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

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

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.
Citation: Christophe Berthon, Rodolphe Turpault. A numerical correction of the $M1$-model in the diffusive limit. Discrete & 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.   Google Scholar

[2]

C. Berthon, J. Dubois and R. Turpault, Numerical approximation of the M1-model,, in, 28 (2009), 55.   Google Scholar

[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.  doi: 10.1007/s10915-006-9108-6.  Google Scholar

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

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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.  doi: 10.1016/S0022-4073(03)00233-4.  Google Scholar

[6]

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

[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, (2003), 103.   Google Scholar

[8]

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

[9]

E. Godlewski and P.-A. Raviart, "Numerical Approximation of Hyperbolic Systems of Conservations Laws,", Applied Mathematical Sciences, 118 (1996).   Google Scholar

[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.  doi: 10.1016/S1631-073X(02)02257-4.  Google Scholar

[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.  doi: 10.1137/0733001.  Google Scholar

[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.  doi: 10.1137/1025002.  Google Scholar

[13]

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

[14]

R. J. LeVeque, "Finite Volume Methods for Hyperbolic Problems,", Cambridge Texts in Applied Mathematics, (2002).  doi: 10.1017/CBO9780511791253.  Google Scholar

[15]

D. Mihalas and B. W. Mihalas, "Foundation of Radiation Hydrodynamics,", Oxford University Press, (1984).   Google Scholar

[16]

G. C. Pomraning, "The Equations of Radiation Hydrodynamics,", Sciences Application, (1973).   Google Scholar

[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.  doi: 10.1016/S0022-4073(03)00102-X.  Google Scholar

[18]

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

[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.  doi: 10.1016/j.jcp.2004.01.011.  Google Scholar

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.   Google Scholar

[2]

C. Berthon, J. Dubois and R. Turpault, Numerical approximation of the M1-model,, in, 28 (2009), 55.   Google Scholar

[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.  doi: 10.1007/s10915-006-9108-6.  Google Scholar

[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.  doi: 10.1016/j.crma.2004.04.006.  Google Scholar

[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.  doi: 10.1016/S0022-4073(03)00233-4.  Google Scholar

[6]

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

[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, (2003), 103.   Google Scholar

[8]

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

[9]

E. Godlewski and P.-A. Raviart, "Numerical Approximation of Hyperbolic Systems of Conservations Laws,", Applied Mathematical Sciences, 118 (1996).   Google Scholar

[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.  doi: 10.1016/S1631-073X(02)02257-4.  Google Scholar

[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.  doi: 10.1137/0733001.  Google Scholar

[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.  doi: 10.1137/1025002.  Google Scholar

[13]

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

[14]

R. J. LeVeque, "Finite Volume Methods for Hyperbolic Problems,", Cambridge Texts in Applied Mathematics, (2002).  doi: 10.1017/CBO9780511791253.  Google Scholar

[15]

D. Mihalas and B. W. Mihalas, "Foundation of Radiation Hydrodynamics,", Oxford University Press, (1984).   Google Scholar

[16]

G. C. Pomraning, "The Equations of Radiation Hydrodynamics,", Sciences Application, (1973).   Google Scholar

[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.  doi: 10.1016/S0022-4073(03)00102-X.  Google Scholar

[18]

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

[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.  doi: 10.1016/j.jcp.2004.01.011.  Google Scholar

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