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Time-Dependent Problems 235
particularly sensitive to the smoothness of the solution. Despite much effort devoted to
this subject, many problems remain. Polynomial approximations by spectral methods
of equation (10.3.21), when A(x,U) := x or A(x,U) := −x, have been considered in
gottlieb and orszag(1977), section 7. Nonlinear examples will be examined in the
coming section.
10.4 Nonlinear time-dependent problems
An interesting family of nonlinear first order problems is represented by conservation
equations. They are written as
(10.4.1)
∂U
∂t
∂F(U)
(x,t) = (x,t), x ∈] − 1,1[, t ∈]0,T],
∂x
where F : R → R is a given function. If F is differentiable, then (10.4.1) is equivalent to
taking A(x,U) := F ′ (U) in (10.3.21). These equations are consequence of conservation
of energy, mass or momentum, in a physics phenomenon. Boundary conditions have to
be imposed at the inflow boundaries, which are determined according to the direction
of the characteristic curves. We note that now the slope of a characteristic curve at a
given point (x,t) depends on the unknown U. The case corresponding to F(U) :=
− 1
2 U2 , U ∈ R (hence A(x,U) = −U), is sufficient to illustrate the possible ways in
which the solution can develop. A theoretical discussion is too lengthy to include here.
We suggest the books of smoller(1983), chapter 15, and kreiss and lorenz(1989),
chapter 4, for a more in-depth analysis.
A large variety of finite-difference schemes has been proposed to solve the above
equations numerically. These are sometimes very sophisticated in order to treat so-
lutions with sharp derivatives or shocks. Among many celebrated papers, we refer to
sod(1985) for a general introductive overview. There is less published material available
in the field of spectral methods. Algorithms are proposed for the approximation of peri-
odic solutions by trigonometric polynomials, but some of the techniques can be adapted