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FIGURE 8.1
Notation for finite difference expressions.
Discretization of the model begins by forming discrete approximations to the derivatives.
Suppose derivatives of the function
u
are to be established. Begin with the mesh of
uniformly spaced points shown in Fig. 8.1 and form finite difference approximations to the
derivatives.
The
central difference
quotient is defined as
(
x
)
u
1
−
u
−
1
u
0
≈
(8.1)
2
h
where the tangent of
u
0
is replaced by the chord between the values of
u
at
1 and 1 (see
Fig. 8.1). The quantity
h
is the length of the interval between two successive grid points,
i.e.,
h
−
=
x
k
+
1
−
x
k
.
Similarly, the difference expressions of higher order can be formed, e.g.,
u
h
2
u
−
1
h
h
2
1
h
2
(
u
0
≈
−
=
u
−
1
−
2
u
0
+
u
1
)
(8.2)
where
u
−
2
≈
,u
2
≈
h
u
0
−
u
−
1
u
1
−
u
0
h
h
1
2
h
[
u
(
1
2
h
3
(
−
u
0
u
(
−
≈
h
)
−
h
)
]
=
u
+
2
u
−
2
u
1
+
u
2
)
−
2
−
1
1
h
2
[
u
(
1
h
4
(
u
i
0
≈
2
u
(
u
(
−
)
−
)
+
)
=
−
+
−
+
)
h
0
h
]
u
−
2
4
u
−
1
6
u
0
4
u
1
u
2
Using such difference expressions for particular derivatives, an ordinary differential equa-
tion is replaced by a system of algebraic equations. The solution of the system of equations
yields values of
u
at particular locations, i.e.,
u
k
=
u
(
x
k
)
.
8.1.1 Derivation of Finite Difference Formulas
The finite difference expressions of Eqs. (8.1) and (8.2) were obtained by geometric consid-
erations. These same expressions can be derived in several other ways. Consider again a
typical point 0 and its immediate neighbors
1 (Fig. 8.2). To
derive second order finite difference formulas, fit a second order polynomial (a parabola)
to the ordinates of the three points. That is, use
−
1 and
+
1or
k, k
−
1
,
and
k
+
a
2
x
2
u
=
a
0
+
a
1
x
+
In order to fit this polynomial through the three points, the constants
a
0
,a
1
,
and
a
2
are
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