Approximating the M_2 method by the extended quadrature method of moments for radiative transfer in slab geometry

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June 2016, 9(2): 237-249. doi: 10.3934/krm.2016.9.237

Approximating the $M_2$ method by the extended quadrature method of moments for radiative transfer in slab geometry

Graham W. Alldredge ^1, , Ruo Li ^2, and Weiming Li ^3,

1.	Department of Mathematics RWTH Aachen University, Aachen, Germany
2.	CAPT, LMAM & School of Mathematical Sciences, Peking University, Beijing, China
3.	School of Mathematical Sciences, Peking University, Beijing, China

Received January 2015 Revised October 2015 Published March 2016

Abstract
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We consider the simplest member of the hierarchy of the extended quadrature method of moments (EQMOM), which gives equations for the zeroth-, first-, and second-order moments of the energy density of photons in the radiative transfer equations in slab geometry. First we show that the equations are well-defined for all moment vectors consistent with a nonnegative underlying distribution, and that the reconstruction is explicit and therefore computationally inexpensive. Second, we show that the resulting moment equations are hyperbolic. These two properties make this moment method quite similar to the attractive but far more expensive $M_2$ method. We confirm through numerical solutions to several benchmark problems that the methods give qualitatively similar results.

Keywords: minimum entropy principle., slab geometry, Radiative transfer, moment model.

Mathematics Subject Classification: Primary: 78M05; Secondary: 82C7.

Citation: Graham W. Alldredge, Ruo Li, Weiming Li. Approximating the $M_2$ method by the extended quadrature method of moments for radiative transfer in slab geometry. Kinetic and Related Models, 2016, 9 (2) : 237-249. doi: 10.3934/krm.2016.9.237

References:

[1]	G. W. Alldredge, C. D. Hauck, D. P. O'Leary and A. L. Tits, Adaptive change of basis in entropy-based moment closures for linear kinetic equations, Journal of Computational Physics, 258 (2014), 489-508. doi: 10.1016/j.jcp.2013.10.049.
[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 Journal on Scientific Computing, 34 (2012), B361-B391. doi: 10.1137/11084772X.
[3]	C. Berthon, P. Charrier and B. Dubroca, An hllc scheme to solve the $M_1$ model of radiative transfer in two space dimensions, Journal of Scientific Computing, 31 (2007), 347-389. doi: 10.1007/s10915-006-9108-6.
[4]	T. A. Brunner, Forms of Approximate Radiation Transport, Tech. Rep SAND2002-1778, 2002.
[5]	T. A. Brunner and J. P. Holloway, One-dimensional riemann solvers and the maximum entropy closure, Journal of Quantitative Spectroscopy and Radiative Transfer, 69 (2001), 543-566. doi: 10.1016/S0022-4073(00)00099-6.
[6]	R. Curto and L. Fialkow, Recursiveness, positivity and truncated moment problems, Houston J. Math, 17 (1991), 603-635.
[7]	B. Dubroca and J.-L. Fuegas, Étude théorique et numérique d'une hiérarchie de modèles aus moments pour le transfert radiatif, C.R. Acad. Sci. Paris, I. 329 (1999), 915-920. doi: 10.1016/S0764-4442(00)87499-6.
[8]	C. K. Garrett, C. Hauck and J. Hill, Optimization and large scale computation of an entropy-based moment closure, Journal of Computational Physics, 302 (2015), 573-590.
[9]	C. D. Levermore, Moment closure hierarchies for kinetic theories, Journal of Statistical Physics, 83 (1996), 1021-1065. doi: 10.1007/BF02179552.
[10]	E. E. Lewis and W. F. Miller, Jr, Computational Methods in Neutron Transport, John Wiley and Sons, New York, 1984.
[11]	R. G. McClarren, J. P. Holloway and T. A. Brunner, On solutions to the $P_n$ equations for thermal radiative transfer, Journal of Computational Physics, 227 (2008), 2864-2885. doi: 10.1016/j.jcp.2007.11.027.
[12]	E. Olbrant, C. D. Hauck and M. Frank, A realizability-preserving discontinuous galerkin method for the m1 model of radiative transfer, Journal of Computational Physics, 231 (2012), 5612-5639. doi: 10.1016/j.jcp.2012.03.002.
[13]	G. C. Pomraning, The Equations of Radiation Hydrodynamics, Courier Dover Publications, 1973.
[14]	B. Su and G. L. Olson, An analytical benchmark for non-equilibrium radiative transfer in an isotropically scattering medium, Annals of Nuclear Energy, 24 (1997), 1035-1055. doi: 10.1016/S0306-4549(96)00100-4.
[15]	E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, Springer Science & Business Media, 2009. doi: 10.1007/b79761.
[16]	V. Vikas, C. D. Hauck, Z. J. Wang and R. O. Fox, Radiation transport modeling using extended quadrature method of moments, Journal of Computational Physics, 246 (2013), 221-241. doi: 10.1016/j.jcp.2013.03.028.

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References:

[1]	G. W. Alldredge, C. D. Hauck, D. P. O'Leary and A. L. Tits, Adaptive change of basis in entropy-based moment closures for linear kinetic equations, Journal of Computational Physics, 258 (2014), 489-508. doi: 10.1016/j.jcp.2013.10.049.
[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 Journal on Scientific Computing, 34 (2012), B361-B391. doi: 10.1137/11084772X.
[3]	C. Berthon, P. Charrier and B. Dubroca, An hllc scheme to solve the $M_1$ model of radiative transfer in two space dimensions, Journal of Scientific Computing, 31 (2007), 347-389. doi: 10.1007/s10915-006-9108-6.
[4]	T. A. Brunner, Forms of Approximate Radiation Transport, Tech. Rep SAND2002-1778, 2002.
[5]	T. A. Brunner and J. P. Holloway, One-dimensional riemann solvers and the maximum entropy closure, Journal of Quantitative Spectroscopy and Radiative Transfer, 69 (2001), 543-566. doi: 10.1016/S0022-4073(00)00099-6.
[6]	R. Curto and L. Fialkow, Recursiveness, positivity and truncated moment problems, Houston J. Math, 17 (1991), 603-635.
[7]	B. Dubroca and J.-L. Fuegas, Étude théorique et numérique d'une hiérarchie de modèles aus moments pour le transfert radiatif, C.R. Acad. Sci. Paris, I. 329 (1999), 915-920. doi: 10.1016/S0764-4442(00)87499-6.
[8]	C. K. Garrett, C. Hauck and J. Hill, Optimization and large scale computation of an entropy-based moment closure, Journal of Computational Physics, 302 (2015), 573-590.
[9]	C. D. Levermore, Moment closure hierarchies for kinetic theories, Journal of Statistical Physics, 83 (1996), 1021-1065. doi: 10.1007/BF02179552.
[10]	E. E. Lewis and W. F. Miller, Jr, Computational Methods in Neutron Transport, John Wiley and Sons, New York, 1984.
[11]	R. G. McClarren, J. P. Holloway and T. A. Brunner, On solutions to the $P_n$ equations for thermal radiative transfer, Journal of Computational Physics, 227 (2008), 2864-2885. doi: 10.1016/j.jcp.2007.11.027.
[12]	E. Olbrant, C. D. Hauck and M. Frank, A realizability-preserving discontinuous galerkin method for the m1 model of radiative transfer, Journal of Computational Physics, 231 (2012), 5612-5639. doi: 10.1016/j.jcp.2012.03.002.
[13]	G. C. Pomraning, The Equations of Radiation Hydrodynamics, Courier Dover Publications, 1973.
[14]	B. Su and G. L. Olson, An analytical benchmark for non-equilibrium radiative transfer in an isotropically scattering medium, Annals of Nuclear Energy, 24 (1997), 1035-1055. doi: 10.1016/S0306-4549(96)00100-4.
[15]	E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, Springer Science & Business Media, 2009. doi: 10.1007/b79761.
[16]	V. Vikas, C. D. Hauck, Z. J. Wang and R. O. Fox, Radiation transport modeling using extended quadrature method of moments, Journal of Computational Physics, 246 (2013), 221-241. doi: 10.1016/j.jcp.2013.03.028.

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