# American Institute of Mathematical Sciences

December  2017, 10(4): 879-900. doi: 10.3934/krm.2017035

## Entropy-based mixed three-moment description of fermionic radiation transport in slab and spherical geometries

 Polish Academy of Sciences, Institute of Fundamental Technological Research, Department of Theory of Continuous Media, Pawinskiego 5B, 02-106 Warsaw, Poland

* Corresponding author: Zbigniew Banach

Received  June 2016 Revised  November 2016 Published  March 2017

The mixed three-moment hydrodynamic description of fermionic radiation transport based on the Boltzmann entropy optimization procedure is considered for the case of one-dimensional flows. The conditions for realizability of the mixed three moments chosen as the energy density and two partial heat fluxes are established. The domain of admissible values of those moments is determined and the existence of the solution to the optimization problem is proved. Here, the standard approaches related to either the truncated Hausdorff or Markov moment problems do not apply because the non-negative fermionic distribution function, denoted $f$, must satisfy the inequality $f≤q 1$ and, at the same time, there are three different intervals of integration in the integral formulae defining the mixed moments. The hydrodynamic equations are obtained in the form of the symmetric hyperbolic system for the Lagrange multipliers of the optimization problem with constraints. The potentials generating this system are explicitly determined as dilogarithm and trilogarithm functions of the Lagrange multipliers. The invertibility of the relation between moments and Lagrange multipliers is proved. However, the inverse relation cannot be determined in a closed analytic form. Using the $H$-theorem for the radiative transfer equation, it is shown that the derived system of hydrodynamic radiation equations has as a consequence an additional balance law with a non-negative source term.

Citation: Zbigniew Banach, Wieslaw Larecki. Entropy-based mixed three-moment description of fermionic radiation transport in slab and spherical geometries. Kinetic & Related Models, 2017, 10 (4) : 879-900. doi: 10.3934/krm.2017035
##### References:
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Larecki, One-dimensional maximum entropy radiation hydrodynamics:Three-moment theory, J. Phys. A:Math. Theor., 45 (2012), 385501, 24pp. doi: 10.1088/1751-8113/45/38/385501. Google Scholar [6] Z. Banach and W. Larecki, Spectral maximum entropy hydrodynamics of fermionic radiation:A three-moment system for one-dimensional flows, Nonlinearity, 26 (2013), 1667-1701. doi: 10.1088/0951-7715/26/6/1667. Google Scholar [7] G. Boillat, Sur l'existence et la recherche d'équations de conservation supplémentaires pour les systémes hyperboliques, C. R. Acad. Sci. Paris A, 278 (1974), 909-912. Google Scholar [8] G. Boillat, Non-linear hyperbolic fields and waves, Recent Mathematical Methods in Nonlinear Wave Propagation, Lecture Notes in Mathematics, 1640 (2006), 1-47. doi: 10.1007/BFb0093705. Google Scholar [9] J. M. Borwein and A. S. Lewis, Duality relationships for entropy-like minimization problems, SIAM J. Control Optim., 29 (1991), 325-338. doi: 10.1137/0329017. Google Scholar [10] J. M. Borwein and W. Huang, Uniform convergence for moment problems with Fermi-Dirac type entropies, Math. Meth. Oper. Res., 40 (1994), 239-252. doi: 10.1007/BF01432968. Google Scholar [11] J. M. Borwein, Maximum entropy and feasibility methods for convex and nonconvex inverse problems, Optimization, 61 (2012), 1-33. doi: 10.1080/02331934.2011.632502. Google Scholar [12] J. Cernohorsky, L. J. van den Horn and J. Cooperstein, Maximum entropy Eddington factors in flux-limited neutrino diffusion, J. Quant. Spectrosc. Radiat. Transfer, 42 (1989), 603-613. doi: 10.1016/0022-4073(89)90054-X. Google Scholar [13] J. Cernohorsky and S. A. Bludman, Maximum entropy distribution and closure for Bose-Einstein and Fermi-Dirac radiation transport, Astrophys. J., 433 (1994), 250-255. doi: 10.1086/174640. Google Scholar [14] 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. Google Scholar [15] B. Dubroca, M. Frank, A. Klar and G. Thömmes, A half space moment approximation to the radiative heat transfer equations, Z. Angew. Math. Phys., 83 (2003), 583-858. doi: 10.1002/zamm.200310055. Google Scholar [16] 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. Google Scholar [17] M. Frank, H. Hensel and A. Klar, A fast and accurate moment method for the Fokker-Planck equation and applications to electron radiotherapy, SIAM J. Appl. Math., 67 (2007), 582-603. doi: 10.1137/06065547X. Google Scholar [18] K. Friedrichs and P. Lax, Systems of conservation equations with a convex extension, Proc. Natl Acad. Sci. USA, 68 (1971), 1686-1688. doi: 10.1073/pnas.68.8.1686. Google Scholar [19] S. K. Godunov, An interesting class of quasilinear systems, Sov. Math. Dokl., 139 (1961), 521-523. Google Scholar [20] C. D. Hauck, High-order entropy-based closures for linear transport in slab geometry, Commun. Math. Sci., 9 (2011), 187-205. doi: 10.4310/CMS.2011.v9.n1.a9. Google Scholar [21] H. T. Janka, R. Dgani and L. J. van den Horn, Fermion angular distribution and maximum entropy Eddington factors, Astron. Astrophys., 265 (1992), 345-354. Google Scholar [22] D. S. Kershaw, Flux Limiting Nature's Own Way: A New Method for Numerical Solution of the Transport Equation UCRL-78378, Lewrence Livermore National laboratory, 1976.Google Scholar [23] M. G. Krein and A. A. Nudelman, The Markov Moment Problem and the Extremal Problems Translations of Mathematical Monographs, Vol. 50, American Mathematical Society, Providence, RI, 1977. Google Scholar [24] W. Larecki and S. Piekarski, Phonon gas hydrodynamics based on the maximum entropy principle and the extended field theory of a rigid conductor of heat, Arch. Mech., 43 (1991), 163-190. Google Scholar [25] W. Larecki and S. Piekarski, Symmetric conservative form of low-temperature phonon gas hydrodynamics Ⅰ:kinetic aspects of the theory, Nuovo Cimento D, 13 (1991), 31-176. Google Scholar [26] W. Larecki and Z. Banach, Entropic derivation of the spectral Eddington factors, J. Quant. Spectrosc. Radiat. Transfer, 112 (2011), 2486-2506. doi: 10.1016/j.jqsrt.2011.06.011. Google Scholar [27] W. Larecki and Z. Banach, Two-field radiation hydrodynamics in $n$ spatial dimensions, J. Phys. A:Math. Theor., 49 (2016), 125501, 23pp. doi: 10.1088/1751-8113/49/12/125501. Google Scholar [28] C. D. Levermore, Relating Eddington factors to flux limiters, J. Quant. Spectrosc. Radiat. Transfer, 31 (1984), 149-160. doi: 10.1016/0022-4073(84)90112-2. Google Scholar [29] C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552. Google Scholar [30] L. Lewin, Polylogarithms and Associated Functions North-Holland, Amsterdam, 1981. Google Scholar [31] D. Mihalas and B. Weibel-Mihalas, Foundations of Radiation Hydrodynamics Oxford University Press, New York, 1984. Google Scholar [32] D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Classical and New Inequalities in Analysis Kluwer, Dordrecht, 1993. doi: 10.1007/978-94-017-1043-5. Google Scholar [33] P. Monreal, Moment Realizability and Kershaw Closures in Radiative Transfer Ph. D thesis, RWTH Aachen University, 2012. Available from: http://publications. rwth-aachen. de/record/210538/files/4482.pdf.Google Scholar [34] I. Müller and T. Ruggeri, Rational Extended Thermodynamics First edition. Springer Tracts in Natural Philosophy, 37. Springer-Verlag, New York, 1993.Google Scholar [35] I. Müller and T. Ruggeri, Rational Extended Thermodynamics Second edition. With supplementary chapters by H. Struchtrup and W. Weiss. Springer Tracts in Natural Philosophy, 37. Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-2210-1. Google Scholar [36] J. Oxenius, Kinetic Theory of Particles and Photons Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-70728-5. Google Scholar [37] G. C. Pomraning, The Equations of Radiation Hydrodynamics Pergamon Press, Oxford, 1973.Google Scholar [38] F. Qi, Inequalities for an integral, Math. Gaz., 80 (1996), 376-377. Google Scholar [39] J. F. Ripoll and A. A. Wray, A half-moment model for radiative transfer in a 3D gray medium and its reduction to a moment model for hot, opaque sources, J. Quant. Spectrosc. Radiat. Transfer, 93 (2005), 473-519. doi: 10.1016/j.jqsrt.2004.09.040. Google Scholar [40] J. F. Ripoll and A. A. Wray, A 3-D multiband closure for radiation and neutron transfer moment models, J. Comput. Phys., 227 (2008), 2212-2237. doi: 10.1016/j.jcp.2007.08.028. Google Scholar [41] T. Ruggeri and A. Strumia, Main field and convex covariant density for quasi-linear hyperbolic systems. Relativistic fluid dynamics, Ann. Inst. Henri Poincaré, 34 (1981), 65-84. Google Scholar [42] M. Schäffer, M. Frank and R. Pinnau, A hierarchy of approximations to the radiative heat transfer equations:Modelling, analysis and simulation, Math. Mod. Meth. Appl. Sci., 15 (2005), 643-665. doi: 10.1142/S0218202505000479. Google Scholar [43] F. Schneider, G. W. Alldredge, M. Frank and A. Klar, Higher order mixed-moment approximations for the Fokker-Planck equation in one space dimension, SIAM J. Appl. Math., 74 (2014), 1087-1114. doi: 10.1137/130934210. Google Scholar [44] F. Schneider, J. Kall and G. W. Alldredge, A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry, Kin. Rel. Mod., 9 (2016), 193-215. doi: 10.3934/krm.2016.9.193. Google Scholar [45] F. Schneider, Kershaw closures for linear transport equations in slab geometry Ⅰ:Model derivation, J. Comput. Phys., 322 (2016), 905-919. doi: 10.1016/j.jcp.2016.02.080. Google Scholar [46] J. A. Shohat and J. D. Tamarkin, The Problem of Moments Mathematical Surveys, Vol. 1, American Mathematical Society, New York, 1943. Google Scholar [47] J. M. Smit, J. Cernohorsky and C. P. Dullemond, Hyperbolicity and critical points in two-moment approximate radiative transfer, Astron. Astrophys., 325 (1997), 203-211. Google Scholar [48] J. M. Smit, L. J. van den Horn and S. A. Bludman, Closure in flux-limited neutrino diffusion and two-moment transport, Astrophys. J., 356 (2000), 559-569. Google Scholar [49] R. Turpault, M. Frank, B. Dubroca and A. Klar, Multigroup half space moment approximations to the radiative heat transfer equations, J. Comput. Phys., 198 (2004), 363-371. Google Scholar [50] R. Turpault, Properties and frequential hybridisation of the multigroup $M$1 model for radiative transfer, Nonlinear Anal. Real World Appl., 11 (2010), 2514-2528. doi: 10.1016/j.nonrwa.2009.08.008. Google Scholar [51] N. M. H. Vaytet, E. Audit, B. Dubroca and F. Delahaye, A numerical model for multigroup radiation hydrodynamics, J. Quant. Spectrosc. Radiat. Transfer, 112 (2011), 1323-1335. doi: 10.1016/j.jqsrt.2011.01.027. Google Scholar [52] V. Vikas, C. D. Hauck, Z. J. Wang and R. O. Fox, Radiation transport modeling using extended quadrature method of moments, J. Comput. Phys., 246 (2013), 221-241. doi: 10.1016/j.jcp.2013.03.028. Google Scholar

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##### References:
 [1] R. P. Agarwal and S. S. Dragomir, An application of Hayashi's inequality for differentiable functions, Comput. Math. Appl., 32 (1996), 95-99. doi: 10.1016/0898-1221(96)00146-0. Google Scholar [2] G. W. 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. Google Scholar [3] G. W. Alldredge and F. Schneider, A realizability-preserving discontinuous Galerkin scheme for entropy-based moment closures for linear kinetic equations in one space dimension, J. Comput. Phys., 295 (2015), 665-684. doi: 10.1016/j.jcp.2015.04.034. Google Scholar [4] G. W. Alldredge, R. Li and W. Li, Approximating the $M2$ method by the extended quadrature method of moments for radiative transfer in slab geometry, Kin. Rel. Mod., 9 (2016), 237-249. doi: 10.3934/krm.2016.9.237. Google Scholar [5] Z. Banach and W. Larecki, One-dimensional maximum entropy radiation hydrodynamics:Three-moment theory, J. Phys. A:Math. Theor., 45 (2012), 385501, 24pp. doi: 10.1088/1751-8113/45/38/385501. Google Scholar [6] Z. Banach and W. Larecki, Spectral maximum entropy hydrodynamics of fermionic radiation:A three-moment system for one-dimensional flows, Nonlinearity, 26 (2013), 1667-1701. doi: 10.1088/0951-7715/26/6/1667. Google Scholar [7] G. Boillat, Sur l'existence et la recherche d'équations de conservation supplémentaires pour les systémes hyperboliques, C. R. Acad. Sci. Paris A, 278 (1974), 909-912. Google Scholar [8] G. Boillat, Non-linear hyperbolic fields and waves, Recent Mathematical Methods in Nonlinear Wave Propagation, Lecture Notes in Mathematics, 1640 (2006), 1-47. doi: 10.1007/BFb0093705. Google Scholar [9] J. M. Borwein and A. S. Lewis, Duality relationships for entropy-like minimization problems, SIAM J. Control Optim., 29 (1991), 325-338. doi: 10.1137/0329017. Google Scholar [10] J. M. Borwein and W. Huang, Uniform convergence for moment problems with Fermi-Dirac type entropies, Math. Meth. Oper. Res., 40 (1994), 239-252. doi: 10.1007/BF01432968. Google Scholar [11] J. M. Borwein, Maximum entropy and feasibility methods for convex and nonconvex inverse problems, Optimization, 61 (2012), 1-33. doi: 10.1080/02331934.2011.632502. Google Scholar [12] J. Cernohorsky, L. J. van den Horn and J. Cooperstein, Maximum entropy Eddington factors in flux-limited neutrino diffusion, J. Quant. Spectrosc. Radiat. Transfer, 42 (1989), 603-613. doi: 10.1016/0022-4073(89)90054-X. Google Scholar [13] J. Cernohorsky and S. A. Bludman, Maximum entropy distribution and closure for Bose-Einstein and Fermi-Dirac radiation transport, Astrophys. J., 433 (1994), 250-255. doi: 10.1086/174640. Google Scholar [14] 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. Google Scholar [15] B. Dubroca, M. Frank, A. Klar and G. Thömmes, A half space moment approximation to the radiative heat transfer equations, Z. Angew. Math. Phys., 83 (2003), 583-858. doi: 10.1002/zamm.200310055. Google Scholar [16] 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. Google Scholar [17] M. Frank, H. Hensel and A. Klar, A fast and accurate moment method for the Fokker-Planck equation and applications to electron radiotherapy, SIAM J. Appl. Math., 67 (2007), 582-603. doi: 10.1137/06065547X. Google Scholar [18] K. Friedrichs and P. Lax, Systems of conservation equations with a convex extension, Proc. Natl Acad. Sci. USA, 68 (1971), 1686-1688. doi: 10.1073/pnas.68.8.1686. Google Scholar [19] S. K. Godunov, An interesting class of quasilinear systems, Sov. Math. Dokl., 139 (1961), 521-523. Google Scholar [20] C. D. Hauck, High-order entropy-based closures for linear transport in slab geometry, Commun. Math. Sci., 9 (2011), 187-205. doi: 10.4310/CMS.2011.v9.n1.a9. Google Scholar [21] H. T. Janka, R. Dgani and L. J. van den Horn, Fermion angular distribution and maximum entropy Eddington factors, Astron. Astrophys., 265 (1992), 345-354. Google Scholar [22] D. S. Kershaw, Flux Limiting Nature's Own Way: A New Method for Numerical Solution of the Transport Equation UCRL-78378, Lewrence Livermore National laboratory, 1976.Google Scholar [23] M. G. Krein and A. A. Nudelman, The Markov Moment Problem and the Extremal Problems Translations of Mathematical Monographs, Vol. 50, American Mathematical Society, Providence, RI, 1977. Google Scholar [24] W. Larecki and S. Piekarski, Phonon gas hydrodynamics based on the maximum entropy principle and the extended field theory of a rigid conductor of heat, Arch. Mech., 43 (1991), 163-190. Google Scholar [25] W. Larecki and S. Piekarski, Symmetric conservative form of low-temperature phonon gas hydrodynamics Ⅰ:kinetic aspects of the theory, Nuovo Cimento D, 13 (1991), 31-176. Google Scholar [26] W. Larecki and Z. Banach, Entropic derivation of the spectral Eddington factors, J. Quant. Spectrosc. Radiat. Transfer, 112 (2011), 2486-2506. doi: 10.1016/j.jqsrt.2011.06.011. Google Scholar [27] W. Larecki and Z. Banach, Two-field radiation hydrodynamics in $n$ spatial dimensions, J. Phys. A:Math. Theor., 49 (2016), 125501, 23pp. doi: 10.1088/1751-8113/49/12/125501. Google Scholar [28] C. D. Levermore, Relating Eddington factors to flux limiters, J. Quant. Spectrosc. Radiat. Transfer, 31 (1984), 149-160. doi: 10.1016/0022-4073(84)90112-2. Google Scholar [29] C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552. Google Scholar [30] L. Lewin, Polylogarithms and Associated Functions North-Holland, Amsterdam, 1981. Google Scholar [31] D. Mihalas and B. Weibel-Mihalas, Foundations of Radiation Hydrodynamics Oxford University Press, New York, 1984. Google Scholar [32] D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Classical and New Inequalities in Analysis Kluwer, Dordrecht, 1993. doi: 10.1007/978-94-017-1043-5. Google Scholar [33] P. Monreal, Moment Realizability and Kershaw Closures in Radiative Transfer Ph. D thesis, RWTH Aachen University, 2012. Available from: http://publications. rwth-aachen. de/record/210538/files/4482.pdf.Google Scholar [34] I. Müller and T. Ruggeri, Rational Extended Thermodynamics First edition. Springer Tracts in Natural Philosophy, 37. Springer-Verlag, New York, 1993.Google Scholar [35] I. Müller and T. Ruggeri, Rational Extended Thermodynamics Second edition. With supplementary chapters by H. Struchtrup and W. Weiss. Springer Tracts in Natural Philosophy, 37. Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-2210-1. Google Scholar [36] J. Oxenius, Kinetic Theory of Particles and Photons Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-70728-5. Google Scholar [37] G. C. Pomraning, The Equations of Radiation Hydrodynamics Pergamon Press, Oxford, 1973.Google Scholar [38] F. Qi, Inequalities for an integral, Math. Gaz., 80 (1996), 376-377. Google Scholar [39] J. F. Ripoll and A. A. Wray, A half-moment model for radiative transfer in a 3D gray medium and its reduction to a moment model for hot, opaque sources, J. Quant. Spectrosc. Radiat. Transfer, 93 (2005), 473-519. doi: 10.1016/j.jqsrt.2004.09.040. Google Scholar [40] J. F. Ripoll and A. A. Wray, A 3-D multiband closure for radiation and neutron transfer moment models, J. Comput. Phys., 227 (2008), 2212-2237. doi: 10.1016/j.jcp.2007.08.028. Google Scholar [41] T. Ruggeri and A. Strumia, Main field and convex covariant density for quasi-linear hyperbolic systems. Relativistic fluid dynamics, Ann. Inst. Henri Poincaré, 34 (1981), 65-84. Google Scholar [42] M. Schäffer, M. Frank and R. Pinnau, A hierarchy of approximations to the radiative heat transfer equations:Modelling, analysis and simulation, Math. Mod. Meth. Appl. Sci., 15 (2005), 643-665. doi: 10.1142/S0218202505000479. Google Scholar [43] F. Schneider, G. W. Alldredge, M. Frank and A. Klar, Higher order mixed-moment approximations for the Fokker-Planck equation in one space dimension, SIAM J. Appl. Math., 74 (2014), 1087-1114. doi: 10.1137/130934210. Google Scholar [44] F. Schneider, J. Kall and G. W. Alldredge, A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry, Kin. Rel. Mod., 9 (2016), 193-215. doi: 10.3934/krm.2016.9.193. Google Scholar [45] F. Schneider, Kershaw closures for linear transport equations in slab geometry Ⅰ:Model derivation, J. Comput. Phys., 322 (2016), 905-919. doi: 10.1016/j.jcp.2016.02.080. Google Scholar [46] J. A. Shohat and J. D. Tamarkin, The Problem of Moments Mathematical Surveys, Vol. 1, American Mathematical Society, New York, 1943. Google Scholar [47] J. M. Smit, J. Cernohorsky and C. P. Dullemond, Hyperbolicity and critical points in two-moment approximate radiative transfer, Astron. Astrophys., 325 (1997), 203-211. Google Scholar [48] J. M. Smit, L. J. van den Horn and S. A. Bludman, Closure in flux-limited neutrino diffusion and two-moment transport, Astrophys. J., 356 (2000), 559-569. Google Scholar [49] R. Turpault, M. Frank, B. Dubroca and A. Klar, Multigroup half space moment approximations to the radiative heat transfer equations, J. Comput. Phys., 198 (2004), 363-371. Google Scholar [50] R. Turpault, Properties and frequential hybridisation of the multigroup $M$1 model for radiative transfer, Nonlinear Anal. Real World Appl., 11 (2010), 2514-2528. doi: 10.1016/j.nonrwa.2009.08.008. Google Scholar [51] N. M. H. Vaytet, E. Audit, B. Dubroca and F. Delahaye, A numerical model for multigroup radiation hydrodynamics, J. Quant. Spectrosc. Radiat. Transfer, 112 (2011), 1323-1335. doi: 10.1016/j.jqsrt.2011.01.027. Google Scholar [52] V. Vikas, C. D. Hauck, Z. J. Wang and R. O. Fox, Radiation transport modeling using extended quadrature method of moments, J. Comput. Phys., 246 (2013), 221-241. doi: 10.1016/j.jcp.2013.03.028. Google Scholar
The set $\Omega$ and its location in $\mathbb{R}^3$. In the 'horizontal' directions, we use the notation $\mbox{'energy'}$ for $\varepsilon$ and $\mbox{'heat'}$ for $q_{ +}$. The tick marks along the 'vertical' direction represent the values of $q_{ -}$
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