Untitled - Cdm.unimo.it

Examples 267
In addition, one can deduce that
(12.1.6)
= λΓ ′ (t) −
= ∂U
(Γ(t),t) −
∂x
d
λ Γ(t) −
dt
Γ(t)
−1
Γ(t)
−1
Γ(t)
This shows that the quantity λΓ(t) − Γ(t)
−1
balance condition).
−1
U(x,t) dx
∂U
∂t (x,t) dx − Γ′ (t) U(Γ(t),t)
∂2U ∂U
(x,t) dx = (−1,t) = 0, ∀t ∈]0,T].
∂x2 ∂x
U(x,t)dx is constant for t ∈]0,T] (heat
We now propose an approximation scheme. We use spectral methods for the space
variable and finite-differences for the time variable. We divide the interval [0,T] in
m ≥ 1 parts of size h := 1/m. Then, the function Γ is approximated at the points
tk := kh, 0 ≤ k ≤ m, by some values γk ∈ R, 0 ≤ k ≤ m, to be determined later.
We assume that the γk’s are increasing and γ0 = 0. At the time tk, 0 ≤ k ≤ m, the
function U is approximated in the interval [−1,γk] by a polynomial p (k)
n ∈ Pn. On
the other hand, according to (12.1.3), the approximating function is equal to zero in the
interval ]γk,1]. The polynomial p (k)
n , 0 ≤ k ≤ m, is defined by its value at the n + 1
points
(12.1.7) θ (n,k)
j
Here, η (n)
0
:= 1
(n)
2 η j + 1 (1 + γk) − 1, 0 ≤ j ≤ n, 0 ≤ k ≤ m.
= −1, η(n) n = 1, and η (n)
j , 1 ≤ j ≤ n − 1, are the zeroes of the derivative
of the Legendre polynomial of degree n (see section 3.1). In practice, the Legendre
Gauss-Lobatto nodes are mapped to the interval [−1,γk] (see (11.2.3)).
From (3.5.1) with w ≡ 1, we get the quadrature formula
(12.1.8)
γk
−1
p dx =
n
j=0
p(θ (n,k)
j ) ˜w (n,k)
j
∀p ∈ P2n−1, 0 ≤ k ≤ m,