Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
106 An Asymptotic Preserving scheme
4.3.3 BV estimates
In this section, we obtain a BV estimate 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 3.4. Assume that u0,v0 are uniformly bounded with respect to ε in BV(R).
For any a0 > 0 and N0 = max supu0L
ε>0
∞,supv0L
ε>0
∞
, we assume that the function
R ∈ C1 (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, we have:
TV(w n+1 ) + TV(z n+1 ) ≤ TV(w n ) + TV(z n ).
Proof. First we note that u 0 , v 0 ∈ BV(R), then by construction, w 0 , z 0 ∈ BV(R) also.
To prove the BV estimate, we proceed in two steps. On the one hand, using the TVD
property of the upwind scheme, we get that
TV(w n+1/2 ) ≤ TV(w n ) and TV(z n+1/2 ) ≤ TV(z n ).
On the other hand, we apply the nonlinear relaxation step (4.3.3) and from Lemma 3.1
(iii) with w n+1/2
1 = wn+1/2 (.) , z n+1/2
1 = zn+1/2 (.) and w n+1/2
2 = wn+1/2 (.+∆x), z n+1/2
2 =
zn+1/2 (.+∆x), it yields for any j ∈ Z,
|w n+1
j+1 −wn+1
j | + |z n+1
j+1 −zn+1
j | ≤ |w n+1/2
j+1 −wn+1/2
j |+|z n+1/2
j+1 −zn+1/2
j |.
Summing over j ∈ Z, we get that
TV(w n+1 ) + TV(z n+1 ) ≤ TV(w n+1/2 ) + TV(z n+1/2 )
4.4 Trend to equilibrium
≤ TV(w n ) + TV(z n ).
In this section we first focus on the asymptotic behavior of the numerical solution to
(4.2.13)-(4.2.14) when ε goes to zero or when times goes to infinity. Then, we prove that
the numerical solution to (4.2.13)-(4.2.14) converges to a consistent approximation of the
conservation laws (4.1.4) when ε goes to zero. It corresponds to the limit Pε h → P0 h , when
ε → 0.
4.4.1 Asymptotic behavior
In this subsection, we drop the subscripts ε for sake of clarity.