# 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 and Related Models, 2017, 10 (4) : 879-900. doi: 10.3934/krm.2017035
##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] S. K. Godunov, An interesting class of quasilinear systems, Sov. Math. Dokl., 139 (1961), 521-523. [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. [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. [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. [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. [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. [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. [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. [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. [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. [29] C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.  doi: 10.1007/BF02179552. [30] L. Lewin, Polylogarithms and Associated Functions North-Holland, Amsterdam, 1981. [31] D. Mihalas and B. Weibel-Mihalas, Foundations of Radiation Hydrodynamics Oxford University Press, New York, 1984. [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. [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. [34] I. Müller and T. Ruggeri, Rational Extended Thermodynamics First edition. Springer Tracts in Natural Philosophy, 37. Springer-Verlag, New York, 1993. [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. [36] J. Oxenius, Kinetic Theory of Particles and Photons Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-70728-5. [37] G. C. Pomraning, The Equations of Radiation Hydrodynamics Pergamon Press, Oxford, 1973. [38] F. Qi, Inequalities for an integral, Math. Gaz., 80 (1996), 376-377. [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. [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. [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. [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. [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. [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. [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. [46] J. A. Shohat and J. D. Tamarkin, The Problem of Moments Mathematical Surveys, Vol. 1, American Mathematical Society, New York, 1943. [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. [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. [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. [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. [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. [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.

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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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] S. K. Godunov, An interesting class of quasilinear systems, Sov. Math. Dokl., 139 (1961), 521-523. [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. [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. [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. [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. [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. [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. [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. [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. [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. [29] C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.  doi: 10.1007/BF02179552. [30] L. Lewin, Polylogarithms and Associated Functions North-Holland, Amsterdam, 1981. [31] D. Mihalas and B. Weibel-Mihalas, Foundations of Radiation Hydrodynamics Oxford University Press, New York, 1984. [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. [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. [34] I. Müller and T. Ruggeri, Rational Extended Thermodynamics First edition. Springer Tracts in Natural Philosophy, 37. Springer-Verlag, New York, 1993. [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. [36] J. Oxenius, Kinetic Theory of Particles and Photons Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-70728-5. [37] G. C. Pomraning, The Equations of Radiation Hydrodynamics Pergamon Press, Oxford, 1973. [38] F. Qi, Inequalities for an integral, Math. Gaz., 80 (1996), 376-377. [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. [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. [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. [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. [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. [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. [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. [46] J. A. Shohat and J. D. Tamarkin, The Problem of Moments Mathematical Surveys, Vol. 1, American Mathematical Society, New York, 1943. [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. [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. [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. [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. [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. [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.
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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