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1 CMLA - Centre de Mathématiques et de Leurs Applications 2 Dipartimento di matematica F.

Abstract : 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.

keyword : Linear Boltzmann equation convergence to equilibrium degenerate cross sections finite volume

Author: Florian De Vuyst - Francesco Salvarani -



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