PENELOPE 2003 - OECD Nuclear Energy Agency

B.1. Cubic spline interpolation 219
and, similarly,
p ′ h i+1 h i+1
i+1(x i+1 ) = −σ i+1 − σ i+2
3 6
Replacing (B.15a) and (B.15c) in (B.5), we obtain
+ δ i+1 . (B.15c)
h i σ i + 2(h i + h i+1 )σ i+1 + h i+1 σ i+2 = 6 (δ i+1 − δ i ) (i = 1, . . . , N − 2). (B.16)
The system (B.16) is linear in the N unknowns σ i (i = 1, . . . , N). However, since
it contains only N − 2 equations, it is underdetermined. This means that we need
either to add two additional (independent) equations or to fix arbitrarily two of the N
unknowns. The usual practice is to adopt endpoint strategies that introduce constraints
on the behaviour of ϕ(x) near x 1 and x N . An endpoint strategy fixes the values of σ 1
and σ N , yielding an (N − 2) × (N − 2) system in the variables σ i (i = 2, . . . , N − 1).
The resulting system is, in matrix form,
⎛
⎞ ⎛ ⎞ ⎛ ⎞
H 2 h 2 0 · · · 0 0 0 σ 2 D 2
h 2 H 3 h 3 · · · 0 0 0
σ 3
D 3
0 h 3 H 4 · · · 0 0 0
σ 4
D 4
. . .
.. . .
. .
.
=
.
, (B.17)
0 0 0 · · · H N−3 h N−3 0
σ N−3
D N−3
⎜
⎝ 0 0 0 · · · h N−3 H N−2 h ⎟ ⎜
N−2 ⎠ ⎝ σ ⎟ ⎜
N−2 ⎠ ⎝ D ⎟ N−2 ⎠
0 0 0 · · · 0 h N−2 H N−1 σ N−1 D N−1
where
and
H i = 2(h i−1 + h i ) (i = 2, . . . , N − 1) (B.18)
D 2 = 6(δ 2 − δ 1 ) − h 1 σ 1
D i = 6(δ i − δ i−1 ) (i = 3, . . . , N − 2) (B.19)
D N−1 = 6(δ N−1 − δ N−2 ) − h N−1 σ N .
(σ 1 and σ N are removed from the first and last equations, respectively). The matrix
of coefficients is symmetric, tridiagonal and diagonally dominant (the larger coefficients
are in the diagonal), so that the system (B.17) can be easily (and accurately) solved
by Gauss elimination. The spline coefficients a i , b i , c i , d i (i = 1, . . . , N − 1) —see eq.
(B.3)— can then be obtained by expanding the expressions (B.12):
a i = 1
6h i
[
σi x 3 i+1 − σ i+1x 3 i + 6 (f ix i+1 − f i+1 x i ) ] + h i
6 (σ i+1x i − σ i x i+1 ),
b i = 1
2h i
[
σi+1 x 2 i − σ i x 2 i+1 + 2 (f i+1 − f i ) ] + h i
6 (σ i − σ i+1 ),
c i = 1
2h i
(σ i x i+1 − σ i+1 x i ),
d i = 1
6h i
(σ i+1 − σ i ).
(B.20)