Untitled - Cdm.unimo.it

248 Polynomial Approximation of Differential Equations
0 ≤ j ≤ m, according to the recursion formula
(10.6.6) p (j)
n (η (n)
i ) := p (j−1)
n
(η (n)
2 d
i ) + ǫh p(j)
dx2 n (η (n)
i ) − h
p (j−1)
n
d
dx p(j−1) n
(η (n)
i ),
1 ≤ i ≤ n − 1, 0 ≤ j ≤ m.
The initial polynomial is p (0)
n (η (n)
i ) := U0(η (n)
i ), 1 ≤ i ≤ n − 1. The derivatives at
the nodes are evaluated with the usual arguments (see section 7.2). Then, the final
polynomial p (m)
n
represents an approximation of the exact solution U(·,T). To avoid
severe time-step restrictions the scheme is implicit. Since a full implicit scheme forces us
to solve a set of nonlinear equations at any step, the nonlinear term is treated explicitly.
A theoretical analysis of convergence of the method for the Chebyshev case is carried
out in bressan and quarteroni(1986a).
The literature on this subject offers an extensive collection of (more or less effi-
cient) algorithms. An overview of the time-discretization techniques most used in spec-
tral methods can be found in gottlieb and orszag(1977), chapter 9, turkel(1980),
canuto, hussaini, quarteroni and zang(1988), chapter 4, boyd(1989), chapter 8.
A deeper analysis is too technical at this point. Nevertheless, this short introduction
should be sufficient for running the first experiments and interpreting the results.