On the linear Boltzmann transport equation: a Monte Carlo algorithm for stationary solutions and residence times in presence of obstacles

We consider the linear Boltzmann transport equation (LBTE) in a 2D strip. We present a Monte Carlo algorithm to directly simulate the motion of single particles following the dynamics prescribed by this equation. We construct the stationary solution of the LBTE in presence of large reflective obstacles in the strip. We compare the simulation steady state with that one obtained as stationary profile of the diffusion equation with mixed boundary conditions on the strip. We present the results of some numerical tests and we show the closeness of the two stationary solutions in the diffusive limit and the efficacy of our algorithm. We introduce the concept of residence time of particles, that is the typical time spent by particles to cross the strip. We show by a numerical simulation that in presence of obstacles the residence time is not monotonic with respect to the obstacles sizes.