# American Institute of Mathematical Sciences

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 & Related Models, 2013, 6 (3) : 557-587. doi: 10.3934/krm.2013.6.557
##### 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). doi: 10.1137/11084772X. Google Scholar [2] A. M. Anile and O. Muscato, Improved hydrodynamical model for carrier transport in semiconductors,, Phys. Rev. B, 51 (1995), 16728. doi: 10.1103/PhysRevB.51.16728. Google Scholar [3] A. M. Anile and S. Pennisi, Thermodynamic derivation of the hydrodynamical model for charge transport in semiconductors,, Phys. Rev. B, 46 (1992), 187. Google Scholar [4] A. M. Anile and V. Romano, Hydrodynamical modeling of charge carrier transport in semiconductors,, Meccanica, 35 (2000), 249. Google Scholar [5] A. M. Anile, W. Allegretto and C. Ringhofer, Mathematical problems in semiconductor physics,, in, 1823 (1998), 15. Google Scholar [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. doi: 10.1006/jcph.1996.5572. Google Scholar [7] R. Biswas, K. D. Devine and J. E. Flaherty, Parallel, adaptive finite element methods for conservation laws,, Applied Numerical Mathematics, 14 (1994), 255. doi: 10.1016/0168-9274(94)90029-9. Google Scholar [8] S. A. Bludman and J. Cernohorsky, Stationary neutrino radiation transport by maximum entropy closure,, Phys. Rep., 256 (1995), 37. doi: 10.1016/0370-1573(94)00100-H. Google Scholar [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. Google Scholar [10] ______, Two-dimensional time-dependent Riemann solvers for neutron transport,, J. Comp Phys., 210 (2005), 386. doi: 10.1016/j.jcp.2005.04.011. Google Scholar [11] T. A. Brunner, "Riemann Solvers for Time-Dependent Transport Based on the Maximum Entropy and Spherical Harmonics Closures,", Ph.D. thesis, (2000). Google Scholar [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. doi: 10.1006/jcph.2001.6718. Google Scholar [13] J. Cernohorsky and S. A. Bludman, Stationary neutrino radiation transport by maximum entropy closure,, Tech. Report LBL-36135, (1994). Google Scholar [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. doi: 10.1016/0022-4073(89)90054-X. Google Scholar [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. doi: 10.2307/2008501. Google Scholar [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, 11 (1999). doi: 10.1007/978-3-642-59721-3. Google Scholar [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. doi: 10.1016/0021-9991(89)90183-6. Google Scholar [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. doi: 10.1137/S0036142997316712. Google Scholar [19] P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle,, J. Stat. Phys., 112 (2003), 587. doi: 10.1023/A:1023824008525. Google Scholar [20] W. Dreyer, Maximisation of the entropy in non-equilibrium,, Journal of Physics A, 20 (1987), 6505. doi: 10.1088/0305-4470/20/18/047. Google Scholar [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. doi: 10.1007/s00161-003-0171-z. Google Scholar [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), 329 (1999), 915. doi: 10.1016/S0764-4442(00)87499-6. Google Scholar [23] B. Dubroca and A. Klar, Half-moment closure for radiative transfer equations,, J. Comput. Phys., 180 (2002), 584. doi: 10.1006/jcph.2002.7106. Google Scholar [24] M. Frank, B. Dubroca and A. Klar, Partial moment entropy approximation to radiative heat transfer,, J. Comput. Phys., 218 (2006), 1. doi: 10.1016/j.jcp.2006.01.038. Google Scholar [25] S. Gottlieb, D. I. Ketcheson and C.-W. Shu, High order strong stability preserving time discretizations,, J. Sci. Comput., 38 (2009), 251. doi: 10.1007/s10915-008-9239-z. Google Scholar [26] C. Groth and J. McDonald, Towards physically realizable and hyperbolic moment closures for kinetic theory,, Continuum Mech. Thermodyn., 21 (2009), 467. doi: 10.1007/s00161-009-0125-1. Google Scholar [27] C. D. Hauck, "Entropy-Based Moment Closures in Semiconductor Models,", Ph.D. thesis, (2006). Google Scholar [28] ______, High-order entropy-based closures for linear transport in slab geometry,, Commun. Math. Sci., 9 (2011), 187. Google Scholar [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. doi: 10.1137/070691139. Google Scholar [30] C. D. Hauck and R. G. McClarren, Positive $P_N$ closures,, SIAM J. Sci. Comput., 32 (2010), 2603. doi: 10.1137/090764918. Google Scholar [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. doi: 10.1137/070683313. Google Scholar [32] M. Junk, Domain of definition of Levermore's five moment system,, J. Stat. Phys., 93 (1998), 1143. doi: 10.1023/B:JOSS.0000033155.07331.d9. Google Scholar [33] _______, Maximum entropy for reduced moment problems,, Math. Mod. Meth. Appl. S., 10 (2000), 1001. doi: 10.1142/S0218202500000513. Google Scholar [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. doi: 10.1007/s00161-004-0201-5. Google Scholar [35] C. D. Levermore, Moment closure hierarchies for kinetic theories,, J. Stat. Phys., 83 (1996), 1021. doi: 10.1007/BF02179552. Google Scholar [36] _______, Moment closure hierarchies for the Boltzmann-Poisson equation,, VLSI Design, 6 (1998), 97. Google Scholar [37] _______, Boundary conditions for moment closures,, Presentation at The Annual Kinetic FRG Meeting, (2009). Google Scholar [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. doi: 10.1063/1.869849. Google Scholar [39] C. D. Levermore and G. C. Pomraning, A flux-limited diffusion theory,, Astrophys. J., 248 (1981), 321. Google Scholar [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. Google Scholar [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. doi: 10.1137/0732039. Google Scholar [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. doi: 10.1016/j.jcp.2008.04.029. Google Scholar [43] G. N. Minerbo, Maximum entropy Eddington factors,, J. Quant. Spectrosc. Radiat. Transfer, 20 (1978), 541. doi: 10.1016/0022-4073(78)90024-9. Google Scholar [44] P. Monreal and M. Frank, Higher order minimum entropy approximations in radiative transfer,, preprint., (). Google Scholar [45] I. Müller and T. Ruggeri, "Rational Extended Thermodynamics,", Second edition, (1993). Google Scholar [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. doi: 10.1016/j.jcp.2009.08.030. Google Scholar [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, (2009). Google Scholar [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. doi: 10.1016/j.jcp.2012.03.002. Google Scholar [49] A. Ore, Entropy of radiation,, Phys. Rev., 98 (1955), 887. doi: 10.1103/PhysRev.98.887. Google Scholar [50] G. C. Pomraning, "Radiation Hydrodynamics,", Pergamon Press, (1973). doi: 10.2172/656708. Google Scholar [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. doi: 10.1137/080714282. Google Scholar [52] P. Rosen, Entropy of radiation,, Phys. Rev., 96 (1954). doi: 10.1103/PhysRev.96.555. Google Scholar [53] K. Salari and P. Knupp, Code verification by the method of manufactured solutions,, Tech. Report SAND2000-1444, (2000), 2000. doi: 10.2172/759450. Google Scholar [54] M. Schäfer, M. Frank and C. D. Levermore, Diffusive corrections to $P_N$ approximations,, Multiscale Model. Simul., 9 (2011), 1. doi: 10.1137/090764542. Google Scholar [55] J. Schneider, Entropic approximation in kinetic theory,, Math. Model. Numer. Anal., 38 (2004), 541. doi: 10.1051/m2an:2004025. Google Scholar [56] C.-W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes,, J. Comput. Phys., 77 (1989), 439. doi: 10.1016/0021-9991(88)90177-5. Google Scholar [57] Y. Shu and C.-W. Shu, Local discontinuous Galerkin methods for high-order time-dependent partial differential equations,, Commun. Comput. Phys., 7 (2010), 1. doi: 10.4208/cicp.2009.09.023. Google Scholar [58] J. M. Smit, J. Cernohorsky and C.-P. Dullemond, Hyperbolicity and critical points in two-moment approximate radiative transfer,, Astrophys. J., 325 (1997), 203. Google Scholar [59] J. M. Smit, L. J. van den Horn and S. A. Bludman, Closure in flux-limited neutrino diffusion and two-moment transport,, Astron. Astrophys., 356 (2000), 559. Google Scholar [60] H. Struchtrup, Linear kinetic heat transfer: Moment equations, boundary conditions, and Knudsen layers,, Physica A Statistical Mechanics and its Applications, 387 (2008), 1750. doi: 10.1016/j.physa.2007.11.044. Google Scholar [61] B. Su and G. L. Olson, An analytical benchmark for non-equilibrium radiative transfer in an isotropically scattering medium,, Ann. Nucl. Energy, 24 (1997), 1035. doi: 10.1016/S0306-4549(96)00100-4. Google Scholar [62] R. Turpault, A consistent 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 [63] D. Wright, M. Frank and A. Klar, The minimum entropy approximation to the radiative transfer equation,, in, 67 (2009), 987. Google Scholar [64] P. Zhang and R.-X. Liu, Hyperbolic conservation laws with space-dependent fluxes. II. General study on numerical fluxes,, J. Comput. Appl. Math., 176 (2005), 105. doi: 10.1016/j.cam.2004.07.005. Google Scholar [65] P. Zhang, S. C. Wong and C.-W. Shu, A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway,, J. Comput. Phys., 212 (2006), 739. doi: 10.1016/j.jcp.2005.07.019. Google Scholar [66] X. Zhang and C.-W. Shu, On maximum-principle-satisfying high order schemes for scalar conservation laws,, J. Comput. Phys., 229 (2010), 3091. doi: 10.1016/j.jcp.2009.12.030. Google Scholar [67] ______, On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes,, J. Comput. Phys., 229 (2010), 8918. doi: 10.1016/j.jcp.2010.08.016. Google Scholar

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##### 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). doi: 10.1137/11084772X. Google Scholar [2] A. M. Anile and O. Muscato, Improved hydrodynamical model for carrier transport in semiconductors,, Phys. Rev. B, 51 (1995), 16728. doi: 10.1103/PhysRevB.51.16728. Google Scholar [3] A. M. Anile and S. Pennisi, Thermodynamic derivation of the hydrodynamical model for charge transport in semiconductors,, Phys. Rev. B, 46 (1992), 187. Google Scholar [4] A. M. Anile and V. Romano, Hydrodynamical modeling of charge carrier transport in semiconductors,, Meccanica, 35 (2000), 249. Google Scholar [5] A. M. Anile, W. Allegretto and C. Ringhofer, Mathematical problems in semiconductor physics,, in, 1823 (1998), 15. Google Scholar [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. doi: 10.1006/jcph.1996.5572. Google Scholar [7] R. Biswas, K. D. Devine and J. E. Flaherty, Parallel, adaptive finite element methods for conservation laws,, Applied Numerical Mathematics, 14 (1994), 255. doi: 10.1016/0168-9274(94)90029-9. Google Scholar [8] S. A. Bludman and J. Cernohorsky, Stationary neutrino radiation transport by maximum entropy closure,, Phys. Rep., 256 (1995), 37. doi: 10.1016/0370-1573(94)00100-H. Google Scholar [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. Google Scholar [10] ______, Two-dimensional time-dependent Riemann solvers for neutron transport,, J. Comp Phys., 210 (2005), 386. doi: 10.1016/j.jcp.2005.04.011. Google Scholar [11] T. A. Brunner, "Riemann Solvers for Time-Dependent Transport Based on the Maximum Entropy and Spherical Harmonics Closures,", Ph.D. thesis, (2000). Google Scholar [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. doi: 10.1006/jcph.2001.6718. Google Scholar [13] J. Cernohorsky and S. A. Bludman, Stationary neutrino radiation transport by maximum entropy closure,, Tech. Report LBL-36135, (1994). Google Scholar [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. doi: 10.1016/0022-4073(89)90054-X. Google Scholar [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. doi: 10.2307/2008501. Google Scholar [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, 11 (1999). doi: 10.1007/978-3-642-59721-3. Google Scholar [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. doi: 10.1016/0021-9991(89)90183-6. Google Scholar [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. doi: 10.1137/S0036142997316712. Google Scholar [19] P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle,, J. Stat. Phys., 112 (2003), 587. doi: 10.1023/A:1023824008525. Google Scholar [20] W. Dreyer, Maximisation of the entropy in non-equilibrium,, Journal of Physics A, 20 (1987), 6505. doi: 10.1088/0305-4470/20/18/047. Google Scholar [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. doi: 10.1007/s00161-003-0171-z. Google Scholar [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), 329 (1999), 915. doi: 10.1016/S0764-4442(00)87499-6. Google Scholar [23] B. Dubroca and A. Klar, Half-moment closure for radiative transfer equations,, J. Comput. Phys., 180 (2002), 584. doi: 10.1006/jcph.2002.7106. Google Scholar [24] M. Frank, B. Dubroca and A. Klar, Partial moment entropy approximation to radiative heat transfer,, J. Comput. Phys., 218 (2006), 1. doi: 10.1016/j.jcp.2006.01.038. Google Scholar [25] S. Gottlieb, D. I. Ketcheson and C.-W. Shu, High order strong stability preserving time discretizations,, J. Sci. Comput., 38 (2009), 251. doi: 10.1007/s10915-008-9239-z. Google Scholar [26] C. Groth and J. McDonald, Towards physically realizable and hyperbolic moment closures for kinetic theory,, Continuum Mech. Thermodyn., 21 (2009), 467. doi: 10.1007/s00161-009-0125-1. Google Scholar [27] C. D. Hauck, "Entropy-Based Moment Closures in Semiconductor Models,", Ph.D. thesis, (2006). Google Scholar [28] ______, High-order entropy-based closures for linear transport in slab geometry,, Commun. Math. Sci., 9 (2011), 187. Google Scholar [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. doi: 10.1137/070691139. Google Scholar [30] C. D. Hauck and R. G. McClarren, Positive $P_N$ closures,, SIAM J. Sci. Comput., 32 (2010), 2603. doi: 10.1137/090764918. Google Scholar [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. doi: 10.1137/070683313. Google Scholar [32] M. Junk, Domain of definition of Levermore's five moment system,, J. Stat. Phys., 93 (1998), 1143. doi: 10.1023/B:JOSS.0000033155.07331.d9. Google Scholar [33] _______, Maximum entropy for reduced moment problems,, Math. Mod. Meth. Appl. S., 10 (2000), 1001. doi: 10.1142/S0218202500000513. Google Scholar [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. doi: 10.1007/s00161-004-0201-5. Google Scholar [35] C. D. Levermore, Moment closure hierarchies for kinetic theories,, J. Stat. Phys., 83 (1996), 1021. doi: 10.1007/BF02179552. Google Scholar [36] _______, Moment closure hierarchies for the Boltzmann-Poisson equation,, VLSI Design, 6 (1998), 97. Google Scholar [37] _______, Boundary conditions for moment closures,, Presentation at The Annual Kinetic FRG Meeting, (2009). Google Scholar [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. doi: 10.1063/1.869849. Google Scholar [39] C. D. Levermore and G. C. Pomraning, A flux-limited diffusion theory,, Astrophys. J., 248 (1981), 321. Google Scholar [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. Google Scholar [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. doi: 10.1137/0732039. Google Scholar [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. doi: 10.1016/j.jcp.2008.04.029. Google Scholar [43] G. N. Minerbo, Maximum entropy Eddington factors,, J. Quant. Spectrosc. Radiat. Transfer, 20 (1978), 541. doi: 10.1016/0022-4073(78)90024-9. Google Scholar [44] P. Monreal and M. Frank, Higher order minimum entropy approximations in radiative transfer,, preprint., (). Google Scholar [45] I. Müller and T. Ruggeri, "Rational Extended Thermodynamics,", Second edition, (1993). Google Scholar [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. doi: 10.1016/j.jcp.2009.08.030. Google Scholar [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, (2009). Google Scholar [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. doi: 10.1016/j.jcp.2012.03.002. Google Scholar [49] A. Ore, Entropy of radiation,, Phys. Rev., 98 (1955), 887. doi: 10.1103/PhysRev.98.887. Google Scholar [50] G. C. Pomraning, "Radiation Hydrodynamics,", Pergamon Press, (1973). doi: 10.2172/656708. Google Scholar [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. doi: 10.1137/080714282. Google Scholar [52] P. Rosen, Entropy of radiation,, Phys. Rev., 96 (1954). doi: 10.1103/PhysRev.96.555. Google Scholar [53] K. Salari and P. Knupp, Code verification by the method of manufactured solutions,, Tech. Report SAND2000-1444, (2000), 2000. doi: 10.2172/759450. Google Scholar [54] M. Schäfer, M. Frank and C. D. Levermore, Diffusive corrections to $P_N$ approximations,, Multiscale Model. Simul., 9 (2011), 1. doi: 10.1137/090764542. Google Scholar [55] J. Schneider, Entropic approximation in kinetic theory,, Math. Model. Numer. Anal., 38 (2004), 541. doi: 10.1051/m2an:2004025. Google Scholar [56] C.-W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes,, J. Comput. Phys., 77 (1989), 439. doi: 10.1016/0021-9991(88)90177-5. Google Scholar [57] Y. Shu and C.-W. Shu, Local discontinuous Galerkin methods for high-order time-dependent partial differential equations,, Commun. Comput. Phys., 7 (2010), 1. doi: 10.4208/cicp.2009.09.023. Google Scholar [58] J. M. Smit, J. Cernohorsky and C.-P. 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