Resolution of the time dependent Pn equations by a Godunov type scheme having the diffusion limit. Part 1 : The case of the time dependent P1 equations

Patricia Cargo (1) , Gerald Samba (1)

1 DAM/DIF - DAM Île-de-France

We consider the P1 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 is an extension of the Godunov type scheme proposed by L.Gosse to solve the P1 model without absorption term. It requires the computation of the solution of the steady P1 equations. 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 positiveness of the zeroth moment.

Mots clés: Time-dependent transport; Pn approximation; Diffusion limit; Finite Volume Method; Riemann Solvers

Domaines: Analyse numérique [math.NA]

Patricia Cargo, Gerald Samba. Resolution of the time dependent Pn equations by a Godunov type scheme having the diffusion limit. Part 1 : The case of the time dependent P1 equations. 2008. ⟨hal-00250203⟩