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Fast diffusion equations: Matching large time asymptotics by relative entropy methods
Optimal prediction for radiative transfer: A new perspective on moment closure
1. | RWTH Aachen University, Department of Mathematics, Schinkelstrasse 2 52062 Aachen, Germany |
2. | Temple University, Department of Mathematics, 1805 North Broad Street Philadelphia, PA 19122, United States |
References:
[1] |
A. M. Anile, S. Pennisi and M. Sammartino, A thermodynamical approach to Eddington factors, J. Math. Phys., 32 (1991), 544-550.
doi: 10.1063/1.529391. |
[2] |
J. Bell, A. J. Chorin and W. Crutchfield, Stochastic optimal prediction with application to averaged Euler equations, Proc. 7th Nat. Conf. CFD, 2000, 1-13. |
[3] |
Y. M. Berezansky and Y. G. Kondratiev, "Spectral Methods in Infinite-Dimensional Analysis," Kluwer Academic Publishers, Dordrecht, 1995. |
[4] |
P. S. Brantley and E. W. Larsen, The simplified $P_3$ approximation, Nucl. Sci. Eng., 134 (2000), 1. |
[5] |
P. N. Brown, B. Chang, U. R. Hanebutte and J. A. Rathkopf, "Spherical Harmonic Solutions to the 3d Kobayashi Benchmark Suite," Technical Report UCRL-VG-135163, Lawrence Livermore National Laboratory, May 2000. |
[6] |
T. A. Brunner, "Forms of Approximate Radiation Transport," Technical Report SAND2002-1778, Sandia National Laboratories, July 2002. |
[7] |
T. A. Brunner and J. P. Holloway, Two-dimensional time dependent Riemann solvers for neutron transport, J. Comput. Phys., 210 (2005), 386-399.
doi: 10.1016/j.jcp.2005.04.011. |
[8] |
S. Chandrasekhar, On the radiative equilibrium of a stellar atmosphere, Astrophys. J., 99 (1944), 180-190.
doi: 10.1086/144606. |
[9] |
_____, "Radiative Transfer," Dover Publications, Inc., New York, 1960. |
[10] |
A. J. Chorin, Conditional expectations and renormalization, Multiscale Model. Simul., 1 (2003), 105-118.
doi: 10.1137/S1540345902405556. |
[11] |
A. J. Chorin and O. H. Hald, "Stochastic Tools in Mathematics and Science," Surveys and Tutorials in the Applied Mathematical Sciences, 1, Springer, New York, 2006. |
[12] |
A. J. Chorin, O. H. Hald, and R. Kupferman, Optimal prediction and the Mori-Zwanzig representation of irreversible processes, Proc. Natl. Acad. Sci. USA, 97 (2000), 2968-2973.
doi: 10.1073/pnas.97.7.2968. |
[13] |
_____, Non-Markovian optimal prediction, Monte Carlo adn Probablilistic Methods for Patial Differential Equations (Monte Carlo, 2000), Monte Carlo Meth. Appl., 7 (2001), 99-109. |
[14] |
_____, Optimal prediction with memory, Physica D, 166 (2002), 239-257.
doi: 10.1016/S0167-2789(02)00446-3. |
[15] |
A. J. Chorin, A. P. Kast and R. Kupferman, Optimal prediction of underresolved dynamics, Proc. Natl. Acad. Sci. USA, 95 (1998), 4094-4098.
doi: 10.1073/pnas.95.8.4094. |
[16] |
_____, Unresolved computation and optimal predictions, Comm. Pure Appl. Math., 52 (1998), 1231-1254. |
[17] |
_____, "On the Prediction of Large-Scale Dynamics using Unresolved Computations," Nonlinear Partial Differential Equations (Evanston, IL, 1998), Contemp. Math., 238, AMS, Providence, RI, (1999), 53-75. |
[18] |
A. J. Chorin and P. Stinis, Problem reduction, renormalization, and memory, Comm. Appl. Math. Comp. Sc., 1 (2006), 1-27.
doi: 10.2140/camcos.2006.1.1. |
[19] |
B. Davison, "Neutron Transport Theory," Clarendon Press, Oxford, 1958. |
[20] |
B. Dubroca and J. L. Feugeas, Theoretical and numerical study of a moment closure hierarchy for the radiative transfer equation, C. R. Acad. Sci. Paris Ser. I, 329 (1999), 915-920. |
[21] |
B. Dubroca, M. Frank, A. Klar and G. Thömmes, A Half space moment approximation to the radiative heat transfer equations, Z. Angew. Math. Mech., 83 (2003), 853-858.
doi: 10.1002/zamm.200310055. |
[22] |
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. |
[23] |
M. Frank, A. Klar, E. W. Larsen and S. Yasuda, Time-dependent simplified $P_N$ approximation to the equations of radiative transfer, J. Comput. Phys., 226 (2007), 2289-2305.
doi: 10.1016/j.jcp.2007.07.009. |
[24] |
M. Frank, A. Klar and R. Pinnau, Optimal control of glass cooling using Simplified $P_n$ theory, Transp. Theory. Stat. Phys., 39 (2010), 282-311.
doi: 10.1080/00411450.2010.533740. |
[25] |
E. M. Gelbard, "Applications of Spherical Harmonics Method to Reactor Problems," Tech. Report WAPD-BT-20, Bettis Atomic Power Laboratory, 1960. |
[26] |
_____, "Simplified Spherical Harmonics Equations and their Use in Shielding Problems," Tech. Report WAPD-T-1182, Bettis Atomic Power Laboratory, 1961. |
[27] |
_____, "Applications of the Simplified Spherical Harmonics Equations in Spherical Geometry," Tech. Report WAPD-TM-294, Bettis Atomic Power Laboratory, 1962. |
[28] |
D. Givon, O. Hald and R. Kupferman, Existence proof for orthogonal dynamics and the Mori-Zwanzig formalism, Israel J. Math., 145 (2005), 221-241.
doi: 10.1007/BF02786691. |
[29] |
T. Hida, H.-H. Kuo, J. Potthoff and L. Streit, "White Noise. An Infinite Dimensional Calculus," Mathematics and its Applications, 253, Kluwer Academic Publishers Group, Dordrecht, 1993. |
[30] |
S. Karlin and L. S. Shapley, Geometry of moment spaces, Mem. Amer. Math. Soc., (1953), 93 pp. |
[31] |
D. S. Kershaw, "Flux Limiting Nature's Own Way," Tech. Report UCRL-78378, Lawrence Livermore National Laboratory, 1976. |
[32] |
D. A. Knoll, W. J. Rider and G. L. Olson, Method for non-equilibrium radiation diffusion, J. Quant. Spectrosc. Radiat. Transfer, 63 (1999), 15-29.
doi: 10.1016/S0022-4073(98)00132-0. |
[33] |
E. W. Larsen and J. R. Keller, Asymptotic solution of neutron transport problems for small mean free paths, J. Math. Phys., 15 (1974), 75-81.
doi: 10.1063/1.1666510. |
[34] |
E. W. Larsen, J. E. Morel and J. M. McGhee, Asymptotic derivation of the multigroup $P_1$ and simplified $P_N$ equations with anisotropic scattering, Nucl. Sci. Eng., 123 (1996), 328-342. |
[35] |
E. W. Larsen and G. C. Pomraning, The $P_N$ theory as an asymptotic limit of transport theory in planar geometry - I: analysis, Nucl. Sci. Eng., 109 (1991), 49-75. |
[36] |
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. |
[37] |
_____, "Transition Regime Models for Radiative Transport," Presentation at IPAM: Grand challenge problems in computational astrophysics workshop on transfer phenomena, 2005. |
[38] |
R. G. McClarren, Theoretical aspects of the Simplified $P_n$ equations, Transp. Theory. Stat. Phys., 39 (2010), 73-109.
doi: 10.1080/00411450.2010.535088. |
[39] |
R. G. McClarren and C. D. Hauck, Robust and accurate filtered spherical Harmonics expansions for radiative transfer, J. Comput. Phys., 229 (2010), 5597-5614.
doi: 10.1016/j.jcp.2010.03.043. |
[40] |
M. F. Modest, "Radiative Heat Transfer," second ed., Academic Press, 1993. |
[41] |
H. Mori, Transport, collective motion and Brownian motion, Prog. Theor. Phys., 33 (1965), 423-455.
doi: 10.1143/PTP.33.423. |
[42] |
I. Müller and T. Ruggeri, "Rational Extended Thermodynamics," second ed., Springer, New York, 1993. |
[43] |
W. H. Reed, Spherical Harmonic solutions of the neutron transport equation from discrete ordinates code, Nucl. Sci. Eng., 49 (1972), 10-19. |
[44] |
M. Schäfer, M. Frank and C. D. Levermore, Diffusive corrections to $P_N$ approximations, Multiscale Model. Simul., 9 (2011), 1-28. |
[45] |
B. Seibold, Optimal prediction in molecular dynamics, Monte Carlo Methods Appl., 10 (2004), 25-50.
doi: 10.1515/156939604323091199. |
[46] |
B. Seibold and M. Frank, Optimal prediction for moment models: Crescendo diffusion and reordered equations, Continuum Mech. Thermodyn., 21 (2009), 511-527.
doi: 10.1007/s00161-009-0111-7. |
[47] |
M. Skibinsky, The range of the $(n+1)$th moment for distributions on $[0,1]$, J. Appl. Probability, 4 (1967), 543-552.
doi: 10.2307/3212220. |
[48] |
B. Su, Variable Eddington factors and flux limiters in radiative transfer, Nucl. Sci. Eng., 137 (2001), 281-297. |
[49] |
D. I. Tomasevic and E. W. Larsen, The simplified $P_2$ approximation, Nucl. Sci. Eng., 122 (1996), 309-325. |
[50] |
R. Turpault, M. Frank, B. Dubroca and A. Klar, Multigroup half space moment appproximations to the radiative heat transfer equations, J. Comput. Phys., 198 (2004), 363-371.
doi: 10.1016/j.jcp.2004.01.011. |
[51] |
R. Zwanzig, Problems in nonlinear transport theory, in "Systems Far from Equilibrium" (Berlin) (ed., L. Garrido), Springer, (1980), 198-221.
doi: 10.1007/BFb0025619. |
show all references
References:
[1] |
A. M. Anile, S. Pennisi and M. Sammartino, A thermodynamical approach to Eddington factors, J. Math. Phys., 32 (1991), 544-550.
doi: 10.1063/1.529391. |
[2] |
J. Bell, A. J. Chorin and W. Crutchfield, Stochastic optimal prediction with application to averaged Euler equations, Proc. 7th Nat. Conf. CFD, 2000, 1-13. |
[3] |
Y. M. Berezansky and Y. G. Kondratiev, "Spectral Methods in Infinite-Dimensional Analysis," Kluwer Academic Publishers, Dordrecht, 1995. |
[4] |
P. S. Brantley and E. W. Larsen, The simplified $P_3$ approximation, Nucl. Sci. Eng., 134 (2000), 1. |
[5] |
P. N. Brown, B. Chang, U. R. Hanebutte and J. A. Rathkopf, "Spherical Harmonic Solutions to the 3d Kobayashi Benchmark Suite," Technical Report UCRL-VG-135163, Lawrence Livermore National Laboratory, May 2000. |
[6] |
T. A. Brunner, "Forms of Approximate Radiation Transport," Technical Report SAND2002-1778, Sandia National Laboratories, July 2002. |
[7] |
T. A. Brunner and J. P. Holloway, Two-dimensional time dependent Riemann solvers for neutron transport, J. Comput. Phys., 210 (2005), 386-399.
doi: 10.1016/j.jcp.2005.04.011. |
[8] |
S. Chandrasekhar, On the radiative equilibrium of a stellar atmosphere, Astrophys. J., 99 (1944), 180-190.
doi: 10.1086/144606. |
[9] |
_____, "Radiative Transfer," Dover Publications, Inc., New York, 1960. |
[10] |
A. J. Chorin, Conditional expectations and renormalization, Multiscale Model. Simul., 1 (2003), 105-118.
doi: 10.1137/S1540345902405556. |
[11] |
A. J. Chorin and O. H. Hald, "Stochastic Tools in Mathematics and Science," Surveys and Tutorials in the Applied Mathematical Sciences, 1, Springer, New York, 2006. |
[12] |
A. J. Chorin, O. H. Hald, and R. Kupferman, Optimal prediction and the Mori-Zwanzig representation of irreversible processes, Proc. Natl. Acad. Sci. USA, 97 (2000), 2968-2973.
doi: 10.1073/pnas.97.7.2968. |
[13] |
_____, Non-Markovian optimal prediction, Monte Carlo adn Probablilistic Methods for Patial Differential Equations (Monte Carlo, 2000), Monte Carlo Meth. Appl., 7 (2001), 99-109. |
[14] |
_____, Optimal prediction with memory, Physica D, 166 (2002), 239-257.
doi: 10.1016/S0167-2789(02)00446-3. |
[15] |
A. J. Chorin, A. P. Kast and R. Kupferman, Optimal prediction of underresolved dynamics, Proc. Natl. Acad. Sci. USA, 95 (1998), 4094-4098.
doi: 10.1073/pnas.95.8.4094. |
[16] |
_____, Unresolved computation and optimal predictions, Comm. Pure Appl. Math., 52 (1998), 1231-1254. |
[17] |
_____, "On the Prediction of Large-Scale Dynamics using Unresolved Computations," Nonlinear Partial Differential Equations (Evanston, IL, 1998), Contemp. Math., 238, AMS, Providence, RI, (1999), 53-75. |
[18] |
A. J. Chorin and P. Stinis, Problem reduction, renormalization, and memory, Comm. Appl. Math. Comp. Sc., 1 (2006), 1-27.
doi: 10.2140/camcos.2006.1.1. |
[19] |
B. Davison, "Neutron Transport Theory," Clarendon Press, Oxford, 1958. |
[20] |
B. Dubroca and J. L. Feugeas, Theoretical and numerical study of a moment closure hierarchy for the radiative transfer equation, C. R. Acad. Sci. Paris Ser. I, 329 (1999), 915-920. |
[21] |
B. Dubroca, M. Frank, A. Klar and G. Thömmes, A Half space moment approximation to the radiative heat transfer equations, Z. Angew. Math. Mech., 83 (2003), 853-858.
doi: 10.1002/zamm.200310055. |
[22] |
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. |
[23] |
M. Frank, A. Klar, E. W. Larsen and S. Yasuda, Time-dependent simplified $P_N$ approximation to the equations of radiative transfer, J. Comput. Phys., 226 (2007), 2289-2305.
doi: 10.1016/j.jcp.2007.07.009. |
[24] |
M. Frank, A. Klar and R. Pinnau, Optimal control of glass cooling using Simplified $P_n$ theory, Transp. Theory. Stat. Phys., 39 (2010), 282-311.
doi: 10.1080/00411450.2010.533740. |
[25] |
E. M. Gelbard, "Applications of Spherical Harmonics Method to Reactor Problems," Tech. Report WAPD-BT-20, Bettis Atomic Power Laboratory, 1960. |
[26] |
_____, "Simplified Spherical Harmonics Equations and their Use in Shielding Problems," Tech. Report WAPD-T-1182, Bettis Atomic Power Laboratory, 1961. |
[27] |
_____, "Applications of the Simplified Spherical Harmonics Equations in Spherical Geometry," Tech. Report WAPD-TM-294, Bettis Atomic Power Laboratory, 1962. |
[28] |
D. Givon, O. Hald and R. Kupferman, Existence proof for orthogonal dynamics and the Mori-Zwanzig formalism, Israel J. Math., 145 (2005), 221-241.
doi: 10.1007/BF02786691. |
[29] |
T. Hida, H.-H. Kuo, J. Potthoff and L. Streit, "White Noise. An Infinite Dimensional Calculus," Mathematics and its Applications, 253, Kluwer Academic Publishers Group, Dordrecht, 1993. |
[30] |
S. Karlin and L. S. Shapley, Geometry of moment spaces, Mem. Amer. Math. Soc., (1953), 93 pp. |
[31] |
D. S. Kershaw, "Flux Limiting Nature's Own Way," Tech. Report UCRL-78378, Lawrence Livermore National Laboratory, 1976. |
[32] |
D. A. Knoll, W. J. Rider and G. L. Olson, Method for non-equilibrium radiation diffusion, J. Quant. Spectrosc. Radiat. Transfer, 63 (1999), 15-29.
doi: 10.1016/S0022-4073(98)00132-0. |
[33] |
E. W. Larsen and J. R. Keller, Asymptotic solution of neutron transport problems for small mean free paths, J. Math. Phys., 15 (1974), 75-81.
doi: 10.1063/1.1666510. |
[34] |
E. W. Larsen, J. E. Morel and J. M. McGhee, Asymptotic derivation of the multigroup $P_1$ and simplified $P_N$ equations with anisotropic scattering, Nucl. Sci. Eng., 123 (1996), 328-342. |
[35] |
E. W. Larsen and G. C. Pomraning, The $P_N$ theory as an asymptotic limit of transport theory in planar geometry - I: analysis, Nucl. Sci. Eng., 109 (1991), 49-75. |
[36] |
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. |
[37] |
_____, "Transition Regime Models for Radiative Transport," Presentation at IPAM: Grand challenge problems in computational astrophysics workshop on transfer phenomena, 2005. |
[38] |
R. G. McClarren, Theoretical aspects of the Simplified $P_n$ equations, Transp. Theory. Stat. Phys., 39 (2010), 73-109.
doi: 10.1080/00411450.2010.535088. |
[39] |
R. G. McClarren and C. D. Hauck, Robust and accurate filtered spherical Harmonics expansions for radiative transfer, J. Comput. Phys., 229 (2010), 5597-5614.
doi: 10.1016/j.jcp.2010.03.043. |
[40] |
M. F. Modest, "Radiative Heat Transfer," second ed., Academic Press, 1993. |
[41] |
H. Mori, Transport, collective motion and Brownian motion, Prog. Theor. Phys., 33 (1965), 423-455.
doi: 10.1143/PTP.33.423. |
[42] |
I. Müller and T. Ruggeri, "Rational Extended Thermodynamics," second ed., Springer, New York, 1993. |
[43] |
W. H. Reed, Spherical Harmonic solutions of the neutron transport equation from discrete ordinates code, Nucl. Sci. Eng., 49 (1972), 10-19. |
[44] |
M. Schäfer, M. Frank and C. D. Levermore, Diffusive corrections to $P_N$ approximations, Multiscale Model. Simul., 9 (2011), 1-28. |
[45] |
B. Seibold, Optimal prediction in molecular dynamics, Monte Carlo Methods Appl., 10 (2004), 25-50.
doi: 10.1515/156939604323091199. |
[46] |
B. Seibold and M. Frank, Optimal prediction for moment models: Crescendo diffusion and reordered equations, Continuum Mech. Thermodyn., 21 (2009), 511-527.
doi: 10.1007/s00161-009-0111-7. |
[47] |
M. Skibinsky, The range of the $(n+1)$th moment for distributions on $[0,1]$, J. Appl. Probability, 4 (1967), 543-552.
doi: 10.2307/3212220. |
[48] |
B. Su, Variable Eddington factors and flux limiters in radiative transfer, Nucl. Sci. Eng., 137 (2001), 281-297. |
[49] |
D. I. Tomasevic and E. W. Larsen, The simplified $P_2$ approximation, Nucl. Sci. Eng., 122 (1996), 309-325. |
[50] |
R. Turpault, M. Frank, B. Dubroca and A. Klar, Multigroup half space moment appproximations to the radiative heat transfer equations, J. Comput. Phys., 198 (2004), 363-371.
doi: 10.1016/j.jcp.2004.01.011. |
[51] |
R. Zwanzig, Problems in nonlinear transport theory, in "Systems Far from Equilibrium" (Berlin) (ed., L. Garrido), Springer, (1980), 198-221.
doi: 10.1007/BFb0025619. |
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