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Time-Dependent Problems 231
= −2ζ
n
j=0
∂pn
∂x pn
(η (n)
j ,t) w (n)
j
1
∂pn
= −2ζ
−1 ∂x pn dx = −ζp 2 n(1,t) ≤ 0.
This shows that the norm pn(·,t)w,n, t ∈ [0,T], is bounded by Ĩw,nU0w,n. This
last term is bounded with respect to n, when U0 is sufficiently regular. A proof of
convergence is given, using similar arguments, by estimating the error pn − Un, where
Un ∈ Pn, n ≥ 1, is a suitable projection of the solution U.
Unfortunately, the same techniques are not effective when different weight functions w
are considered. As pointed out in gottlieb and orszag(1977), p.89, in the Chebyshev
case (w(x) = 1/ √ 1 − x 2 , x ∈] − 1,1[), we cannot expect convergence of pn to U in
the norm · w. In the papers of gottlieb and turkel(1985) and salomonoff
and turkel(1989), convergence estimates for Chebyshev-collocation approximations
are proven in a norm weighted by the function w(x) := (1 − x)/(1 + x), x ∈] − 1,1[,
but the theoretical analysis is harder than in the Legendre case.
A different treatment of the boundary conditions is possible. We modify (10.3.6)
as follows:
(10.3.9)
∂pn
∂t (η(n)
0 ,t) = −ζ ∂pn
∂x (η(n) 0 ,t) − γ pn(η (n)
0 ,t), ∀t ∈]0,T],
where γ > 0. Basically, we are trying to force the polynomial pn to satisfy, at the
point x = −1, both the boundary constraint and the differential equation (the same
trick used in section 9.4 for the Neumann problem). This leads to a (n + 1) × (n + 1)
differential system, where the corresponding matrix is considered in section 7.4 (see
(7.4.6) for the case n = 3). The eigenvalues of this matrix are examined in section
8.1. When γ is proportional to n 2 , a proof of convergence can be given. The Legendre
and Chebyshev cases are examined in funaro and gottlieb(1989) and funaro and
gottlieb(1988) respectively.
Let us now introduce the tau method for σ ≡ 0. This is obtained by projecting
equation (10.3.1) onto the space Pn−1, n ≥ 1. Then, we are concerned with finding
pn(·,t) ∈ Pn, t ∈ [0,T], such that
∂pn
(10.3.10) Πw,n−1 (x,t) = −ζ
∂t
∂pn
(x,t), ∀x ∈] − 1,1], ∀t ∈]0,T].
∂x