Resolution of the time dependent Pn equations by a Godunov type scheme having the diffusion limit

摘要

We consider the Pn model to approximate the time dependent transport equation in one dimension of space. In a diffusive regime, the solution of this system is solution of a diffusion equation. We are looking for a numerical scheme having the diffusion limit property: in a diffusive regime, it has to give the solution of the limiting diffusion equation on a mesh at the diffusion scale. The numerical scheme proposed is an extension of the Godunov type scheme proposed by Gosse to solve the P1 model without absorption term. It requires the computation of the solution of the steady state Pn equations. This is made by one Monte-Carlo simulation performed outside the time loop. Using formal expansions with respect to a small parameter representing the inverse of the number of mean free path in each cell, the resulting scheme is proved to have the diffusion limit. In order to avoid the CFL constraint on the time step, we give an implicit version of the scheme which preserves the positivity of the zeroth moment.