1 Idea and Derivation of the Method
3
u i+1 − u i =
du
dX
i
+
d
2 u
dX 2
i
2
2
+ · · · ,
(1.7)
or
du
dX
i
=
u i+1 − u i
X
−
d
2 u
dX 2
i
− · · ·
O((X )
,
(1.8)
which gives an expression for the forward difference or forward Euler approximation of the first order derivative. In a similar way, Eq. (1.2) can be rearranged to
obtain the backward difference or backward Euler approximation of the first order
derivative as:
du
dX
i
=
u i − u i−1
+
d
2 u
dX 2
i
− · · ·
O((X )
.
(1.9)
Graphical representations of the finite difference approximations of first order derivatives are shown in Fig. 1.2.
Further derivatives of u(X ) of different order can be derived if u is evaluated at
nodes i + 2, i − 2 etc. through a Taylor’s series about the node i. Finite difference
formulae of the first order derivative with truncation error of order
2 , based on
a forward or backward difference approximation, can be obtained in the following
way: Let us consider in Eq. (1.2) the second order derivative, i.e.
u i−1 − u i = −
du
dX
i
+
1
2
d
2 u
dX 2
i
2
−
1
6
d
3 u
dX 3
i
3
+ · · · (1.10)
Fig. 1.2 Graphical representation of first order derivative approximations: c: centered; f: forward;
b: backward difference