Geoscience Reference
In-Depth Information
The forward and backward difference schemes (4.11) and (4.12) are first-order
accurate, whereas the central difference schemes (4.13) and (4.14) are second-order
accurate. They are bases for many widely used difference schemes.
4.2.1.2 Discretization of 1-D steady problems
Numerical schemes often used in the discretization of 1-D steady problems include the
central, upwind, and exponential difference schemes, which are introduced below.
Central and upwind difference schemes
Consider the 1-D steady convection-diffusion equation:
d
2
f
dx
2
+
u
df
dx
=
ε
S
(4.15)
c
where
u
is the velocity; and
c
is the diffusion coefficient, which is positive.
Applying the central difference schemes (4.13) and (4.14) to the convection and
diffusion terms in Eq. (4.15), respectively, yields
ε
u
f
i
+
1
−
f
i
−
1
f
i
−
1
−
2
f
i
+
f
i
+
1
=
ε
+
S
i
(4.16)
c
x
2
2
x
The central difference scheme (4.14) is adequate for discretizing the diffusion term.
However, the use of the central difference scheme (4.13) for the convection term may
result in numerical oscillations. Upwind difference schemes are usually preferred for
the convection term. The first-order upwind scheme uses the backward or forward
difference scheme for the convection term, depending on whether the velocity
u
is
positive or negative, i.e.,
⎨
u
f
i
−
f
i
−
1
u
∂
i
=
x
(
u
≥
0
)
f
(4.17)
u
f
i
+
1
−
f
i
∂
x
⎩
(
u
<
0
)
x
Applying the upwind scheme (4.17) to the convection term and the central difference
scheme (4.14) to the diffusion term in Eq. (4.15) yields
⎧
⎨
u
f
i
−
f
i
−
1
f
i
−
1
−
2
f
i
+
f
i
+
1
=
ε
+
S
i
(
u
≥
0
)
c
x
2
x
(4.18)
⎩
u
f
i
+
1
−
f
i
−
1
−
2
f
i
+
f
i
f
i
+
1
=
ε
+
S
i
(
u
<
0
)
c
x
x
2
Exponential difference scheme
Assuming constant
u
,
ε
c
, and
S
in the segment
x
i
−
1
≤
x
≤
x
i
+
1
and imposing boundary
conditions
f
=
f
i
−
1
at
x
=
x
i
−
1
and
f
=
f
i
+
1
at
x
=
x
i
+
1
, one obtains the following
