Recherche - Valorisation
Numerical simulations of degenerate transport problems
Pré-print du CMLA 2013-09 - version du 22 juin 2013
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Auteurs : Florian De Vuyst, Francesco Salvarani
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.
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.
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