this paper.
For instance, if the reconstructed flux includes the flow gradients in other directions
using the multi-dimensional scheme [4], it can be shown in (29) that the y-derivatives
and z-derivatives will be recovered as in the NS equations.
FGKS
U




 U 2  p   4     U   2      V  W  

B
B 



3
 x  3
  y z  


 F ,  
V
U



UV  





x
y
 F u ,  


2
2
2
2

W
U
  t  F v ,    O  t , x , y , z 



UW






x
z
 F w, 


F 
U
V
W
4





  E  p U      U

  E , 
 V
 W
 
B



x
x
x



3




T  2
U
U  

  V W 
   x  3    B  U  y  z   V y  W z  

 



(29)
Third, the EDE is 2nd-order accurate in space and 1st-order accurate in time. The
first-order terms in (28) are related to the y-derivative, z-derivative, and gas viscosity.
These y-derivatives and z-derivatives, however, can be removed if multi-dimensional
splitting schemes are used as shown in (29). The viscous related terms can be traced
back to (15) where they will show up when the inviscid fluxes are converted from t=0
to time t. This is the same as the way to evaluate A in (6) using (9), which is
equivalent to the Euler equations. In other words, the viscous terms in the NS
equations have no contribution to the time evolution of the flow variables at the
interface. If these terms are to be cancelled, the reconstructed distribution function (5)
should contain a nonlinear term t . In other words, a high-order expansion beyond
the traditional Chapman-Enskog for the NS needs to be used. This has been done in
[9,10]. Therefore, the EDE is 2nd-order accurate in both space and time at the level of
Euler equations, but is 1st-order accurate in time when the flow is viscous. In practical
applications, however, the time step is usually much larger than the relaxation time  .
Thus the 1st-order time error that originated from the nonlinear term t may be
included in the 2nd-order time error. The numerical time error can still be 2nd-order for
the GKS scheme when simulating high-Reynolds viscous flows.
It should mention that Ohwada [7] found that the solution of the GKS scheme was
2nd-order accurate in space and time, which is consistent with the present result since
the error in conservative variables Q can be easily obtained by integrating (25) over
the time step. The expressions of fluxes given in [7], however, show difference in the
time error term. This is because Ohwada’s analysis is mainly for illustration purpose
where specific expressions of the error in terms of flow variables have not been
presented.
5 Conclusion
In this paper we used a simplified version of the evolution solution of the gas
distribution function at the interface, formulated the fluxes with instantaneous values
of macroscopic flow variables using Taylor expansions, and derived the equivalent
partial differential equation for the gas-kinetic BGK scheme. It is confirmed that the
GKS scheme solves the Navier-Stokes equations with a predetermined bulk viscosity.
The derived equation shows that multi-dimensional splitting schemes are desired to
eliminate flux error from the directional splitting method for viscous flow simulations.
The GKS scheme is 2nd-order accurate in both space and time at the level of Euler
equation. However, there is 1st-order error in time for viscous flows.
To our knowledge, numerical schemes using the finite volume method are seldom
studied to obtain the equivalent partial differential equation. The analysis presented in
this paper can then be used to study similar schemes. For instance, with the same GKS
scheme (multi-dimensional one), it is unclear how the order of scheme will be
improved by using a high-order initial reconstruction for the slopes in space, such as
the use of high-order WENO methods. What is the equivalent differential equation of
the high-order GKS schemes in [9,10]? What will happen if the fluxes are solved
using the Riemann solver such as the exact one? Answering these questions may be
useful to the CFD community.
Acknowledgments
This work has been supported by the National Natural Science Foundation
of China through grants No. 91116013 and No. 11372325.
Appendix: Some Auxiliary Results
A.1 Taylor expansions at the interface for some variables
b1 

1   x 2  3 

 O  x 4   ,

3
0  x 24 x

(A1)
b2 

1  U x 2  3 U

 O  x 4   ,

3
0  x
24 x

(A2)
b3 

1  V x 2  3 V

 O  x 4   ,

3
0  x
24 x

(A3)
b4 

1  W x 2  3 W

 O  x 4   ,

3
0  x
24 x

(A4)
b5 

1   E x 2  3  E

 O  x 4   ,

3
0  x
24 x

(A5)
x 2  2 
 O  x 4 
8 x 2
(A6)

1  x 2  2 
 O  x 4  
1 
2
  8 x

(A7)
0   
1
0

U0  U 
V0  V 
W0  W 
x 2   2 U
2  
4

U

  O  x 
x 2 
8  x 2
(A8)
x 2   2 V
2  
 V 2   O  x 4 

2
x 
8  x
(A9)
x 2   2 W
2  
4

W

  O  x 
x 2 
8  x 2
(A10)
x 2  2  U U V V W W
1
 RT 




20
8  K  3  x x x x x x
 2  T  2T


 2
R



  x x x

4
   O  x 

(A11)
0   
x 2   2 1  U U V V W W




8 T  K  3 R  x x x x x x
2
  2  T  T  


 O  x 4  (A12)
 
2 
   x x x  
0 2 x2    2 1  U U V V W W   2  T 2T    2 
4
 




 
  O x  (A13)
 
0  4  T  K  3 R  x x x x x x    x x x2    x2 
1 1
t U
 
 O x 2  O(t 2 )
Q1   2 x
(A14)
 U 1 p 
2
2

U t  U  t U
  O  x   O  t 


x
x



(A15)


V
 O  x 2   O  t 2 
x
W
 O  x 2   O  t 2 
Wt  W  tU
x
(A16)
Vt  V  tU
q
(A17)
 2K  4 U  U 1 p 
T
K 5
 V V W W  
2
2


R  t 
U
 U 
   O x  O t
x
2
 x x x x  
 K  3 x  x  x 
   
(A18)
A.2 Some terms contributed to fluxes
F ,   
F , y 

 UV  ,
y
(A19)
F , z 

 UW  ,
z
(A20)
2K  4   U    U    U
 

 
K  3 x  x  y  y  z  z
(A21)
U 2V
p K  5 V ,
V

p
y
y K  3 y
(A22)
U 2W
p K  5 W
,
W

p
z
z K  3 z
(A23)
Fu , y 
F u , z 
,


2   T  4K  8   U  2  2 K  4 U U V V W W 
U 







K  3 x  x  K  3 x  x  K  3  K  3 x x x x x x 
2   T 
  U  2  U U 2 K  4 V V W W 




  2U  



,
K  3 y  y 
K  3 y y y y 
y  y  K  3  y y
F u ,   
2   T 
  U  2   U U V V 2 K  4 W W 



  2U  



K  3 z z 
K  3 z  z 
z  z  K  3  z z z z
2 K  4   U 



K  3 t  x 

(A24)
  U

t  x
 2 K  4   U    U




K  3 x  x  y  y
2
T
U
U









K  3 T x x
x 

U
U
U p
V
W

 U
x
y
z x

   U  



 z  z  



 K 3
 T
 U V W  
T
T 
 R U
V
W



  p

2
y
z 
 x
 x y z  

 
 
  2 K  4  U U  V V  W W   V V   , (A25)
2
 U   K  3  x x y y z z  x x
 

  

 K  3  R T x 
W W U U W W U U V V  






  x x
y y
y y
z z z z  


   T    T    T 

  x   x   y   y   z   z 










 O t , x 2 , y 2 , z 2

F v , y 
UV 2
p ,
U
y
y
F v , z 
(A26)
UVW
,
z
(A27)
  U  2 K  4   V 

U 


 V  
y  y  K  3 y  y  ,

  U 
  V    V 
V  
 U  
 

z  z 
z  z  t  x 
F v ,   
2 K  4   U
V 
K  3 x  x
  V

 U  
x  x

   V


  V 
  x  x

 
t  x 
x
V
V
 U
x

 2 K  4   V


 K  3 y  y
V
V p
W

y
z y
(A28)
   V  



 z  z  



 K 3
 T
 U V W  
T
T 
V
W


 R U

  p

y
z 
2
 x
 x y z  

 
 
  2 K  4  U U  V V  W W   V V  
, (A29)
z z  x x
 V   K  3  x x y y
2
 



 K  3  R T x   W W U U W W U U V V  






  x x
y y
y y
z z z z  


   T    T    T 

  x   x   y   y   z   z 










 O t , x 2 , y 2 , z 2

F w, y 
F w, z 
UVW
,
y
UW 2
p
,
U
z
z
(A30)
(A31)
  U 
  W 

 U  

 W  
y  y 
y  y  ,

  U  2 K  4   W    W 
W  
U 

 

z  z  K  3 z  z  t  x 
F w,   
2K  4
  U
W 
K 3
x  x
   W


  W 
  x  x

 
t  x 
x
W
V
 U
x

  W

 U  
x  x

(A32)
   W  2 K  4   W  





 y  y  K  3 z  z  

W
W p
W


y
z z

 K 3
 T
 U V W  
T
T 
V
W


 R U

  p

y
z 
2
 x
 x y z  

 
 
  2 K  4  U U  V V  W W   V V  
, (A33)
 W   K  3  x x y y z z  x x
2
 

  

 K  3  R T x 
W W U U W W U U V V  






  x x
y y
y y
z z z z  


   T    T    T 

  x   x   y   y   z   z 










 O t , x 2 , y 2 , z 2
1 
p K  5 pUV K  5
V
,
UV U 2  V 2  W 2   UV 

pU
2 y
y
2
y
K 3
y
(A34)
1 
p K  5 pUW K  5
W
,
UW U 2  V 2  W 2   UW 

pU
2 z
2
z
z
K 3
z
(A35)
F E , y 
F E , z 
F E ,  


2K  4

U
 2K  4   U 
  V 
  W  
U 
 E  RT      U 
 V  
 W  

3
x  x 






K 3
K
x
x
x
x
x  x  





K  5   2K  4 U U V V W W    T  


U 
  

K  3   K  3 x x x x x x  x  x  
  E  RT 
 U U 2K  4 V V W W 
  U  K  5
U 





y  y  K  3
K  3 y y y y 
 y y
   U  2K  4   V 
  W  K  5   T  
 U U  
V 

 W  



y  y  K  3 y  y  
 y  y  K  3 y  y 
  U  K  5
 U U V V 2K  4 W W 
U 
  E  RT   




z  z  K  3
 z z z z K  3 z z 
   U 
  V  2K  4   W  K  5   T  
W 
 U U  




 V  

z  z  K  3 z  z  K  3 z  z  


z
z





    K  2 2 V 2 W 2 K  5

U 
RT  
 
t  x  K  3
2
2Pr

, (A36)

    K  2 2 V 2 W 2 K  5

U 
RT  
  
t  x  K  3
2
2 Pr

   T    T    T 
T
 
  
  
 U

x

x

y

y

z

z
x








T
T
2T  U V W 
 V
W





y
z K  3  x y z 



   K  2 2 V 2  W 2 K  5
2
2 K  4  U U V V W W 


U 

RT    




K  3  x x y y z z 
2
2 Pr
 K  3  R T x  K  3

  V V W W U U 


  

y y 
  x x x x
  W W U U V V 
   



z z z z 
  y y

















   T    T    T  K  3
 
 T
T
T 
V
W
 R U

 
 
  

  

y
2
z 
 x

 x  x  y  y  z  z 
 


 .



K  5   U V W 
2 K  4  U U V V W W  V V  
 
p












x   K  3  Pr   x y z 
 K  3  x x y y z z  x x   


 

  W W U U W W U U V V 
 








  x x
 
y
y
y
y
z
z
z
z









 
 



  U 2 
 U 2 
 U 2   2K  4 U p V p W p
K
2





 U 
  
 V 
 W 
 K  3   x 

 y 
 z   K  3  x  y  z


2
2
2
2
2
2
W  V  V
W  W  V
W 
 U  V












 2  x

x  2  y
y  2  z
z 




 2 K  4   2 K  4   U    U    U  




W










U 
 W  

 1  K  3   K  3 x  x  y  y  z  z  
z  z  

 

   2 K  4   V 
  V 
  V 
  W 
  W   

V
V
W
W




V














 K  3 y  y 
x  x 
z  z 
x  x 
y  y   





 O t , x 2 , y 2 , z 2
(A37)

References
1. Xu, K.: Gas-kinetic schemes for unsteady compressible flow simulations. von
Karman Institute report, (1998)
2. Xu, K.: A gas-kinetic BGK scheme for the Navier-Stokes equations, and its
connection with artificial dissipation and Godunov method. J. Comput. Phys. 171,
289-335 (2001)
3. Ohwada, T., Fukata, S.: Simple derivation of high-resolution schemes for
compressible flows by kinetic approach. J. Comput. Phys. 211, 424-447 (2006)
4. Xu, K., Mao, M., Tang, L.: A multidimensional gas-kinetic BGK scheme for
hypersonic viscous flow. J. Comput. Phys. 203, 405-421 (2005)
5. May, G., Srinivasan, B., Jameson, A.: An improved gas kinetic BGK finite volume
method for three dimensional transonic flow. J. Comput. Phys. 220, 856–878 (2007)
6. Li, J., Li, Q., Xu, K.: Comparison of the generalized Riemann solver and the
gas-kinetic scheme for inviscid compressible flow simulations, J. Comput. Phys.
230, 5080-5099 (2011)
7. Ohwada, T.: On the construction of kinetic schemes. J. Comput. Phys. 177,
156–175 (2002)
8. Li, Q., Fu, S.: On the multidimensional gas-kinetic BGK scheme. J. Comput. Phys.
220, 532-548 (2006)
9 Li, Q., Xu, K., Fu, S.: A high-order gas-kinetic Navier-Stokes solver. J. Comput.
Phys. 229, 6715-6731 (2010)
10 Luo, J., Xu, K.: A high-order multidimensional gas-kinetic scheme for
hydrodynamic equations. Science China Technological Sciences 56, 2370-2384
(2013)
11 Xu, K., Huang, J.: A unified gas-kinetic scheme for continuum and rarefied flows.
J. Comput. Phys. 229, 7747-7764 (2010)
12 Chen, S., Xu, K., Lee, C., Cai, Q.: A unified gas kinetic scheme with moving mesh
and velocity space adaptation. J. Comput. Phys. 231, 6643-6664 (2012)
13 Vincenti, W.G., Kruger, C.H.: Introduction to Physical Gas Dynamics. John Wiley
& Sons, New York (1965)