Untitled - Cdm.unimo.it

230 Polynomial Approximation of Differential Equations
where ζ > 0, U0 : [−1,1] → R, σ : [0,T] → R, are given. We assume that U0 and
σ are continuous and σ(0) = U0(−1).
All the standard spectral schemes can be considered to approximate U. Neverthe-
less, the theoretical analysis of convergence is more difficult than the analysis of the
problem in section 10.2. We discuss some examples.
Take σ ≡ 0 . In this case, we can explicitly compute the solution, given by U(x,t) =
U0(x − ζt) if − 1 < x − ζt ≤ 1, U(x,t) = 0 if x − ζt ≤ −1. Consider the collocation
method. We must find pn(·,t) ∈ Pn, n ≥ 1, t ∈]0,T], such that
(10.3.4)
∂pn
∂t (η(n)
i ,t) = −ζ ∂pn
∂x (η(n)
i ,t), 1 ≤ i ≤ n, ∀t ∈]0,T],
(10.3.5) pn(η (n)
i ,0) = U0(η (n)
i ), 0 ≤ i ≤ n,
(10.3.6) pn(η (n)
0 ,t) = 0, ∀t ∈]0,T].
Again, these equations can be reduced to a n ×n linear system of ordinary differential
equations. For instance, using the notations of section 7.2, in the case n = 3, we have
(10.3.7)
d
dt
⎡
pn(η
⎢
⎣
(n)
1 ,t)
pn(η (n)
⎤
⎥
2 ,t) ⎥
⎦
pn(η (n)
3 ,t)
⎡
⎢
= −ζ ⎢
⎣
˜d (1)
11
˜d (1)
21
˜d (1)
31
˜d (1)
12
˜d (1)
22
˜d (1)
32
˜d (1)
13
˜d (1)
23
˜d (1)
33
⎤⎡
pn(η
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎦⎣
(n)
1 ,t)
pn(η (n)
⎤
⎥
2 ,t) ⎥
⎥,
t ∈]0,T].
⎦
pn(η (n)
3 ,t)
The eigenvalues of the matrix in (10.3.7) (see also (7.4.3)) have been studied in section
8.1. We will return to this point in section 10.6.
In the Legendre case (w ≡ 1), we can provide a simple analysis of stability. In fact,
formula (3.5.1) allows us to write for any t ∈]0,T]
(10.3.8)
d
dt pn(·,t) 2 w,n = 2
n
j=0
∂pn
∂t pn
(η (n)
j ,t) w (n)
j
=