Volume 27, Issue 6
A Uniform First-Order Method for the Discrete Ordinate Transport Equation with Interfaces in X,Y-Geometry

Min Tang

J. Comp. Math., 27 (2009), pp. 764-786.

Published online: 2009-12

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  • Abstract

A uniformly first-order convergent numerical method for the discrete-ordinate transport equation in the rectangle geometry is proposed in this paper. Firstly we approximate the scattering coefficients and source terms by piecewise constants determined by their cell averages. Then for each cell, following the work of De Barros and Larsen [1, 19], the solution at the cell edge is approximated by its average along the edge. As a result, the solution of the system of equations for the cell edge averages in each cell can be obtained analytically. Finally, we piece together the numerical solution with the neighboring cells using the interface conditions. When there is no interface or boundary layer, this method is asymptotic-preserving, which implies that coarse meshes (meshes that do not resolve the mean free path) can be used to obtain good numerical approximations. Moreover, the uniform first-order convergence with respect to the mean free path is shown numerically and the rigorous proof is provided.  

  • Keywords

Transport equation Interface Diffusion limit Asymptotic preserving Uniform numerical convergence X Y-geometry

  • AMS Subject Headings

41A30 41A60 65D25.

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COPYRIGHT: © Global Science Press

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@Article{JCM-27-764, author = {}, title = {A Uniform First-Order Method for the Discrete Ordinate Transport Equation with Interfaces in X,Y-Geometry}, journal = {Journal of Computational Mathematics}, year = {2009}, volume = {27}, number = {6}, pages = {764--786}, abstract = {

A uniformly first-order convergent numerical method for the discrete-ordinate transport equation in the rectangle geometry is proposed in this paper. Firstly we approximate the scattering coefficients and source terms by piecewise constants determined by their cell averages. Then for each cell, following the work of De Barros and Larsen [1, 19], the solution at the cell edge is approximated by its average along the edge. As a result, the solution of the system of equations for the cell edge averages in each cell can be obtained analytically. Finally, we piece together the numerical solution with the neighboring cells using the interface conditions. When there is no interface or boundary layer, this method is asymptotic-preserving, which implies that coarse meshes (meshes that do not resolve the mean free path) can be used to obtain good numerical approximations. Moreover, the uniform first-order convergence with respect to the mean free path is shown numerically and the rigorous proof is provided.  

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2009.09-m2894}, url = {http://global-sci.org/intro/article_detail/jcm/8602.html} }
TY - JOUR T1 - A Uniform First-Order Method for the Discrete Ordinate Transport Equation with Interfaces in X,Y-Geometry JO - Journal of Computational Mathematics VL - 6 SP - 764 EP - 786 PY - 2009 DA - 2009/12 SN - 27 DO - http://doi.org/10.4208/jcm.2009.09-m2894 UR - https://global-sci.org/intro/article_detail/jcm/8602.html KW - Transport equation KW - Interface KW - Diffusion limit KW - Asymptotic preserving KW - Uniform numerical convergence KW - X KW - Y-geometry AB -

A uniformly first-order convergent numerical method for the discrete-ordinate transport equation in the rectangle geometry is proposed in this paper. Firstly we approximate the scattering coefficients and source terms by piecewise constants determined by their cell averages. Then for each cell, following the work of De Barros and Larsen [1, 19], the solution at the cell edge is approximated by its average along the edge. As a result, the solution of the system of equations for the cell edge averages in each cell can be obtained analytically. Finally, we piece together the numerical solution with the neighboring cells using the interface conditions. When there is no interface or boundary layer, this method is asymptotic-preserving, which implies that coarse meshes (meshes that do not resolve the mean free path) can be used to obtain good numerical approximations. Moreover, the uniform first-order convergence with respect to the mean free path is shown numerically and the rigorous proof is provided.  

Min Tang. (2019). A Uniform First-Order Method for the Discrete Ordinate Transport Equation with Interfaces in X,Y-Geometry. Journal of Computational Mathematics. 27 (6). 764-786. doi:10.4208/jcm.2009.09-m2894
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