Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
104 An Asymptotic Preserving scheme
Proof. We will proceed in two steps :
− find a particular solution (w n ,z n ) ∈ R 2 to the scheme (4.3.2)-(4.3.3) which is uniformly
bounded,
− apply the comparison principle on the compact set I(N0,a0) to prove an L ∞ bound
on (u n ,v n ).
For R0 = (1 + √ a)N0, we consider the numerical solution (w n ,z n ) to (4.3.2)-(4.3.3)
with the particular initial data (w 0 ,z 0 ) = (R0,R0), which does not depend on the space
variable so that the transport step (4.3.2) is invariant. Then we apply the relaxation
scheme (4.3.3), which yields
where (u n ,v n ) are only given by
⎧
⎪⎨
⎪⎩
u n = u 0 = − R0
√a =
v n =
w n = −v n − √ au n , z n = +v n − √ au n
1+ 1
√ a
N0,
1+ β∆t
e
ε
−β∆t/ε v n−1 +
− ∆t
ε e−β∆t/ε R u 0 ,v n−1 .
Then, we proceed by induction to show that:
1− 1+ β∆t
e
ε
−β∆t/ε
A u 0
∀n ∈ {0,...,N}, (w n ,z n ) ∈ I(N0,a0).
We assume that (wn−1 ,zn−1 ) ∈ I(N0,a0), for some n ≥ 1. Let us prove that (wn ,zn ) ∈
I(N0,a0). On the one hand since un = u0 , it yields
u n L∞ = u0L∞ ≤
1 + 1
√
a
N0 ≤ U(N0,a0).
On the other hand, using a first order Taylor expansion of the source term R(u0 ,.), we get
that there exists ˜v n−1 ∈ R such that
v n
= 1+ β −∂vR(u0 ,˜v k
)
∆t e
ε
−β∆t/ε v n−1
+ 1− 1+ β −∂vR(u0 ,˜v k
)
∆t e
ε
−β∆t/ε
A u 0 .
Therefore, denoting by λk ∈ R, the real number such that
λk := 1+ β −∂vR(u0 ,˜v k
)
∆t e
ε
−β∆t/ε , ∀k ∈ N,