Untitled - Cdm.unimo.it

Examples 269
Then, we construct the polynomial p (k+1)
n
consider the set of equations
(12.1.14)
⎧
⎪⎨
⎪⎩
p (k+1)
n
(θ (n,k+1)
i ) = q (k)
n + h d2
dx
p (k+1)
n (−1) = q (k)
n + h d2
dx
by discretizing the heat equation. Thus, we
p(k+1)
2 n
p(k+1)
2 n
(θ (n,k+1)
i ), 1 ≤ i ≤ n,
+ h[ ˜w (n,k+1)
0 ] −1 d
dx p(k+1)
n (−1).
Equation (12.1.1) has been collocated at all the points θ (n,k+1)
i , 0 ≤ i ≤ n, and for
i = 0 we imposed the boundary condition as suggested in (9.4.23). This choice will
allow us to obtain a discrete counterpart of relation (12.1.6).
Basically, we used the backward Euler method (see (10.6.5)). Hence, the scheme is
implicit and p (k+1)
n
is recovered by solving a linear system. To determine the corre-
sponding matrix, we first note that, by (12.1.12) and (12.1.13), one has
(12.1.15)
d
dx p(k+1)
n
=
(θ (n,k+1)
i ) =
2
1 + γk+1
n
j=1
˜d (1)
n
j=0
p (k+1)
n
(θ (n,k+1)
d
j )
dx ˜l (n,k+1)
j
ij p(k+1) n (θ (n,k+1)
j ), 0 ≤ i ≤ n,
(θ (n,k+1)
i )
where we use the notations of section 7.2. Therefore, the matrix takes the form
In −
4h
(1 + γk+1) 2 ˜ D 2 n −
2h
˜w (n,k+1)
0 (1 + γk+1)
⎡
⎢
⎣
˜d (1)
00
0
·
·
·
·
·
·
d ˜(1) 0n
0
· ·
· ·
0 · · · 0
where In is the (n + 1) × (n + 1) identity matrix and ˜ D 2 n is the second derivative
matrix at the collocation points (see section 7.2).
Once we obtain p (k+1)
n , we can finally update the value of the free boundary location
in (12.1.10) with the help of (12.1.15) for i = n.
⎤
⎥
⎦ ,