Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
112 An Asymptotic Preserving scheme
Integrating over x ∈ Cj (4.5.3) and dividing by ∆x, it yields
⎧
⎪⎨
⎪⎩
˜w n+1
j
˜z n+1
j
= ˜wn+1/2
j
= ˜zn+1/2
j
with ⎧⎪ ⎨
⎪⎩
+ Gε,∆t
− Gε,∆t
˜w n+1/2
j
˜z n+1/2
j
˜w n+1/2
j ,˜z n+1/2
j + ∆tE n 1,j + ∆tEn 2,j ,
˜w n+1/2
j ,˜z n+1/2
j + ∆tE n 3,j + ∆tE n 4,j,
= ˜w n j −√ a ∆t
∆x
= ˜z n j +√ a ∆t
∆x
n
˜w j − ˜w n
j−1 ,
n
˜z j+1 − ˜z n j .
The consistency errors related to the transport operator E n 1,j , En 3,j
by
where ε n 1,j+1/2 and εn 3,j+1/2
by
∆tE n 1,j = εn 1,j+1/2 −εn 1,j−1/2
∆x
⎧
⎪⎨
⎪⎩
ε n 1,j+1/2
ε n 3,j+1/2
, ∆tE n 3,j = εn 3,j+1/2 −εn 3,j−1/2
,
∆x
(4.5.5)
are respectively defined
are the consistency errors of the numerical flux and are given
= −
= +
whereas the consistency errors ∆tE n 2,j and ∆tEn 4,j
√ a∆t
wδ(t
0
n ,xj+1/2 −s)ds + √ a∆t ˜w n j ,
√ a∆t
zδ(t
0
n ,xj+1/2 +s)ds − √ a∆t˜z n j+1 ,
correspond to the stiff source term and
are given by
⎧
∆tE
⎪⎨
n 2,j = + 1
∆t 1
∆x Cj 0 ε Rδ(u,v)(t n +s,x− √
a(∆t−s))ds − Gε,∆t ˜w n+1/2
j ,˜z n+1/2
j dx,
⎪⎩
∆tE n 4,j = − 1
∆t 1
∆x Cj 0 ε Rδ(u,v)(t n +s,x+ √
a(∆t−s))ds − Gε,∆t ˜w n+1/2
j ,˜z n+1/2
j dx.
We then evaluate successively each consistency error term. On the one hand, we prove
the following consistency error for smooth solutions, which is related to the transport
approximation.
Proposition 5.2. Let (w,z) be given by (4.3.1), where (u,v) is the exact solution to
(4.1.1)-(4.1.2) and such that w, z ∈ L ∞ (R + ,BV(R)). Then the consistency error related
to the transport part satisfies
j∈Z
∆x |E n 1,j| + |E n 3,j| ≤ C ∆x
δ
(TV(w(t n )) + TV(z(t n )).