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

CEA, DAM, DIF, 91297 Arpajon, France. gerald.samba@cea.fr

Received: 5 March 2009
Revised: 26 October 2009

Abstract

We consider the P_n 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 P_n 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.