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236 Polynomial Approximation of Differential Equations
to algebraic polynomials. If the solution displays a smooth behavior, a straightforward
application of the usual methods (tau and collocation) generally provides good results.
For the treatment of other situations, a lot of extra care is needed. Many research groups
are working with great effort on this subject, following different ideas and devising new
schemes. A detailed analysis of these techniques is an ambitious project that we prefer
to avoid in this book. We will only outline the main patterns. For a closer approach, an
update review and numerous references are given in canuto, hussaini, quarteroni
and zang(1988), chapters 8 and 12.
One strategy commonly used in computations is filtering. The procedure consists in
appropriately damping the higher Fourier coefficients of the approximating polynomial,
to control oscillations near the points of discontinuity of the solution. A drawback to
using filters is that such a smoothing affects the convergence, and reduces the accuracy
of the scheme.
Another approach is to apply domain decomposition methods (see chapter eleven),
either to improve the resolution in the critical zones, or to allow discontinuities of the
approximating functions in accordance with those of the solution. These methods have
been employed for instance in kopriva(1986), macaraeg and streett(1986). The
location of the singular points can be estimated by a procedure called shock-fitting.
Numerical codes based on cell-averaging offer another promising field of research.
Results for Chebyshev approximations are provided in cai, gottlieb and harten
(1990).
As far as the theory is concerned, very little is known about these techniques,
especially in the case of algebraic polynomials.
Let us examine some other equations. An interesting variation of (10.4.1) is
(10.4.2)
∂U
(x,t) =
∂t
where ǫ > 0 is a given parameter.
ǫ ∂2
U ∂F(U)
+ (x,t), x ∈] − 1,1[, t ∈]0,T],
∂x2 ∂x