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Time-Dependent Problems 233
Thus, we can finally view pn(·,t) ∈ Pn, with pn(−1,t) = 0, ∀t ∈ [0,T], as the solution
of the collocation scheme
(10.3.17)
∂pn
∂t (ξ(n)
i ,t) = −ζ ∂pn
∂x (ξ(n)
i ,t), 1 ≤ i ≤ n, ∀t ∈]0,T],
where the ξ (n)
i ’s are the zeroes of un (see section 3.1). By defining qn := pn/r, where
r(x) := (1 + x), x ∈ [−1,1], we have qn(·,t) ∈ Pn−1, ∀t ∈ [0,T]. Using (10.3.17), qn
also satisfies
(10.3.18)
∂qn
∂t (ξ(n)
i ,t) = −ζ
∂qn
∂x (ξ(n)
(n)
qn(ξ i ,t)
i ,t) +
r(ξ (n)
, 1 ≤ i ≤ n, ∀t ∈]0,T].
i )
This is equivalent to a n × n linear system of differential equations in the unknowns
qn(ξ (n)
i , ·), 1 ≤ i ≤ n. The entries of the corresponding matrix are (see (7.2.5)):
−ζ[d (1)
ij + δij/(1 + ξ (n)
i )], 1 ≤ i ≤ n, 1 ≤ j ≤ n.
We can prove stability when α > −1 and −1 < β ≤ 1. We first note that
1−x
1+x pn = −cnun + {lower degree terms}. Then, (9.3.16) yields
(10.3.19)
= −2ζ
= −2ζ
1
1−x
1+x
−1
1
1−x
1+x
−1
1
d 1−x
dt 1+x
−1
p2nw dx + c2 n(t)un2
w
1
1−x = 2 1+x
−1
∂pn
∂x pnw dx + 2c ′ n(t)
∂pn
∂x pnw dx = ζ
∂pn
∂t pnw dx + 2c ′ n(t)cn(t)un 2 w
1
−1
1
p
−1
2 n
1−x
1+x unpnw dx + 2c ′ n(t)cn(t)un 2 w
d
1−x
dx 1+xw
dx ≤ 0, ∀t ∈]0,T].
The last integral is negative by virtue of the restrictions on α and β. This shows that
a certain weighted norm of pn(·,t) is bounded by the initial data for any t ∈]0,T].
Further details and improvements are given in gottlieb and tadmor(1990).
A proof of stability in a norm which also controls the derivative ∂pn ∂x has been
outlined in gottlieb and orszag(1977), section 8, for the Chebyshev case. The same
proof can be extended to the other Jacobi cases, provided −1 < β ≤ 0 and α > −1.