Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
120 An Asymptotic Preserving scheme
We integrate the Broadwell system over the space domain[−1,1], with reflecting boundary
condition and a Courant number λ = 0.9 and output the solution at different times.
We used only 100 grid points, so that the time step is fixed equal to 0.002, and compare
the results with a fully explicit solver (for which the time step has to be of the order of ε)
for different values of the relaxation parameter.
In Figure 4.1, we take ε = 0.5 and 0.1. For such values of ε, the problem is not stiff
and this test is performed to compare the accuracy of our AP scheme with a fully explicit
scheme (global Lax-Friedrichs method with slope limiters and explicit Euler discretization
in time). The density (u1 = ρ), the momentum (u2 = m), v = z, and the deviation to
the equilibrium v − A(u) are plotted at different times t = 0.05, 0.2, 0.35 and 0.5. At
the kinetic regime, we can observe that our method gives the same accuracy as a standard
fully explicit scheme.
Next we investigate the cases of small values of ε. The same time step for the AP
scheme is used, whereas the fully explicit scheme requires it to be of order O(ε). We report
the numerical results for ε = 10 −2 and ε = 10 −3 in Figure 4.6.1. In this case, we add in the
comparison the numerical solution to the Euler system (4.6.4), obtained with a standard
first order finite volume scheme.
We observe that the AP scheme and the fully explicit scheme still agree, aven if the
time step is at least ten times larger with our method. Moreover, now that we are closer
to the fluid regime, we see that the macroscopic quantities are in good agreement with
the ones obtained with the limit problem. Yet some differences between the AP and the
explicit schemes can be observed for ε = 10 −3 , this comes from the fact that we used a
very small number of points for both discretizations.
Finally, an approximation of the L 1 error is plotted in logscale in Figure 4.2, that is,
we compute
E = (u ε h −uε 2h ,vε h −vε 2h ),
where h is the discretization parameter and · is the discrete L 1 norm.
We observe that for discontinuous initial data, we get an order of convergence which is
not better than √ h, as in the estimate (4.5.11).
4.6.2 Approximation of smooth solutions
For this test, we considered the smooth initial data [158]:
ρ0(x) = 1+0.2 sin(πx),
m0(x) = 0,
ρ0 +m2 0 /ρ0
,
z0(x) = 1
2
(4.6.6)
which is again in the local equilibrium. We used periodic boundary condition and λ = 0.9.