Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
4.2 Numerical schemes and main results 97
where β0, g and h are some constants depending only on U0.
Hence the subcharacteristic condition reads, in our case:
∂uR(u,v)
∂vR(u,v)
< √ a.
For the sequel, we define for any N > 0 and α > 0,
⎧
⎪⎨
⎪⎩
U(N,α) :=
1+ 1
√ N , α
F(N,α) := sup |A(ξ)|,
|ξ|≤U(N,α)
V(N,α) := U(N,α) + 1
√ α F (N,α) .
We also denote by I(N,α) the compact set
(4.2.3)
I(N,α) := − √ aV(N,α), √ aV(N,α) 2 . (4.2.4)
Moreover, we assume that the initial conditions u ε 0 ,vε 0
L ∞ (R), such that:
are bounded independently of ε in
N0 := max supu
ε>0
ε 0L∞,supv ε>0
ε 0L∞
< ∞. (4.2.5)
Consider any a0 > 0 and assume that the function R ∈ 1 (R×R,R) satisfies (4.1.3) and
(4.2.2). We choose the characteristic speed √ a > 0 and the parameter β > 0 such that
⎧
⎪⎨
⎪⎩
where V is given by (4.2.3).
√ √a0,
a > max
g(V(N0,a0))
,
β0(V(N0,a0))
β = h(V(N0,a0)),
Remark 2.1. Note that if we differentiate with respect to u the equilibrium equation
we obtain
R(u,A(u)) = 0,
A ′ (u) = − ∂uR(u,A(u))
∂vR(u,A(u))
(4.2.6)
(4.2.7)
and thus recover the well known subcharacteristic condition in the case of semi-linear
relaxation, namely:
|A ′ (u)| < √ a.