Numerical simulations of degenerate transport problems

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.