Resolution of the time dependent Pn equations by a Godunov type scheme having the diffusion limit
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 6, p. 1193-1224

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.

DOI : https://doi.org/10.1051/m2an/2010027
Classification:  82C70,  35B40,  74S10
Keywords: time-dependent transport, Pn model, diffusion limit, finite volume method, Riemann solver
@article{M2AN_2010__44_6_1193_0,
author = {Cargo, Patricia and Samba, G\'erald},
title = {Resolution of the time dependent Pn equations by a Godunov type scheme having the diffusion limit},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {44},
number = {6},
year = {2010},
pages = {1193-1224},
doi = {10.1051/m2an/2010027},
zbl = {pre05835018},
mrnumber = {2769054},
language = {en},
url = {http://www.numdam.org/item/M2AN_2010__44_6_1193_0}
}

Cargo, Patricia; Samba, Gérald. Resolution of the time dependent Pn equations by a Godunov type scheme having the diffusion limit. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 6, pp. 1193-1224. doi : 10.1051/m2an/2010027. http://www.numdam.org/item/M2AN_2010__44_6_1193_0/

[1] G. Bell and S. Glasstone, Nuclear Reactor Theory. Van Nostrand, Princeton (1970).

[2] C. Berthon, P. Charrier and B. Dubroca, An HLLC scheme to solve the M1 model of radiative transfer in two space dimensions. J. Sci. Comp. 31 (2007) 347-389. | Zbl 1133.85003

[3] T.A. Brunner, Riemann solvers for time-dependent transport based on the maximum entropy and spherical harmonics closures. Ph.D. Thesis, University of Michigan (2000).

[4] C. Buet and B. Despres, Asymptotic preserving and positive schemes for radiation hydrodynamics. J. Comput. Phys. 215 (2006) 717-740. | Zbl 1090.76046

[5] C. Buet and S. Cordier, Asymptotic preserving scheme and numerical methods for radiative hydrodynamic models. C. R. Acad. Sci., Sér. 1 Math. 338 (2004) 951-956. | Zbl 1149.76649

[6] K.M. Case and P.F. Zweifel, Linear Transport Theory. Addison-Wesley Publishing Co., Inc. Reading (1967). | Zbl 0162.58903

[7] S. Chandrasekhar, Radiative transfer. Dover, New York (1960). | Zbl 0037.43201

[8] R. Dautray and J.L. Lions, Analyse mathématique et calcul numérique. Chap. 21, Masson, Paris (1988).

[9] J.J. Duderstadt and W.R. Martin, Transport theory. Wiley-Interscience, New York (1979). | Zbl 0407.76001

[10] E.M. Gelbard, Simplified spherical harmonics equations and their use in shielding problems. Technical report WAPD-T-1182, Bettis Atomic Power Laboratory, USA (1961).

[11] L. Gosse and G. Toscani, An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations. C. R. Acad. Sci., Sér. 1 Math. 334 (2002) 337-342. | Zbl 0996.65093

[12] L. Gosse and G. Toscani, Space localization and well-balanced schemes for discrete kinetic models in diffusive regimes. SIAM J. Numer. Anal. 41 (2003) 641-658. | Zbl 1130.82340

[13] J.M. Greenberg and A.Y. Leroux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1-16. | Zbl 0876.65064

[14] H.B. Keller, Approximate solutions of transport problems. II. Convergence and applications of the discrete-ordinate method. J. Soc. Indust. Appl. Math. 8 (1960) 43-73. | Zbl 0116.33301

[15] E.W. Larsen, On numerical solutions of transport problems in the diffusion limit. Nucl. Sci. Eng. 83 (1983) 90.

[16] E.W. Larsen and J.B. Keller, Asymptotic solution of neutron transport problems for small mean free paths. J. Math. Phys. 15 (1974) 75.

[17] E.W. Larsen, G.C. Pomraning and V.C. Badham, Asymptotic analysis of radiative transfer problems. J. Quant. Spectrosc. Radiat. Transfer 29 (1983) 285.

[18] K.D. Lathrop, Ray effects in discrete ordinates equations. Nucl. Sci. Eng. 32 (1968) 357.

[19] C.D. Levermore, Relating Eddington factors to flux limiters. J. Quant. Spec. Rad. Transfer. 31 (1984) 149-160.

[20] R. Mcclarren, J.P. Holloway, T.A. Brunner and T. Melhorn, An implicit Riemann solver for the time-dependent Pn equations, in International Topical Meeting on Mathematics and Computation, Supercomputing, Reactor Physics and Nuclear and Biological Applications, American Nuclear Society, Avignon, France (2005).

[21] R. Mcclarren, J.P. Holloway and T.A. Brunner, Establishing an asymptotic diffusion limit for Riemann solvers on the time-dependent Pn equations, in International Topical Meeting on Mathematics and Computation, Supercomputing, Reactor Physics and Nuclear and Biological Applications, American Nuclear Society, Avignon, France (2005).

[22] R. Mcclarren, J.P. Holloway and T.A. Brunner, On solutions to the Pn equations for thermal radiative transfer. J. Comput. Phys. 227 (2008) 2864-2885. | Zbl 1142.65108

[23] G.C. Pomraning, Diffusive limits for linear transport equations. Nucl. Sci. Eng. 112 (1992) 239-255.