Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
4.6 Numerical simulations for the Broadwell system 119
The system can then be written as follows.
Hence, denoting
⎧
⎨
⎩
∂tρ+∂xm = 0,
∂tm+∂xz = 0,
∂tz +∂xm = − 1
ε
ρz − 1
2
ρ 2 +m 2 .
u = (u1,u2) := (ρ,m) and v := z,
we can rewrite the system under the form (4.1.1) as follows.
⎧
⎨
⎩
∂tu1 +∂xu2 = 0,
∂tu2 +∂xv = 0,
∂tv +∂xu2 = −1 ε R(u,v) ,
where
R(u,v) = u1v − 1
2
The local equilibrium is defined by
v = A(u) , where A(u) = 1
2
2
u1 +u 2
2 .
u1 + u2 2
u1
.
Hence, when ε goes to zero, we obtain the following “Euler" system:
with
(4.6.2)
(4.6.3)
∂tu+∂xF(u) = 0, (4.6.4)
F(u) = u2 A(u) T .
Here, we have to examine the generalization of the subcharateristic condition (4.3.4).
Indeed, the new stability criterion is expressed as follows. The eigenvalues of the limit
problem (4.6.4) are required to be enterlaced between the ones of the relaxation problem
(4.6.3) [29, 145, 129], that is:
where λ± = 1
2
u2
u1
±
2− u2 2
u 2 1
4.6.1 The Riemann problem
−1 < λ− < 0 < λ+ < 1,
.
We present first several simulations for the problem (4.6.3) with different relaxation
parameters ε, from the rarefied regime to the fluid regime. The initial data is given by the
local equilibrium:
⎧
⎨ (1,0,1), if −1 ≤ x ≤ 0,
(ρ0,m0,z0) =
⎩
(0.25,0,0.125), if 0 < x ≤ 1.
(4.6.5)