1 Idea and Derivation of the Method
7
Fig. 1.4 Interpolation
functions for cell collocation
to derive a finite difference
scheme, adapted from [3]
∞
−∞
δ(X − X k ) dX =
X k +ε
X k −ε
δ(X − X k ) dX = 1 ,
(1.20)
∞
−∞
f (X )δ(X − X k ) dX =
X k +ε
X k −ε
f (X )δ(X − X k ) dX = f (X k ) ,
(1.21)
the last equation gives the statement
1
X 2 (1u i−1 − 2u i + 1u i+1 ) = 0 ,
(1.22)
which is equivalent to the centered difference scheme as given in Table 1.1.
Instead of using the weighted residual method in the form of cell collocation, the
method of collocation by subregions can be alternatively used to derive the finite
difference method, cf. [3]. For this approach, the weight function is chosen as a
step-type function as shown in Fig. 1.5.
Let us consider again a second-order derivative
d
2 u
dX 2 for which the weighted residual statement can be written as:
7
Fig. 1.4 Interpolation
functions for cell collocation
to derive a finite difference
scheme, adapted from [3]
∞
−∞
δ(X − X k ) dX =
X k +ε
X k −ε
δ(X − X k ) dX = 1 ,
(1.20)
∞
−∞
f (X )δ(X − X k ) dX =
X k +ε
X k −ε
f (X )δ(X − X k ) dX = f (X k ) ,
(1.21)
the last equation gives the statement
1
X 2 (1u i−1 − 2u i + 1u i+1 ) = 0 ,
(1.22)
which is equivalent to the centered difference scheme as given in Table 1.1.
Instead of using the weighted residual method in the form of cell collocation, the
method of collocation by subregions can be alternatively used to derive the finite
difference method, cf. [3]. For this approach, the weight function is chosen as a
step-type function as shown in Fig. 1.5.
Let us consider again a second-order derivative
d
2 u
dX 2 for which the weighted residual statement can be written as: