Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
4.3 A priori estimates 103
Hence, from Remark 2.1 and the subcharacteristic condition (4.3.4), we obtain that for
all (u,v) ∈ I(N0,a0)
⎪⎩
∂wGε,s(w,z) ≤ 0 and ∂zGa,s(w,z) ≥ 0.
Moreover, still using condition (4.3.4), we also get for all (w,z) ∈ I(N0,a0)
⎧
⎪⎨
∂wGε,s(w,z) ≥ − 1− βs
ε e−βs/ε
− ∂vR(u,v) s
ε e−βs/ε ,
∂zGε,s(w,z) ≤
1− βs
ε e−βs/ε
+ ∂vR(u,v) s
ε e−βs/ε .
Now since |u| ≤ V(N0,a0) and from the choice of the parameter β in (4.2.6), it yields
|∂vR(u,v)| ≤ β. Therefore, we conclude that
−1 ≤ ∂wGε,s(w,z) and ∂zGε,s(w,z) ≤ 1, ∀(w,z) ∈ I(N0,a0).
Finally (iii) follows from a first order Taylor expansion of Gε,s.
This Lemma allows to obtain the following comparison principle.
Corollary 3.2. Consider for i = 1,2, two initial data (w n+1/2
fying the monotononicity condition
w n+1/2
1
≤ w n+1/2
2
and z n+1/2
1
i ,z n+1/2
≤ z n+1/2
2 .
i ) ∈ I(N0,a0) satis-
Then, the numerical solution (w n+1
i ,z n+1
i ), given by (4.3.3) corresponding to the inital data
(w n+1/2
i ,z n+1/2
i ) for i = 1,2, satisfies
w n+1
1
≤ wn+1 2 and z n+1
1 ≤ zn+1 2 .
Proof. Starting from the equality (4.3.6), it yields to the result applying the estimates
(4.3.5).
4.3.2 L ∞ estimates
In this section, we establish a uniform bound on the numerical solution to the scheme
(4.2.13)-(4.2.14), that is, equivalently the scheme (4.3.2)-(4.3.3), with the time-space step
h = (∆t,∆x) such that (4.2.8) is satisfied.
Proposition 3.3. Consider any a0 > 0 and
N0 = max supu0L
ε>0
∞,supv0L
ε>0
∞
.
We assume that the function R ∈ C 1 (R×R,R) satisfies (4.1.3)-(4.2.2) and choose a > 0,
β > 0 such that (4.2.6) is verified. Then, for all n ∈ N
u n L ∞ ≤ V(N0,a0), v n L ∞ ≤ √ aV(N0,a0).