Untitled - Cdm.unimo.it

Time-Dependent Problems 237
For F(U) := − 1
2 U2 , U ∈ R, (10.4.2) is known as Burgers equation. Due to the presence
of the second-order derivative (which, in analogy with fluid dynamics, corresponds
to viscosity in the physical model problem), solutions of (10.4.2) are smoother than
solutions corresponding to (10.4.1). This definitely helps the theoretical analysis. In
addition, one expects that solutions of (10.4.2) for a small ǫ are in some way close to
solution of (10.4.1). We refer to smoller, p.257, for a theoretical explanation of this
fact. For boundary conditions of the form U(±1,t) = 0, t ∈]0,T], approximations
of the Burgers equation by Galerkin and collocation methods are respectively consid-
ered in maday and quarteroni(1981), and maday and quarteroni(1982), for the
Chebyshev and Legendre cases. The analysis is carried out for the steady equation,
which means that U does not depend on t (see (9.8.5)). Other results are provided in
bressan and quarteroni(1986a).
Finally, results are also available for the Korteveg-De Vries equation (see pavoni
(1988), bressan and pavoni(1990)):
(10.4.3)
∂U
(x,t) =
∂t
where ǫ = 0 is a given constant.
ǫ ∂3U ∂U
− U
∂x3 ∂x
10.5 Approximation of the wave equation
(x,t), x ∈] − 1,1[, t ∈]0,T],
Another classical time-dependent problem is given by the wave equation. Here, the
unknown is the function U : [−1,1] × [0,T] → R satisfying
(10.5.1)
∂2U ∂t2 (x,t) = ζ2 ∂2U (x,t), x ∈] − 1,1[, t ∈]0,T],
∂x2