# 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:
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##### References:
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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), 841-864. 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), 7561-7586. 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), 541-545. 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. 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