Numerical simulations of degenerate transport problems

Florian de Vuyst (1) , Francesco Salvarani (2)

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

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.

Mots clés: Linear Boltzmann equation; convergence to equilibrium; degenerate cross sections; finite volume

Domaines: Modélisation et simulation; Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Florian de Vuyst, Francesco Salvarani. Numerical simulations of degenerate transport problems. 2013. ⟨hal-00837511⟩