September  2013, 6(3): 557-587. doi: 10.3934/krm.2013.6.557

Perturbed, entropy-based closure for radiative transfer

1. 

RWTH Aachen University, Department of Mathematics & Center for Computational Engineering Science, Schinkelstrasse 2, D-52062 Aachen, Germany

2. 

Computer Science and Mathematics Division, Oak Ridge National Laboratory, 1 Bethel Valley Road, Oak Ridge, TN 37831, United States

Received  August 2012 Revised  February 2013 Published  May 2013

We derive a hierarchy of closures based on perturbations of well-known entropy-based closures; we therefore refer to them as perturbed entropy-based models. Our derivation reveals final equations containing an additional convective and diffusive term which are added to the flux term of the standard closure. We present numerical simulations for the simplest member of the hierarchy, the perturbed $M_1$ or $PM_1$ model, in one spatial dimension. Simulations are performed using a Runge-Kutta discontinuous Galerkin method with special limiters that guarantee the realizability of the moment variables and the positivity of the material temperature. Improvements to the standard $M_1$ model are observed in cases where unphysical shocks develop in the $M_1$ model.
Citation: Martin Frank, Cory D. Hauck, Edgar Olbrant. Perturbed, entropy-based closure for radiative transfer. Kinetic and Related Models, 2013, 6 (3) : 557-587. doi: 10.3934/krm.2013.6.557
References:
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G. Alldredge, C. D. Hauck and A. L. Tits, High-order, entropy-based closures for linear transport in slab geometry II: A computational study of the optimization problem, SIAM J. Sci. Comput., 34 (2012), B361-B391. doi: 10.1137/11084772X.

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A. M. Anile and S. Pennisi, Thermodynamic derivation of the hydrodynamical model for charge transport in semiconductors, Phys. Rev. B, 46 (1992), 187-193.

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A. M. Anile and V. Romano, Hydrodynamical modeling of charge carrier transport in semiconductors, Meccanica, 35 (2000), 249-296.

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A. M. Anile, W. Allegretto and C. Ringhofer, Mathematical problems in semiconductor physics, in "Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy on July 15-22, 1998" (ed. Anile), Lecture Notes in Mathematics, 1823, Springer-Verlag, Berlin, 2003.

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F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131 (1997), 267-279. doi: 10.1006/jcph.1996.5572.

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R. Biswas, K. D. Devine and J. E. Flaherty, Parallel, adaptive finite element methods for conservation laws, Applied Numerical Mathematics, 14 (1994), 255-283. doi: 10.1016/0168-9274(94)90029-9.

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S. A. Bludman and J. Cernohorsky, Stationary neutrino radiation transport by maximum entropy closure, Phys. Rep., 256 (1995), 37-51. doi: 10.1016/0370-1573(94)00100-H.

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T. A. Brunner and J. P. Holloway, One-dimensional Riemann solvers and the maximum entropy closure, J. Quant Spect. and Radiative Trans., 69 (2001), 543-566.

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______, Two-dimensional time-dependent Riemann solvers for neutron transport, J. Comp Phys., 210 (2005), 386-399. doi: 10.1016/j.jcp.2005.04.011.

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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.

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A. Burbeau, P. Sagaut and Ch-H. Bruneau, A problem-independent limiter for high-order Runge-Kutta discontinuous Galerkin methods, J. Comput. Phys., 169 (2001), 111-150. doi: 10.1006/jcph.2001.6718.

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J. Cernohorsky and S. A. Bludman, Stationary neutrino radiation transport by maximum entropy closure, Tech. Report LBL-36135, Lawrence Berkely National Laboratory, 1994.

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J. Cernohorsky, L. J. van den Horn and J. Cooperstein, Maximum entropy eddington factors in flux-limited neutrino diffusion, Journal of Quantitative Spectroscopy and Radiative Transfer, 42 (1989), 603-613. doi: 10.1016/0022-4073(89)90054-X.

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B. Cockburn, S. Hou and C.-W. Shu, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case, Mathematics of Computation, 54 (1990), 545-581. doi: 10.2307/2008501.

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B. Cockburn, G. Karniadakis and C.-W. Shu, eds., "Discontinuous Galerkin Methods. Theory, Computation and Applications," Papers from the 1st International Symposium held in Newport, RI, May 24–26, 1999, Lecture Notes in Computational Science and Engineering, 11, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-642-59721-3.

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P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys., 112 (2003), 587-628. doi: 10.1023/A:1023824008525.

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W. Dreyer, Maximisation of the entropy in non-equilibrium, Journal of Physics A, 20 (1987), 6505-6517. doi: 10.1088/0305-4470/20/18/047.

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W. Dreyer, M. Herrmann and M. Kunik, Kinetic solutions of the Boltzmann-Peierls equation and its moment systems, Continuum Mechanics and Thermodynamics, 16 (2004), 453-469. doi: 10.1007/s00161-003-0171-z.

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B. Dubroca and J.-L. Feugeas, Étude théorique et numérique d'une hiérarchie de modèles aus moments pour le transfert radiatif, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), 915-920. doi: 10.1016/S0764-4442(00)87499-6.

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B. Dubroca and A. Klar, Half-moment closure for radiative transfer equations, J. Comput. Phys., 180 (2002), 584-596. doi: 10.1006/jcph.2002.7106.

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M. Frank, B. Dubroca and A. Klar, Partial moment entropy approximation to radiative heat transfer, J. Comput. Phys., 218 (2006), 1-18. doi: 10.1016/j.jcp.2006.01.038.

[25]

S. Gottlieb, D. I. Ketcheson and C.-W. Shu, High order strong stability preserving time discretizations, J. Sci. Comput., 38 (2009), 251-289. doi: 10.1007/s10915-008-9239-z.

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C. Groth and J. McDonald, Towards physically realizable and hyperbolic moment closures for kinetic theory, Continuum Mech. Thermodyn., 21 (2009), 467-493. doi: 10.1007/s00161-009-0125-1.

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C. D. Hauck, "Entropy-Based Moment Closures in Semiconductor Models," Ph.D. thesis, University of Maryland, College Park, 2006.

[28]

______, High-order entropy-based closures for linear transport in slab geometry, Commun. Math. Sci., 9 (2011), 187-205.

[29]

C. D. Hauck, C. D. Levermore and A. L. Tits, Convex duality and entropy-based moment closures: Characterizing degenerate densities, SIAM J. Control Optim., 47 (2008), 1977-2015. doi: 10.1137/070691139.

[30]

C. D. Hauck and R. G. McClarren, Positive $P_N$ closures, SIAM J. Sci. Comput., 32 (2010), 2603-2626. doi: 10.1137/090764918.

[31]

A. Jüngel, S. Krause and P. Pietra, A hierarchy of diffusive higher-order moment equations for semiconductors, SIAM Journal on Applied Mathematics, 68 (2007), 171-198. doi: 10.1137/070683313.

[32]

M. Junk, Domain of definition of Levermore's five moment system, J. Stat. Phys., 93 (1998), 1143-1167. doi: 10.1023/B:JOSS.0000033155.07331.d9.

[33]

_______, Maximum entropy for reduced moment problems, Math. Mod. Meth. Appl. S., 10 (2000), 1001-1025. doi: 10.1142/S0218202500000513.

[34]

M. Junk and V. Romano, Maximum entropy moment system of the semiconductor Boltzmann equation using Kane's dispersion relation, Continuum Mechanics and Thermodynamics, 17 (2005), 247-267. doi: 10.1007/s00161-004-0201-5.

[35]

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

[36]

_______, Moment closure hierarchies for the Boltzmann-Poisson equation, VLSI Design, 6 (1998), 97-101.

[37]

_______, Boundary conditions for moment closures, Presentation at The Annual Kinetic FRG Meeting, 2009.

[38]

C. D. Levermore, W. J. Morokoff and B. T. Nadiga, Moment realizability and the validity of the Navier-Stokes equations for rarefied gas dynamics, Phys. Fluids, 10 (1998), 3214-3226. doi: 10.1063/1.869849.

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C. D. Levermore and G. C. Pomraning, A flux-limited diffusion theory, Astrophys. J., 248 (1981), 321-334.

[40]

H. Liu and J. Yan, The direct discontinuous Galerkin (DDG) methods for diffusion problems,, SIAM J. Numer. Anal., 47 (): 675.  doi: 10.1137/080720255.

[41]

L. W. Lin, B. Temple and J. H. Wang, Suppression of oscillations in Godunov's method for a resonant non-strictly hyperbolic system, SIAM J. Numer. Anal., 32 (1995), 841-864. doi: 10.1137/0732039.

[42]

R. G. McClarren, T. M. Evans, R. B. Lowrie and J. D. Densmore, Semi-implicit time integration for $P_N$ thermal radiative transfer, J. Comput. Phys., 227 (2008), 7561-7586. doi: 10.1016/j.jcp.2008.04.029.

[43]

G. N. Minerbo, Maximum entropy Eddington factors, J. Quant. Spectrosc. Radiat. Transfer, 20 (1978), 541-545. doi: 10.1016/0022-4073(78)90024-9.

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P. Monreal and M. Frank, Higher order minimum entropy approximations in radiative transfer,, preprint., (). 

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I. Müller and T. Ruggeri, "Rational Extended Thermodynamics," Second edition, Springer Tracts in Natural Philosophy, Vol. 37, Springer-Verlag, New York, 1993.

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N. C. Nguyen, J. Peraire and B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations, J. Comput. Phys., 228 (2009), 8841-8855. doi: 10.1016/j.jcp.2009.08.030.

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K. S. Oh and J. P. Holloway, A quasi-static closure for $3^{rd}$ order spherical harmonics time-dependent radiation transport in 2-d, Joint International Topical Meeting on Mathematics and Computing and Supercomputing in Nuclear Applications, on CD-ROM, 2009.

[48]

E. Olbrant, C. D. Hauck and M. Frank, A realizability-preserving discontinuous Galerkin method for the M1 model of radiative transfer, J. Comput. Phys., 231 (2012), 5612-5639. doi: 10.1016/j.jcp.2012.03.002.

[49]

A. Ore, Entropy of radiation, Phys. Rev., 98 (1955), 887-888. doi: 10.1103/PhysRev.98.887.

[50]

G. C. Pomraning, "Radiation Hydrodynamics," Pergamon Press, New York, 1973. doi: 10.2172/656708.

[51]

S. La Rosa, G. Mascali and V. Romano, Exact maximum entropy closure of the hydrodynamical model for Si semiconductors: The 8-moment case, SIAM Journal on Applied Mathematics, 70 (2009), 710-734. doi: 10.1137/080714282.

[52]

P. Rosen, Entropy of radiation, Phys. Rev., 96 (1954), 555. doi: 10.1103/PhysRev.96.555.

[53]

K. Salari and P. Knupp, Code verification by the method of manufactured solutions, Tech. Report SAND2000-1444, Sandia National Laboratories, 2000. doi: 10.2172/759450.

[54]

M. Schäfer, M. Frank and C. D. Levermore, Diffusive corrections to $P_N$ approximations, Multiscale Model. Simul., 9 (2011), 1-28. doi: 10.1137/090764542.

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J. Schneider, Entropic approximation in kinetic theory, Math. Model. Numer. Anal., 38 (2004), 541-561. doi: 10.1051/m2an:2004025.

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C.-W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. Comput. Phys., 77 (1989), 439-471. doi: 10.1016/0021-9991(88)90177-5.

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Y. Shu and C.-W. Shu, Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Commun. Comput. Phys., 7 (2010), 1-46. doi: 10.4208/cicp.2009.09.023.

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show all references

References:
[1]

G. Alldredge, C. D. Hauck and A. L. Tits, High-order, entropy-based closures for linear transport in slab geometry II: A computational study of the optimization problem, SIAM J. Sci. Comput., 34 (2012), B361-B391. doi: 10.1137/11084772X.

[2]

A. M. Anile and O. Muscato, Improved hydrodynamical model for carrier transport in semiconductors, Phys. Rev. B, 51 (1995), 16728-16740. doi: 10.1103/PhysRevB.51.16728.

[3]

A. M. Anile and S. Pennisi, Thermodynamic derivation of the hydrodynamical model for charge transport in semiconductors, Phys. Rev. B, 46 (1992), 187-193.

[4]

A. M. Anile and V. Romano, Hydrodynamical modeling of charge carrier transport in semiconductors, Meccanica, 35 (2000), 249-296.

[5]

A. M. Anile, W. Allegretto and C. Ringhofer, Mathematical problems in semiconductor physics, in "Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy on July 15-22, 1998" (ed. Anile), Lecture Notes in Mathematics, 1823, Springer-Verlag, Berlin, 2003.

[6]

F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131 (1997), 267-279. doi: 10.1006/jcph.1996.5572.

[7]

R. Biswas, K. D. Devine and J. E. Flaherty, Parallel, adaptive finite element methods for conservation laws, Applied Numerical Mathematics, 14 (1994), 255-283. doi: 10.1016/0168-9274(94)90029-9.

[8]

S. A. Bludman and J. Cernohorsky, Stationary neutrino radiation transport by maximum entropy closure, Phys. Rep., 256 (1995), 37-51. doi: 10.1016/0370-1573(94)00100-H.

[9]

T. A. Brunner and J. P. Holloway, One-dimensional Riemann solvers and the maximum entropy closure, J. Quant Spect. and Radiative Trans., 69 (2001), 543-566.

[10]

______, Two-dimensional time-dependent Riemann solvers for neutron transport, J. Comp Phys., 210 (2005), 386-399. doi: 10.1016/j.jcp.2005.04.011.

[11]

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.

[12]

A. Burbeau, P. Sagaut and Ch-H. Bruneau, A problem-independent limiter for high-order Runge-Kutta discontinuous Galerkin methods, J. Comput. Phys., 169 (2001), 111-150. doi: 10.1006/jcph.2001.6718.

[13]

J. Cernohorsky and S. A. Bludman, Stationary neutrino radiation transport by maximum entropy closure, Tech. Report LBL-36135, Lawrence Berkely National Laboratory, 1994.

[14]

J. Cernohorsky, L. J. van den Horn and J. Cooperstein, Maximum entropy eddington factors in flux-limited neutrino diffusion, Journal of Quantitative Spectroscopy and Radiative Transfer, 42 (1989), 603-613. doi: 10.1016/0022-4073(89)90054-X.

[15]

B. Cockburn, S. Hou and C.-W. Shu, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case, Mathematics of Computation, 54 (1990), 545-581. doi: 10.2307/2008501.

[16]

B. Cockburn, G. Karniadakis and C.-W. Shu, eds., "Discontinuous Galerkin Methods. Theory, Computation and Applications," Papers from the 1st International Symposium held in Newport, RI, May 24–26, 1999, Lecture Notes in Computational Science and Engineering, 11, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-642-59721-3.

[17]

B. Cockburn, S.-Y. Lin and C.-W. Shu, TVB runge-kutta local projection Discontinuous Galerkin Finite Element method for conservation laws. III: One-dimensional systems, J. Comput. Phys., 84 (1989), 90-113. doi: 10.1016/0021-9991(89)90183-6.

[18]

B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), 2440-2463. doi: 10.1137/S0036142997316712.

[19]

P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys., 112 (2003), 587-628. doi: 10.1023/A:1023824008525.

[20]

W. Dreyer, Maximisation of the entropy in non-equilibrium, Journal of Physics A, 20 (1987), 6505-6517. doi: 10.1088/0305-4470/20/18/047.

[21]

W. Dreyer, M. Herrmann and M. Kunik, Kinetic solutions of the Boltzmann-Peierls equation and its moment systems, Continuum Mechanics and Thermodynamics, 16 (2004), 453-469. doi: 10.1007/s00161-003-0171-z.

[22]

B. Dubroca and J.-L. Feugeas, Étude théorique et numérique d'une hiérarchie de modèles aus moments pour le transfert radiatif, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), 915-920. doi: 10.1016/S0764-4442(00)87499-6.

[23]

B. Dubroca and A. Klar, Half-moment closure for radiative transfer equations, J. Comput. Phys., 180 (2002), 584-596. doi: 10.1006/jcph.2002.7106.

[24]

M. Frank, B. Dubroca and A. Klar, Partial moment entropy approximation to radiative heat transfer, J. Comput. Phys., 218 (2006), 1-18. doi: 10.1016/j.jcp.2006.01.038.

[25]

S. Gottlieb, D. I. Ketcheson and C.-W. Shu, High order strong stability preserving time discretizations, J. Sci. Comput., 38 (2009), 251-289. doi: 10.1007/s10915-008-9239-z.

[26]

C. Groth and J. McDonald, Towards physically realizable and hyperbolic moment closures for kinetic theory, Continuum Mech. Thermodyn., 21 (2009), 467-493. doi: 10.1007/s00161-009-0125-1.

[27]

C. D. Hauck, "Entropy-Based Moment Closures in Semiconductor Models," Ph.D. thesis, University of Maryland, College Park, 2006.

[28]

______, High-order entropy-based closures for linear transport in slab geometry, Commun. Math. Sci., 9 (2011), 187-205.

[29]

C. D. Hauck, C. D. Levermore and A. L. Tits, Convex duality and entropy-based moment closures: Characterizing degenerate densities, SIAM J. Control Optim., 47 (2008), 1977-2015. doi: 10.1137/070691139.

[30]

C. D. Hauck and R. G. McClarren, Positive $P_N$ closures, SIAM J. Sci. Comput., 32 (2010), 2603-2626. doi: 10.1137/090764918.

[31]

A. Jüngel, S. Krause and P. Pietra, A hierarchy of diffusive higher-order moment equations for semiconductors, SIAM Journal on Applied Mathematics, 68 (2007), 171-198. doi: 10.1137/070683313.

[32]

M. Junk, Domain of definition of Levermore's five moment system, J. Stat. Phys., 93 (1998), 1143-1167. doi: 10.1023/B:JOSS.0000033155.07331.d9.

[33]

_______, Maximum entropy for reduced moment problems, Math. Mod. Meth. Appl. S., 10 (2000), 1001-1025. doi: 10.1142/S0218202500000513.

[34]

M. Junk and V. Romano, Maximum entropy moment system of the semiconductor Boltzmann equation using Kane's dispersion relation, Continuum Mechanics and Thermodynamics, 17 (2005), 247-267. doi: 10.1007/s00161-004-0201-5.

[35]

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

[36]

_______, Moment closure hierarchies for the Boltzmann-Poisson equation, VLSI Design, 6 (1998), 97-101.

[37]

_______, Boundary conditions for moment closures, Presentation at The Annual Kinetic FRG Meeting, 2009.

[38]

C. D. Levermore, W. J. Morokoff and B. T. Nadiga, Moment realizability and the validity of the Navier-Stokes equations for rarefied gas dynamics, Phys. Fluids, 10 (1998), 3214-3226. doi: 10.1063/1.869849.

[39]

C. D. Levermore and G. C. Pomraning, A flux-limited diffusion theory, Astrophys. J., 248 (1981), 321-334.

[40]

H. Liu and J. Yan, The direct discontinuous Galerkin (DDG) methods for diffusion problems,, SIAM J. Numer. Anal., 47 (): 675.  doi: 10.1137/080720255.

[41]

L. W. Lin, B. Temple and J. H. Wang, Suppression of oscillations in Godunov's method for a resonant non-strictly hyperbolic system, SIAM J. Numer. Anal., 32 (1995), 841-864. doi: 10.1137/0732039.

[42]

R. G. McClarren, T. M. Evans, R. B. Lowrie and J. D. Densmore, Semi-implicit time integration for $P_N$ thermal radiative transfer, J. Comput. Phys., 227 (2008), 7561-7586. doi: 10.1016/j.jcp.2008.04.029.

[43]

G. N. Minerbo, Maximum entropy Eddington factors, J. Quant. Spectrosc. Radiat. Transfer, 20 (1978), 541-545. doi: 10.1016/0022-4073(78)90024-9.

[44]

P. Monreal and M. Frank, Higher order minimum entropy approximations in radiative transfer,, preprint., (). 

[45]

I. Müller and T. Ruggeri, "Rational Extended Thermodynamics," Second edition, Springer Tracts in Natural Philosophy, Vol. 37, Springer-Verlag, New York, 1993.

[46]

N. C. Nguyen, J. Peraire and B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations, J. Comput. Phys., 228 (2009), 8841-8855. doi: 10.1016/j.jcp.2009.08.030.

[47]

K. S. Oh and J. P. Holloway, A quasi-static closure for $3^{rd}$ order spherical harmonics time-dependent radiation transport in 2-d, Joint International Topical Meeting on Mathematics and Computing and Supercomputing in Nuclear Applications, on CD-ROM, 2009.

[48]

E. Olbrant, C. D. Hauck and M. Frank, A realizability-preserving discontinuous Galerkin method for the M1 model of radiative transfer, J. Comput. Phys., 231 (2012), 5612-5639. doi: 10.1016/j.jcp.2012.03.002.

[49]

A. Ore, Entropy of radiation, Phys. Rev., 98 (1955), 887-888. doi: 10.1103/PhysRev.98.887.

[50]

G. C. Pomraning, "Radiation Hydrodynamics," Pergamon Press, New York, 1973. doi: 10.2172/656708.

[51]

S. La Rosa, G. Mascali and V. Romano, Exact maximum entropy closure of the hydrodynamical model for Si semiconductors: The 8-moment case, SIAM Journal on Applied Mathematics, 70 (2009), 710-734. doi: 10.1137/080714282.

[52]

P. Rosen, Entropy of radiation, Phys. Rev., 96 (1954), 555. doi: 10.1103/PhysRev.96.555.

[53]

K. Salari and P. Knupp, Code verification by the method of manufactured solutions, Tech. Report SAND2000-1444, Sandia National Laboratories, 2000. doi: 10.2172/759450.

[54]

M. Schäfer, M. Frank and C. D. Levermore, Diffusive corrections to $P_N$ approximations, Multiscale Model. Simul., 9 (2011), 1-28. doi: 10.1137/090764542.

[55]

J. Schneider, Entropic approximation in kinetic theory, Math. Model. Numer. Anal., 38 (2004), 541-561. doi: 10.1051/m2an:2004025.

[56]

C.-W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. Comput. Phys., 77 (1989), 439-471. doi: 10.1016/0021-9991(88)90177-5.

[57]

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