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September  2014, 7(3): 463-476. doi: 10.3934/krm.2014.7.463

Numerical simulations of degenerate transport problems

 1 Centre de Mathématiques et de leurs Applications, CMLA UMR 8536, École Normale Supérieure de Cachan, 61 avenue du Président Wilson, 94235 Cachan, France 2 Dipartimento di Matematica, Università degli Studi di Pavia, Via Ferrata 1 - 27100 Pavia

Received  September 2013 Revised  February 2014 Published  July 2014

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 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.
Citation: Florian De Vuyst, Francesco Salvarani. Numerical simulations of degenerate transport problems. Kinetic & Related Models, 2014, 7 (3) : 463-476. doi: 10.3934/krm.2014.7.463
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References:
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