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Volume 31, Issue 2
Discrete Duality Finite Volume Discretization of the Thermal-$P_N$ Radiative Transfer Equations on General Meshes

Francois Hermeline

DOI: 10.4208/cicp.OA-2021-0084

Commun. Comput. Phys., 31 (2022), pp. 398-448.

Published online: 2022-01

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  • Abstract

The discrete duality finite volume method has proven to be a practical tool for discretizing partial differential equations coming from a wide variety of areas of physics on nearly arbitrary meshes. The main ingredients of the method are: (1) use of three meshes, (2) use of the Gauss-Green theorem for the approximation of derivatives, (3) discrete integration by parts. In this article we propose to extend this method to the coupled grey thermal-$P_N$ radiative transfer equations in Cartesian and cylindrical coordinates in order to be able to deal with two-dimensional Lagrangian approximations of the interaction of matter with radiation. The stability under a Courant-Friedrichs-Lewy condition and the preservation of the diffusion asymptotic limit are proved while the experimental second-order accuracy is observed with manufactured solutions. Several numerical experiments are reported which show the good behavior of the method.

  • Keywords

Hybrid multiscale models, radiative transfer equation, grey $P_N$ approximation, discrete duality finite volume method.

  • AMS Subject Headings

65N06, 65N08, 65N12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-31-398, author = {Hermeline , Francois}, title = {Discrete Duality Finite Volume Discretization of the Thermal-$P_N$ Radiative Transfer Equations on General Meshes}, journal = {Communications in Computational Physics}, year = {2022}, volume = {31}, number = {2}, pages = {398--448}, abstract = {

The discrete duality finite volume method has proven to be a practical tool for discretizing partial differential equations coming from a wide variety of areas of physics on nearly arbitrary meshes. The main ingredients of the method are: (1) use of three meshes, (2) use of the Gauss-Green theorem for the approximation of derivatives, (3) discrete integration by parts. In this article we propose to extend this method to the coupled grey thermal-$P_N$ radiative transfer equations in Cartesian and cylindrical coordinates in order to be able to deal with two-dimensional Lagrangian approximations of the interaction of matter with radiation. The stability under a Courant-Friedrichs-Lewy condition and the preservation of the diffusion asymptotic limit are proved while the experimental second-order accuracy is observed with manufactured solutions. Several numerical experiments are reported which show the good behavior of the method.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0084}, url = {http://global-sci.org/intro/article_detail/cicp/20211.html} }
TY - JOUR T1 - Discrete Duality Finite Volume Discretization of the Thermal-$P_N$ Radiative Transfer Equations on General Meshes AU - Hermeline , Francois JO - Communications in Computational Physics VL - 2 SP - 398 EP - 448 PY - 2022 DA - 2022/01 SN - 31 DO - http://doi.org/10.4208/cicp.OA-2021-0084 UR - https://global-sci.org/intro/article_detail/cicp/20211.html KW - Hybrid multiscale models, radiative transfer equation, grey $P_N$ approximation, discrete duality finite volume method. AB -

The discrete duality finite volume method has proven to be a practical tool for discretizing partial differential equations coming from a wide variety of areas of physics on nearly arbitrary meshes. The main ingredients of the method are: (1) use of three meshes, (2) use of the Gauss-Green theorem for the approximation of derivatives, (3) discrete integration by parts. In this article we propose to extend this method to the coupled grey thermal-$P_N$ radiative transfer equations in Cartesian and cylindrical coordinates in order to be able to deal with two-dimensional Lagrangian approximations of the interaction of matter with radiation. The stability under a Courant-Friedrichs-Lewy condition and the preservation of the diffusion asymptotic limit are proved while the experimental second-order accuracy is observed with manufactured solutions. Several numerical experiments are reported which show the good behavior of the method.

Hermeline , Francois. (2022). Discrete Duality Finite Volume Discretization of the Thermal-$P_N$ Radiative Transfer Equations on General Meshes. Communications in Computational Physics. 31 (2). 398-448. doi:10.4208/cicp.OA-2021-0084
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