Untitled - Cdm.unimo.it

Time-Dependent Problems 239
Since (10.5.4) is in a diagonal form, V and W are also called characteristic variables.
From (10.5.2), the two unknowns are coupled via the boundary relations
(10.5.5)
where R = L = −1.
⎧
⎨V
(−1,t) = L W(−1,t)
⎩
W(1,t) = R V (1,t)
∀t ∈]0,T],
The physical interpretation of formula (10.5.5) is that the travelling waves are reflected
at the endpoints of the vibrating string. First-order systems, such as (10.5.4), are often
encountered in fluid dynamics models (see kreiss and lorenz(1989), section 7.6).
For spectral type approximations, results are generally obtained for hyperbolic
systems written in terms of characteristic variables which are combined together at the
boundary points by (10.5.5), with R and L satisfying |RL| < 1 (note that for the wave
equation one has instead |RL| = 1). In this situation, the amplitude of the oscillations
decays for t → +∞ (energy dissipation due to friction effects).
The collocation method is obtained in the following way. We are concerned with
finding pn(·,t) ∈ Pn, qn(·,t) ∈ Pn, n ≥ 1, t ∈]0,T], such that
(10.5.6)
⎧
⎪⎨
⎪⎩
∂pn
∂t (η(n)
i ,t) = −ζ ∂pn
∂x (η(n)
i ,t) 1 ≤ i ≤ n, t ∈]0,T],
∂qn
∂t (η(n)
i ,t) = ζ ∂qn
∂x (η(n)
i ,t) 0 ≤ i ≤ n − 1, t ∈]0,T].
At the boundaries, we require that
⎧
⎨pn(−1,t)
= L qn(−1,t)
(10.5.7)
⎩
qn(1,t) = R pn(1,t)
∀t ∈]0,T].
When R and L in (10.5.7) satisfy |LR| < 1, the polynomials pn(·,t) and qn(·,t)
decay to zero when t → +∞ (see lustman(1986)). It is an easy exercise to set up
the 2n × 2n system of ordinary differential equations corresponding to (10.5.6). We
note that two unknowns can be eliminated by virtue of (10.5.7). For n = 2, we get for
example