Analyse asymptotique de schémas de résolution de l'équation du transport en régime diffusif

The Symbolic Implicit Monte Carlo method (SIMC) gives an approximation of the transport equation. In the original method, this function was supposed constant on each cell of the mesh. We have demonstrated that by taking piecewise linear function, this method becomes asymptotically preserving unlike the constant SIMC method. It means that in a diffusive medium where the collisions are predominant, it gives a correct solution even if the mesh size is large in regard of the mean free path but small enough to solve the diffusion scale. Boundary layers arise in diffusive medium when the incident intensity is anisotropic. We demonstrate and verify numerically that the results of the linear SIMC method can be quite good even if the boundary layers are not meshed at the mean free path scale. We study also linear discontinuous finite elements schemes and demonstrate that these schemes verify the saure asymptotic limit and possess the saure boundary conditions in diffusive medium as the SIMC method.La méthode Symbolique Implicite Monte Carlo permet d'obtenir une approximation de la solution de l'équation du transport. Dans la méthode originelle, les fonctions d'approximation étaient choisies constantes par morceaux. On démontre qu'en prenant des fonctions linéaires par morceaux, cette méthode possède alors la limite diffusion, c'est à dire qu'en milieu diffusif, elle approche la solution de l'équation de diffusion même lorsque la taille des mailles est grande vis à vis du libre parcours, à condition que celle ci reste suffisante pour résoudre l'échelle de la diffusion. On montre que les conditions aux limites en milieu diffusif approchent, sous certaines hypothèses, les conditions aux limites exactes, ce qui autorise un traitement précis des couches limites sans devoir les mailler finement. On présente des tests numériques étayant cette analyse. On étudie également des schémas aux éléments finis linéaires discontinus et on explique pourquoi ces schémas possèdent la même limite diffusion ainsi que les mêmes conditions aux limites en milieu diffusif que la méthode Symbolique Implicite Monte Carlo

Resolution of the time dependent

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

An asymptotic preserving scheme for

A new scheme for discretizing the P1 model on unstructured polygonal meshes is proposed. This scheme is designed such that its limit in the diffusion regime is the MPFA-O scheme which is proved to be a consistent variant of the Breil-Maire diffusion scheme. Numerical tests compare this scheme with a derived GLACE scheme for the P1 system

Moment models for an axisymmetric inertial connement experiment and one dimensional numerical study

In Inertial Connement Fusion (ICF) experiments, radiation is well described by a kinetic model (radiative transfer equation). This model is usually too expensive to be used in numerical simulations of such phenomena. Hence, approximations are used. A common one is to use a moment model, in which the radiative transfer equation is replaced by its rst and second order (in velocity) moments, together with a closure assumption. In this article, we propose a closure for 2D and 3D geometries, which are extensions of a one-dimensional radially symmetric model called P'1. This model has proved to be very accurate in the study of ICF, which makes the models we propose promising in this respect. The closure is based on the fact that the model should, if possible, reproduce the exact solutions of radiative transfer equation in vacuum. We also design a numerical scheme for the radially symmetric case. This scheme is well-balanced and satises the diusion limit of the model. This scheme is validated by various numerical tests