Geology Reference
In-Depth Information
h
3
fa
f
(
a
+
h
) -
f
(
a
-
h
)= 2
hf
(
a
) +
+ -----------
3!
fa h
(
h
+
O
(
h
2
)
)
fa
( )
So
f
(
a
)=
(23)
2
This is the
central difference approximation.
Now we have four equations with us:
fa h
(
h
+
O
(
h
) forward difference approximation
)
fa
( )
f
(
a
) =
fa
()
h
+
O
(
h
) backward difference approximation
fa h
(
)
f
(
a
) =
fa h
(
+
O
(
h
2
) central difference approximation
)
fa h
(
)
f
(
a
) =
2
h
fa h
(
+
O
(
h
2
)
)
fa h
(
)
2
fa
( )
f
(
a
) =
h
2
So finite difference formulas for the first- and second-order derivatives at
(
i
,
j
) is,
f
f
i1,j
i,j
f
(
a
)=
+
O
(
h
)
(24)
h
f
f
i, j
i 1, j
f
(
a
)=
+
O
(
h
)
(25)
h
f
f
i 1,j
i 1,j
+
O
(
h
2
)
f
(
a
)=
(26)
2
h
f
+
O
(
h
2
)
2
f
f
i1,j
i,j
i1,j
f
(
a
)=
(27)
h
2
Method of Solution
The first step in obtaining a finite difference solution to eq.(17) is to superpose
a grid on the region of interest in the
x
-
t
plain. Each intersection is called
a 'node' or a mesh point.
An Explicit Approximation:
Let us take a nodal point (
u
i
n
) anywhere on
the grid, where '
n
' is the time interval and '
i
' is the space interval. Now
using forward difference approximation method we have
