Geoscience Reference
In-Depth Information
4.2 FINITE DIFFERENCE METHOD
4.2.1 Finite difference method for 1-D problems
4.2.1.1 Taylor-series formulation of finite difference
schemes
Fig. 4.3 shows the 1-D computational grid used in the finite difference method. The
values of function
f
at
x
=
x
i
+
1
and
x
=
x
i
−
1
can be expanded as Taylor series about
the point
x
=
x
i
:
i
∂
i
2
f
3
f
f
1
2
∂
1
6
∂
x
2
x
3
f
i
+
1
=
f
i
+
x
+
i
+
+···
(4.9)
x
2
x
3
∂
x
∂
∂
i
∂
i
2
f
3
f
f
1
2
∂
1
6
∂
x
2
x
3
f
i
−
1
=
f
i
−
x
+
i
−
+···
(4.10)
∂
x
∂
x
2
∂
x
3
where
x
is assumed to be uniform on the entire
computational grid for convenience in the following analyses.
x
=
x
i
+
1
−
x
i
or
x
=
x
i
−
x
i
−
1
.
Figure 4.3
1-D finite difference grid.
Ignoring the high-order terms in Eq. (4.9), the first derivative of function
f
can be
approximated as
∂
i
≈
f
f
i
+
1
−
f
i
(4.11)
∂
x
x
Eq. (4.11) is called the forward difference scheme. Similarly, from Eq. (4.10), the
backward difference scheme can be obtained as
∂
i
≈
f
f
i
−
f
i
−
1
(4.12)
∂
x
x
Subtracting Eqs. (4.9) and (4.10) yields the central difference scheme for the first
derivative:
∂
f
f
i
+
1
−
f
i
−
1
i
≈
(4.13)
∂
x
2
x
Summing Eqs. (4.9) and (4.10) yields the central difference scheme widely used for
the second derivative:
i
≈
2
f
∂
f
i
−
1
−
2
f
i
+
f
i
+
1
(4.14)
x
2
x
2
∂
