Geoscience Reference
In-Depth Information
Figure 4.4
Finite difference grid in (
x
,
t
) plane.
scheme:
f
n
+
1
i
f
i
−
∂
f
=
(4.23)
∂
t
t
t
n
+
1
t
n
is the time step length, and the superscript
n
is the time step
where
t
=
−
index.
Scheme (4.23) can be the forward or backward difference scheme in time. Applying
the forward difference scheme for the time-derivative term in Eq. (4.22) leads to an
explicit scheme:
f
n
+
1
i
u
f
i
f
i
−
1
f
i
−
−
S
i
+
=
(
u
>
0
)
(4.24)
t
x
whereas applying the backward difference scheme leads to an implicit scheme:
f
n
+
1
i
u
f
n
+
1
f
n
+
1
i
f
i
−
−
i
−
1
S
n
+
1
i
+
=
(
u
>
0
)
(4.25)
t
x
Note that the convection term in Eqs. (4.24) and (4.25) is discretized using the
upwind scheme (4.17) rather than the central difference scheme for better stability, as
described in Section 4.2.1.2.
The forward difference scheme, with all other terms evaluated at time level
t
n
,is
also known as the Euler method.
