Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
4.6 Numerical simulations for the Broadwell system 121
We first plot the L 1 , L 2 and L ∞ errors at time t = 1 (Figures 4.3, 4.4 and 4.5), for
different relaxation regimes. We observe that the order of accuracy of our method is not
deteriorated, from big values to very small values of the relaxation parameter ε.
Next our goal is to investigate numerically the long-time behavior of the solution to
the Broadwell system. If f = (f+,f0,f−) T is a reasonable smooth solution to (4.6.1),
then it is expected to converge to the (unique) global equilibrium Mg as t goes to +∞.
(Desvillettes-Villani, Filbet-Jin, Filbet-Mouhot-Pareschi, Guo-Strain for the Boltzmann
equation). In order to observe this damping phenomenon for the simpler Broadwell model,
as well as the time oscillations conjectured by Desvillettes and Villani, we investigate the
behavior of the quantities
where, in our case, the global equilibrium is:
E1 = ρ(t)−ρg L 1, E2 = m(t)−mg L 1,
ρg = 1, mg = 0.
Therefore the initial local equilibrium data (4.6.6) is a perturbation of the global equilibrium.
In Figure 4.6, one can then observe oscillations of E1, E2 for the relaxation model and
the Euler system. The frequency of oscillations does not depend on ε, contrary to the
slope of the envelop curve is smaller when ε decreases, that is when we get closer to the
hydrodynamic regime. In other words, it appears that the equilibration is much more
rapid in the rarefied regime (ε large), and the convergence seems exponential. While there
is no equilibration in the hydrodynamic regime, where the quantities E1, E2 are simply
transported.