We consider in this article the monokinetic linear Boltzmann equation in two space dimensions with degenerate cross section and produce, by means of a finite-volume method, numerical simulations of the large-time asymptotics of the solution. The numerical computations are performed in the $2Dx-1Dv$ phase space on Cartesian grids of size $256^3$ and deal with both cross sections satisfying the geometrical condition and cross sections that do not satisfy it. The numerical simulations confirm the theoretical results on the long-time behaviour of degenerate kinetic equations for cross sections satisfying the geometrical condition. Moreover, they suggest that, for general non-trivial degenerate cross sections whose support contains a ball, the theoretical upper bound of order $t^{-1/2}$ for the time decay rate (in $L^2$-sense) can actually be reached.
Publié le : 2013-06-21
Classification:
Linear Boltzmann equation,
convergence to equilibrium,
degenerate cross sections,
finite volume,
[INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation,
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP],
[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph]
@article{hal-00837511,
author = {De Vuyst, Florian and Salvarani, Francesco},
title = {Numerical simulations of degenerate transport problems},
journal = {HAL},
volume = {2013},
number = {0},
year = {2013},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00837511}
}
De Vuyst, Florian; Salvarani, Francesco. Numerical simulations of degenerate transport problems. HAL, Tome 2013 (2013) no. 0, . http://gdmltest.u-ga.fr/item/hal-00837511/