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

Patricia Cargo; Gérald Samba

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 6, page 1193-1224
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Cargo, Patricia, and Samba, Gérald. "Resolution of the time dependent Pn equations by a Godunov type scheme having the diffusion limit." ESAIM: Mathematical Modelling and Numerical Analysis 44.6 (2010): 1193-1224. <http://eudml.org/doc/250800>.

@article{Cargo2010,
abstract = { 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. },
author = {Cargo, Patricia, Samba, Gérald},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Time-dependent transport; Pn model; diffusion limit; finite volume method; Riemann solver; time-dependent transport; -model; neutron transport equations},
language = {eng},
month = {10},
number = {6},
pages = {1193-1224},
publisher = {EDP Sciences},
title = {Resolution of the time dependent Pn equations by a Godunov type scheme having the diffusion limit},
url = {http://eudml.org/doc/250800},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Cargo, Patricia
AU - Samba, Gérald
TI - Resolution of the time dependent Pn equations by a Godunov type scheme having the diffusion limit
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/10//
PB - EDP Sciences
VL - 44
IS - 6
SP - 1193
EP - 1224
AB - 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.
LA - eng
KW - Time-dependent transport; Pn model; diffusion limit; finite volume method; Riemann solver; time-dependent transport; -model; neutron transport equations
UR - http://eudml.org/doc/250800
ER -

References

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