Geoscience Reference
In-Depth Information
The following difference schemes for the first and second derivatives of
f
can be
derived from Eq. (4.96) (Wu, 1996b):
9
9
∂
f
∂
f
x
=
a
k
f
k
,
y
=
b
k
f
k
(4.97)
∂
∂
k
=
1
k
=
1
9
9
2
f
2
f
∂
∂
=
c
k
f
k
,
=
d
k
f
k
(4.98)
∂
x
2
∂
y
2
k
=
1
k
=
1
where
a
k
,
b
k
,
c
k
, and
d
k
are coefficients:
1
∂ϕ
1
∂ϕ
2
∂ϕ
2
∂ϕ
k
∂ξ
+
α
k
∂η
k
∂ξ
+
α
k
∂η
1
2
1
2
a
k
=
α
,
b
k
=
α
,
1
1
∂ξ
+
α
1
1
∂η
2
2
2
1
∂α
1
∂α
1
∂
ϕ
1
∂
ϕ
1
∂
ϕ
∂ϕ
k
k
∂ξ∂η
+
α
k
k
∂ξ
1
1
1
1
1
2
2
1
2
1
2
c
k
=
α
α
+
2
α
α
α
+
α
2
2
∂ξ
∂η
α
2
1
∂ξ
+
α
2
1
∂η
1
∂α
1
∂α
∂ϕ
k
∂η
1
2
+
,
and
α
2
2
2
1
2
1
2
∂η
2
∂
ϕ
k
∂ξ
2
∂
ϕ
k
∂ξ∂η
+
α
2
∂
ϕ
k
∂η
2
∂α
2
∂α
∂ϕ
k
∂ξ
1
2
1
1
2
2
2
2
2
1
2
d
k
=
α
α
+
2
α
α
α
+
∂ξ
+
α
2
2
2
2
2
2
∂η
2
∂α
2
∂α
∂ϕ
k
∂η
1
2
+
α
∂ξ
+
α
.
Eqs. (4.97) and (4.98) can be applied to any point in the local element. For example,
for the central point 5, one can obtain
a
k
,
b
k
,
c
k
, and
d
k
by specifying
0.
The isoparametric interpolation formula (4.96) can be extended to the 3-D case using
the interpolation functions (4.88) based on the 27-point element shown in Fig. 4.9, and
similar difference schemes for the 3-D first and second derivatives can be easily derived.
Note that the difference schemes (4.97) are similar to the central difference scheme
(4.13); thus, they are not as adequate for strong convection problems as the exponen-
tial difference scheme (4.95) and the upwind interpolation method introduced in the
next subsection.
ξ
=
η
=
4.2.4.2 Upwind interpolation method on fixed grids
Wang and Hu (1993) analytically solved the following convection-diffusion equation
with constant velocity, diffusivity, and source term in the 1-D local element shown
in Fig. 4.12:
d
2
f
d
u
df
d
S
∗
ˆ
ξ
=
ε
ξ
2
+
(4.99)
ξ
and derived the upwind interpolation functions:
