American Institute of Mathematical Sciences

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April  2012, 5(2): 245-255. doi: 10.3934/dcdss.2012.5.245

A numerical correction of the $M1$-model in the diffusive limit

Christophe Berthon 1, and  Rodolphe Turpault 1,

1. 

Université de Nantes - Laboratoire de Mathématiques Jean Leray, 2, rue de la Houssinière, 44322 Nantes Cedex 03, France, France

Received  July 2009 Revised  March 2010 Published  September 2011

The present work concerns the numerical simulations of radiative transfer. To address such an issue, the $M1$-model is here adopted. Indeed, this moment model is known to preserve several essential physical properties about radiative energy and radiative flux. In addition, it reduces drastically the numerical cost of the simulations. Unfortunately, the model is not able to restore the expected diffusive regime as prescribed by physics. To correct such a failure, a suitable numerical procedure is derived. The proposed approximation technique enforces, in a sense to be specified, a numerical diffusive regime governed by the Rosseland's mean value of the opacity as imposed by the radiative transfer equation. Numerical experiments issuing from relevant physical benchmarks, illustrate the interest of the derived method.
Keywords: radiative transfer, asymptotic preserving, Hyperbolic conservation laws with source term, Rosseland mean value..
Mathematics Subject Classification: Primary: 76M12; Secondary: 65M1.
Citation: Christophe Berthon, Rodolphe Turpault. A numerical correction of the $M1$-model in the diffusive limit. Discrete and Continuous Dynamical Systems - S, 2012, 5 (2) : 245-255. doi: 10.3934/dcdss.2012.5.245
References:
[1]

E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comp., 25 (2004), 2050-2065.

[2]

C. Berthon, J. Dubois and R. Turpault, Numerical approximation of the M1-model, in "Mathematical Models and Numerical Methods for Radiative Transfer" (ed. T. Goudon), Panor. Synthèses, 28, Soc. Math. France, Paris, (2009), 55-86.

[3]

C. Berthon, P. Charrier and B. Dubroca, An HLLC scheme to solve the M1 Model of radiative transfer in two space dimensions, J. Scie. Comput., 31 (2007), 347-389. doi: 10.1007/s10915-006-9108-6.

[4]

C. Buet and S. Cordier, Asymptotic preserving scheme and numerical methods for radiative hydrodynamic models, C. R. Math. Acad. Sci. Paris, 338 (2004), 951-956. doi: 10.1016/j.crma.2004.04.006.

[5]

C. Buet and B. Després, Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics, J. Quant. Spectrosc. Radiat. Transfer, 85 (2004), 385-418. doi: 10.1016/S0022-4073(03)00233-4.

[6]

C. Buet and B. Després, Asymptotic preserving and positive schemes for radiation hydrodynamics, J. Compt. Phy., 215 (2006), 717-740. doi: 10.1016/j.jcp.2005.11.011.

[7]

P. Charrier, B. Dubroca, G. Duffa and R. Turpault, "Multigroup Model for Radiating Flows during Atmospheric Hypersonic Re-Entry," proceedings of International Workshop on Radiation of High Temperature Gases in Atmospheric Entry, Lisbonne, Portugal, (2003), 103-110.

[8]

B. Dubroca and J.-L. Feugeas, Entropic moment closure hierarchy for the radiative transfer equation, C. R. Acad. Sci. Paris, Ser. I, 329 (1999), 915-920.

[9]

E. Godlewski and P.-A. Raviart, "Numerical Approximation of Hyperbolic Systems of Conservations Laws," Applied Mathematical Sciences, 118, Springer-Verlag, New York, 1996.

[10]

L. Gosse and G. Toscani, Asymptotic-preserving well-balanced scheme for the hyperbolic heat equations, C. R. Math. Acad. Sci. Paris, 334 (2002), 337-342. doi: 10.1016/S1631-073X(02)02257-4.

[11]

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. doi: 10.1137/0733001.

[12]

A. Harten, P. D. Lax and B. van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Review, 25 (1983), 35-61. doi: 10.1137/1025002.

[13]

C. D. Levermore, Moment closure hierarchies for kinetic theory, J. Statist. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552.

[14]

R. J. LeVeque, "Finite Volume Methods for Hyperbolic Problems," Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253.

[15]

D. Mihalas and B. W. Mihalas, "Foundation of Radiation Hydrodynamics," Oxford University Press, New York, 1984.

[16]

G. C. Pomraning, "The Equations of Radiation Hydrodynamics," Sciences Application, 1973.

[17]

J.-F. Ripoll, An averaged formulation of the M1 radiation model with presumed probability density function for turbulent flows, J. Quant. Spectrosc. Radiat. Transfer, 83 (2004), 493-517. doi: 10.1016/S0022-4073(03)00102-X.

[18]

R. Turpault, A consistant multigroup model for radiative transfer and its underlying mean opacities, J. Quant. Spectrosc. Radiat. Transfer, 94 (2005), 357-371. doi: 10.1016/j.jqsrt.2004.09.042.

[19]

R. Turpault, B. Dubroca, M. Frank and A. Klar, Multigroup half space moment approximations to the radiative heat transfer equations, J. Comp. Phys., 198 (2004), 363-371. doi: 10.1016/j.jcp.2004.01.011.

show all references

References:
[1]

E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comp., 25 (2004), 2050-2065.

[2]

C. Berthon, J. Dubois and R. Turpault, Numerical approximation of the M1-model, in "Mathematical Models and Numerical Methods for Radiative Transfer" (ed. T. Goudon), Panor. Synthèses, 28, Soc. Math. France, Paris, (2009), 55-86.

[3]

C. Berthon, P. Charrier and B. Dubroca, An HLLC scheme to solve the M1 Model of radiative transfer in two space dimensions, J. Scie. Comput., 31 (2007), 347-389. doi: 10.1007/s10915-006-9108-6.

[4]

C. Buet and S. Cordier, Asymptotic preserving scheme and numerical methods for radiative hydrodynamic models, C. R. Math. Acad. Sci. Paris, 338 (2004), 951-956. doi: 10.1016/j.crma.2004.04.006.

[5]

C. Buet and B. Després, Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics, J. Quant. Spectrosc. Radiat. Transfer, 85 (2004), 385-418. doi: 10.1016/S0022-4073(03)00233-4.

[6]

C. Buet and B. Després, Asymptotic preserving and positive schemes for radiation hydrodynamics, J. Compt. Phy., 215 (2006), 717-740. doi: 10.1016/j.jcp.2005.11.011.

[7]

P. Charrier, B. Dubroca, G. Duffa and R. Turpault, "Multigroup Model for Radiating Flows during Atmospheric Hypersonic Re-Entry," proceedings of International Workshop on Radiation of High Temperature Gases in Atmospheric Entry, Lisbonne, Portugal, (2003), 103-110.

[8]

B. Dubroca and J.-L. Feugeas, Entropic moment closure hierarchy for the radiative transfer equation, C. R. Acad. Sci. Paris, Ser. I, 329 (1999), 915-920.

[9]

E. Godlewski and P.-A. Raviart, "Numerical Approximation of Hyperbolic Systems of Conservations Laws," Applied Mathematical Sciences, 118, Springer-Verlag, New York, 1996.

[10]

L. Gosse and G. Toscani, Asymptotic-preserving well-balanced scheme for the hyperbolic heat equations, C. R. Math. Acad. Sci. Paris, 334 (2002), 337-342. doi: 10.1016/S1631-073X(02)02257-4.

[11]

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. doi: 10.1137/0733001.

[12]

A. Harten, P. D. Lax and B. van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Review, 25 (1983), 35-61. doi: 10.1137/1025002.

[13]

C. D. Levermore, Moment closure hierarchies for kinetic theory, J. Statist. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552.

[14]

R. J. LeVeque, "Finite Volume Methods for Hyperbolic Problems," Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253.

[15]

D. Mihalas and B. W. Mihalas, "Foundation of Radiation Hydrodynamics," Oxford University Press, New York, 1984.

[16]

G. C. Pomraning, "The Equations of Radiation Hydrodynamics," Sciences Application, 1973.

[17]

J.-F. Ripoll, An averaged formulation of the M1 radiation model with presumed probability density function for turbulent flows, J. Quant. Spectrosc. Radiat. Transfer, 83 (2004), 493-517. doi: 10.1016/S0022-4073(03)00102-X.

[18]

R. Turpault, A consistant multigroup model for radiative transfer and its underlying mean opacities, J. Quant. Spectrosc. Radiat. Transfer, 94 (2005), 357-371. doi: 10.1016/j.jqsrt.2004.09.042.

[19]

R. Turpault, B. Dubroca, M. Frank and A. Klar, Multigroup half space moment approximations to the radiative heat transfer equations, J. Comp. Phys., 198 (2004), 363-371. doi: 10.1016/j.jcp.2004.01.011.

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