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Volume 30, Issue 4
A Sparse Grid Discrete Ordinate Discontinuous Galerkin Method for the Radiative Transfer Equation

Jianguo Huang & Yue Yu

DOI: 10.4208/cicp.OA-2020-0248

Commun. Comput. Phys., 30 (2021), pp. 1009-1036.

Published online: 2021-08

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  • Abstract

The radiative transfer equation is a fundamental equation in transport theory and applications, which is a 5-dimensional PDE in the stationary one-velocity case, leading to great difficulties in numerical simulation. To tackle this bottleneck, we first use the discrete ordinate technique to discretize the scattering term, an integral with respect to the angular variables, resulting in a semi-discrete hyperbolic system. Then, we make the spatial discretization by means of the discontinuous Galerkin (DG) method combined with the sparse grid method. The final linear system is solved by the block Gauss-Seidal iteration method. The computational complexity and error analysis are developed in detail, which show the new method is more efficient than the original discrete ordinate DG method. A series of numerical results are performed to validate the convergence behavior and effectiveness of the proposed method.

  • Keywords

Radiative transfer equation, sparse grid method, discrete ordinate method, discontinuous Galerkin method.

  • AMS Subject Headings

65N30, 65N35, 65N15, 65R20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-30-1009, author = {Huang , Jianguo and Yu , Yue}, title = {A Sparse Grid Discrete Ordinate Discontinuous Galerkin Method for the Radiative Transfer Equation}, journal = {Communications in Computational Physics}, year = {2021}, volume = {30}, number = {4}, pages = {1009--1036}, abstract = {

The radiative transfer equation is a fundamental equation in transport theory and applications, which is a 5-dimensional PDE in the stationary one-velocity case, leading to great difficulties in numerical simulation. To tackle this bottleneck, we first use the discrete ordinate technique to discretize the scattering term, an integral with respect to the angular variables, resulting in a semi-discrete hyperbolic system. Then, we make the spatial discretization by means of the discontinuous Galerkin (DG) method combined with the sparse grid method. The final linear system is solved by the block Gauss-Seidal iteration method. The computational complexity and error analysis are developed in detail, which show the new method is more efficient than the original discrete ordinate DG method. A series of numerical results are performed to validate the convergence behavior and effectiveness of the proposed method.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0248}, url = {http://global-sci.org/intro/article_detail/cicp/19392.html} }
TY - JOUR T1 - A Sparse Grid Discrete Ordinate Discontinuous Galerkin Method for the Radiative Transfer Equation AU - Huang , Jianguo AU - Yu , Yue JO - Communications in Computational Physics VL - 4 SP - 1009 EP - 1036 PY - 2021 DA - 2021/08 SN - 30 DO - http://doi.org/10.4208/cicp.OA-2020-0248 UR - https://global-sci.org/intro/article_detail/cicp/19392.html KW - Radiative transfer equation, sparse grid method, discrete ordinate method, discontinuous Galerkin method. AB -

The radiative transfer equation is a fundamental equation in transport theory and applications, which is a 5-dimensional PDE in the stationary one-velocity case, leading to great difficulties in numerical simulation. To tackle this bottleneck, we first use the discrete ordinate technique to discretize the scattering term, an integral with respect to the angular variables, resulting in a semi-discrete hyperbolic system. Then, we make the spatial discretization by means of the discontinuous Galerkin (DG) method combined with the sparse grid method. The final linear system is solved by the block Gauss-Seidal iteration method. The computational complexity and error analysis are developed in detail, which show the new method is more efficient than the original discrete ordinate DG method. A series of numerical results are performed to validate the convergence behavior and effectiveness of the proposed method.

Huang , Jianguo and Yu , Yue. (2021). A Sparse Grid Discrete Ordinate Discontinuous Galerkin Method for the Radiative Transfer Equation. Communications in Computational Physics. 30 (4). 1009-1036. doi:10.4208/cicp.OA-2020-0248
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