-5-
can be derived from Eq. (2) through the Chapman-Enskog expansion
21
, where
p
RT
ρ
= and
'
(23)
B
Puxuxux ux
α
βαββααβχχαβχχ
μδμδ
=− ∂ ∂ +∂ ∂ − ∂ ∂ − ∂ ∂（）
.
2.2. Time discretization
To solve the hyperbolic Eq. (2) numerically, we adopt the IMEX Runge-Kutta schemes
38,40
that consist
of an implicit time discretization for the relaxation term and an explicit one for the other terms. This is
because with the high performance of the IMEX Runge-Kutta scheme, the restriction of
t
τ
Δ< can be
relaxed. And
tΔ
can be set even much bigger than
τ
with a moderate
CFL
C .
To advance the distribution function
i
f
from time level n to
1n
+
in Eq. (2) by the IMEX
Runge-Kutta schemes, the following computations are conducted:
()
()
1
()
()
()
11
()
eq k
k
jj
nk
j
ii
ii jkii jk
k
kk
f
f
fftm f tm
τ
−
==
−
=−Δ ⋅∇ +Δ
∑∑
c
, (10)
and
()
()
1()
()
11
()
eq j
j
rr
nn j
ii
ii jii j
j
jj
ff
fftnf tn
τ
+
==
−
=−Δ ⋅∇ +Δ
∑∑
c
, (11)
where
()
i
f
,
()eq
i
f
,
()
τ
and r are the stage distribution function, the stage local equilibrium
distribution function, the relaxation time and the stage number, respectively. The two
rr× matrices
()
j
k
M
m=
( 0
jk
m =
for k
j
≥ ) and ()
j
k
M
m
=
( 0
jk
m
=
for k
j
> ), and the two vectors
1
( ,..., )
T
r
nn n=
and
1
( ,..., )
T
r
nn n=
characterize the IMEX Runge-Kutta schemes. Equations (10)
and (11) are, respectively, the implicit and explicit part of the schemes.
For the implicit Eq. (10), it is natural to adopt diagonally implicit Ruge-Kutta schemes for the
relaxation term.
38
This will be an arduous procedure because not only
()
j
i
f
but also
()eq j
i
f are needed
to be fixed in iteration. Fortunately, the characteristic of the collision invariants
ϕ
of the LBM brings
us a simple trick to update
()
j
i
f
without iteration. For
()
j
i
f
, we rewrite Eq. (10) as:
1
() ()
() ()
()
11
( ( )) ( ( ))
jj
nkeqk
jk
j k
ii jkii ii
k
iiki ki
m
fftm ft ff
ϕ
ϕϕ ϕ
τ
−
==
=−Δ ⋅∇+Δ −
∑∑ ∑∑ ∑∑
c
NN N N
, (12)
and from
()
()
()0
eq k
k
ii
i
ff
ϕ
−
=
∑
N
, (13)
we can further obtain:
1
()
()
1
(( ))
j
nk
j
ii jkii
iiki
fftm f
ϕ
ϕϕ
−
=
=−Δ ⋅∇
∑∑ ∑∑
c
NN N
. (14)
Equation (14) means the jth stage macroscopical variables (
()
j
ρ
,
()
j
u
α
and
()
j
T
) can be calculated
explicitly from
()
i
f
of the previous stages. With these stage variables, we can update the corresponding
()eq j
i
f
and
()
j
τ
. As a result, the
()
j
i
f
can be given explicitly by:
()
()
11
() ()
() ()
()
11
()
()
1
eq k
k
jj
nk eqj
ii
ijkii jk jji
kj
j
kk
i
jj
j
ff
t
ftm f tm mf
f
t
m
ττ
τ
−−
==
−
Δ
−Δ ⋅∇ +Δ +
=
Δ
+
∑∑
c
. (15)
In summary, Eqs. (15) and (11) can be regarded as the governing equations of the IMEX Runge-Kutta
schemes based on the LBM. And implicitness is completely eliminated, although IMEX is used. As a
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