Skip to main content Accessibility help
×
Home
Hostname: page-component-888d5979f-lgdn2 Total loading time: 0.159 Render date: 2021-10-25T11:49:37.511Z Has data issue: false Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

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

Published online by Cambridge University Press:  15 April 2010

Patricia Cargo
Affiliation:
CEA, DAM, DIF, 91297 Arpajon, France. gerald.samba@cea.fr
Gérald Samba
Affiliation:
CEA, DAM, DIF, 91297 Arpajon, France. gerald.samba@cea.fr
Get access

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 P 1 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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

G. Bell and S. Glasstone, Nuclear Reactor Theory. Van Nostrand, Princeton (1970).
Berthon, C., Charrier, P. and Dubroca, B., HLLC, An scheme to solve the M 1 model of radiative transfer in two space dimensions. J. Sci. Comp. 31 (2007) 347389. CrossRef
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).
Buet, C. and Despres, B., Asymptotic preserving and positive schemes for radiation hydrodynamics. J. Comput. Phys. 215 (2006) 717740. CrossRef
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.
K.M. Case and P.F. Zweifel, Linear Transport Theory. Addison-Wesley Publishing Co., Inc. Reading (1967).
S. Chandrasekhar, Radiative transfer. Dover, New York (1960).
R. Dautray and J.L. Lions, Analyse mathématique et calcul numérique. Chap. 21, Masson, Paris (1988).
J.J. Duderstadt and W.R. Martin, Transport theory. Wiley-Interscience, New York (1979).
E.M. Gelbard, Simplified spherical harmonics equations and their use in shielding problems. Technical report WAPD-T-1182, Bettis Atomic Power Laboratory, USA (1961).
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.
Gosse, L. and Toscani, G., Space localization and well-balanced schemes for discrete kinetic models in diffusive regimes. SIAM J. Numer. Anal. 41 (2003) 641658. CrossRef
Greenberg, J.M. and Leroux, A.Y., A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 116. CrossRef
Keller, H.B., Approximate solutions of transport problems. II. Convergence and applications of the discrete-ordinate method. J. Soc. Indust. Appl. Math. 8 (1960) 4373. CrossRef
Larsen, E.W., On numerical solutions of transport problems in the diffusion limit. Nucl. Sci. Eng. 83 (1983) 90. CrossRef
Larsen, E.W. and Keller, J.B., Asymptotic solution of neutron transport problems for small mean free paths. J. Math. Phys. 15 (1974) 75. CrossRef
Larsen, E.W., Pomraning, G.C. and Badham, V.C., Asymptotic analysis of radiative transfer problems. J. Quant. Spectrosc. Radiat. Transfer 29 (1983) 285. CrossRef
Lathrop, K.D., Ray effects in discrete ordinates equations. Nucl. Sci. Eng. 32 (1968) 357. CrossRef
Levermore, C.D., Relating Eddington factors to flux limiters. J. Quant. Spec. Rad. Transfer. 31 (1984) 149160. CrossRef
R. McClarren, J.P. Holloway, T.A. Brunner and T. Melhorn, An implicit Riemann solver for the time-dependent P n equations, in International Topical Meeting on Mathematics and Computation, Supercomputing, Reactor Physics and Nuclear and Biological Applications, American Nuclear Society, Avignon, France (2005).
R. McClarren, J.P. Holloway and T.A. Brunner, Establishing an asymptotic diffusion limit for Riemann solvers on the time-dependent P n equations, in International Topical Meeting on Mathematics and Computation, Supercomputing, Reactor Physics and Nuclear and Biological Applications, American Nuclear Society, Avignon, France (2005).
McClarren, R., Holloway, J.P. and Brunner, T.A., On solutions to the P n equations for thermal radiative transfer. J. Comput. Phys. 227 (2008) 28642885. CrossRef
Pomraning, G.C., Diffusive limits for linear transport equations. Nucl. Sci. Eng. 112 (1992) 239255. CrossRef

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Resolution of the time dependent P n equations by a Godunov type scheme having the diffusion limit
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Resolution of the time dependent P n equations by a Godunov type scheme having the diffusion limit
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Resolution of the time dependent P n equations by a Godunov type scheme having the diffusion limit
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *