An Unsplit Monte-Carlo scheme for the resolution of the linear Boltzmann equation coupled to (stiff) Bateman equations

In this paper, we are interested in the resolution of the time-dependent problem of particle transport in a media whose composition evolves with time due to interactions, the only constraint being the use of Monte-Carlo (MC) scheme for the transport phase. A common resolution strategy consists in a splitting between the MC/transport phase and the time discretization scheme/matter evolution phase. After going over and illustrating the main drawbacks of split schemes in a simplified configuration (monokinet-ic/monocomposition problem), we progressively build a new Unsplit MC (UMC) scheme improving the accuracy of the solutions, avoiding numerical instabilities, and less sensitive to time discretization. The new scheme needs two key ingredients: an Unsplit MC solver and a specific change of variable implying an on the fly resolution of a system of ODEs for each MC particle in order to describe along their flight path the time evolution of the matter