Geoscience Reference
In-Depth Information
(
ξ
,
η
) coordinate system by these coordinate transformations:
2
f
∂ξ
2
f
∂η
u
ξ
∂
f
∂ξ
+ˆ
u
η
∂
∂η
=
ε
ξξ
∂
f
2
+
ε
ηη
∂
S
∗
ˆ
2
+
(4.94)
1
1
u
x
1
2
u
y
,
2
1
u
x
2
2
u
y
,
1
1
1
1
1
2
1
2
where
u
ξ
=
α
ˆ
+
α
u
η
=
α
ˆ
+
α
ε
ξξ
=
ε
(α
α
+
α
α
)
,
ε
ηη
=
c
2
1
2
1
2
2
2
2
ε
(α
α
+
α
α
)
, and
c
1
1
∂ξ
+
α
1
1
∂η
+
α
1
2
∂ξ
+
α
1
2
∂η
2
f
∂ξ∂η
+
ε
1
∂α
1
∂α
2
∂α
2
∂α
)
∂
∂
f
∂ξ
S
∗
=
1
1
2
1
1
2
2
2
1
2
1
2
S
+
2
ε
(α
α
+
α
α
α
c
c
2
1
∂ξ
+
α
2
1
∂η
+
α
2
2
∂ξ
+
α
2
2
∂η
1
∂α
1
∂α
2
∂α
2
∂α
∂
f
∂η
1
2
1
2
+
ε
α
.
c
The transformed equation (4.94) is still a convection-diffusion equation, which can
be easily discretized using the upwind difference scheme (4.17), exponential difference
scheme (4.21), or another scheme on the rectangular logical domain or element. For
example, using the exponential difference scheme (4.21) for
2
f
2
u
ξ
∂
ˆ
f
/∂ξ
−
ε
ξξ
∂
/∂ξ
2
f
2
u
η
∂
ˆ
/∂η
−
ε
ηη
∂
/∂η
and
f
in Eq. (4.94) yields Eq. (4.53) with coefficients (Wu,
1996b):
u
ξ
ˆ
u
ξ
ˆ
a
W
=
exp
(
P
ξ
/
2
)/
sinh
(
P
ξ
/
2
)
,
a
E
=
exp
(
−
P
ξ
/
2
)/
sinh
(
P
ξ
/
2
)
,
2
ξ
2
ξ
u
η
ˆ
u
η
2
ˆ
a
S
=
exp
(
P
η
/
2
)/
sinh
(
P
η
/
2
)
,
a
N
=
exp
(
−
P
η
/
2
)/
sinh
(
P
η
/
2
)
,
(4.95)
2
η
η
a
P
=
a
W
+
a
E
+
a
S
+
a
N
,
and the source term replaced by
S
∗
. Here,
P
ξ
=ˆ
u
ξ
ξ/ε
ξξ
and
P
η
=ˆ
u
η
η/ε
ηη
, with
ξ
and
η
being the grid lengths in the (
ξ
,
η
) system. For the local element shown in
Fig. 4.8,
ξ
=
η
=
1.
4.2.4 Interpolation method
4.2.4.1 Isoparametric interpolation method on fixed grids
At the nine-point isoparametric element shown in Fig. 4.8, the function
f
can be
approximated by interpolation:
9
f
=
f
k
ϕ
k
(4.96)
k
=
1
where
f
k
are the values of
f
on grid points, and
ϕ
k
are the interpolation functions given
in Eq. (4.84).
