2
1 Idea and Derivation of the Method
Fig. 1.1 Finite difference model of a one-dimensional problem
The infinite series of Eqs. (1.1) and (1.2) are truncated for practical use after a certain
number of terms. As a result of this approximation, the so-called truncation errors
occurs. Summing up the expression of Eqs. (1.1) and (1.2) gives
u i+1 + u i−1 = 2u i +
d
2 u
dX 2
i
X
2
+
1
12
d
4 u
dX 4
i
4
+ · · · ,
(1.3)
or rearranged for the second order derivative:
d
2 u
dX 2
i
=
u i+1 − 2u i + u i−1
2
−
1
12
d
4 u
dX 4
i
2
− · · ·
O((X 2 )
.
(1.4)
The symbol ‘O’ in Eq. (1.4) reads ‘order of’ and states that if the second order
derivative of u(X ) is approximated by the first expression on the right-hand side
of Eq. (1.4), then the truncation error is of order of
2 . This approximation is a
second order accurate approximation because of the truncated terms and is called the
centered difference scheme. In a similar way, other derivatives can be derived from
Eqs. (1.1) and (1.2). Subtracting of Eq. (1.2) from (1.1) gives
u i+1 − u i−1 = 2
du
dX
i
+
1
3
d
3 u
dX 3
i
3
+ · · ·
(1.5)
or rearranged for the first order derivative
du
dX
i
=
u i+1 − u i−1
2
−
1
6
d
3 u
dX 3
i
2
− · · ·
O((X 2 )
.
(1.6)
This last approximation of the first order derivative is called centered difference or
centered Euler and the truncation error is of order
2 . Rearranging Eq. (1.1), i.e.