Volume 24, Issue 4
A Uniformly Convergent Scheme for Radiative Transfer Equation in the Diffusion Limit Up to the Boundary and Interface Layers

Hongfei Chen, Gaoyu Chen, Xiang Hong, Hao Gao & Min Tang

Commun. Comput. Phys., 24 (2018), pp. 1021-1048.

Published online: 2018-06

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  • Abstract

In this paper, we present a numerical scheme for the steady-state radiative transfer equation (RTE) with Henyey-Greenstein scattering kernel. The scattering kernel is anisotropic but not highly forward peaked. On the one hand, for the velocity discretization, we approximate the anisotropic scattering kernel by a discrete matrix that can preserve the diffusion limit. On the other hand, for the space discretization, a uniformly convergent scheme up to the boundary or interface layer is proposed. The idea is that we first approximate the scattering coefficients as well as source by piecewise constant functions, then, in each cell, the true solution is approximated by the summation of a particular solution and a linear combinations of general solutions to the homogeneous RTE. Second-order accuracy can be observed, uniformly with respect to the mean free path up to the boundary and interface layers. The scheme works well for heterogenous medium, anisotropic sources as well as for the strong source regime.

  • Keywords

Radiative transfer equation, discrete Ordinates method, tailored finite point method, anisotropic scattering, asymptotic-preserving.

  • AMS Subject Headings

65L12, 76N20, 35Q70

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-24-1021, author = {}, title = {A Uniformly Convergent Scheme for Radiative Transfer Equation in the Diffusion Limit Up to the Boundary and Interface Layers}, journal = {Communications in Computational Physics}, year = {2018}, volume = {24}, number = {4}, pages = {1021--1048}, abstract = {

In this paper, we present a numerical scheme for the steady-state radiative transfer equation (RTE) with Henyey-Greenstein scattering kernel. The scattering kernel is anisotropic but not highly forward peaked. On the one hand, for the velocity discretization, we approximate the anisotropic scattering kernel by a discrete matrix that can preserve the diffusion limit. On the other hand, for the space discretization, a uniformly convergent scheme up to the boundary or interface layer is proposed. The idea is that we first approximate the scattering coefficients as well as source by piecewise constant functions, then, in each cell, the true solution is approximated by the summation of a particular solution and a linear combinations of general solutions to the homogeneous RTE. Second-order accuracy can be observed, uniformly with respect to the mean free path up to the boundary and interface layers. The scheme works well for heterogenous medium, anisotropic sources as well as for the strong source regime.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.2018.hh80.06}, url = {http://global-sci.org/intro/article_detail/cicp/12316.html} }
TY - JOUR T1 - A Uniformly Convergent Scheme for Radiative Transfer Equation in the Diffusion Limit Up to the Boundary and Interface Layers JO - Communications in Computational Physics VL - 4 SP - 1021 EP - 1048 PY - 2018 DA - 2018/06 SN - 24 DO - http://dor.org/10.4208/cicp.2018.hh80.06 UR - https://global-sci.org/intro/cicp/12316.html KW - Radiative transfer equation, discrete Ordinates method, tailored finite point method, anisotropic scattering, asymptotic-preserving. AB -

In this paper, we present a numerical scheme for the steady-state radiative transfer equation (RTE) with Henyey-Greenstein scattering kernel. The scattering kernel is anisotropic but not highly forward peaked. On the one hand, for the velocity discretization, we approximate the anisotropic scattering kernel by a discrete matrix that can preserve the diffusion limit. On the other hand, for the space discretization, a uniformly convergent scheme up to the boundary or interface layer is proposed. The idea is that we first approximate the scattering coefficients as well as source by piecewise constant functions, then, in each cell, the true solution is approximated by the summation of a particular solution and a linear combinations of general solutions to the homogeneous RTE. Second-order accuracy can be observed, uniformly with respect to the mean free path up to the boundary and interface layers. The scheme works well for heterogenous medium, anisotropic sources as well as for the strong source regime.

Hongfei Chen, Gaoyu Chen, Xiang Hong, Hao Gao & Min Tang. (2020). A Uniformly Convergent Scheme for Radiative Transfer Equation in the Diffusion Limit Up to the Boundary and Interface Layers. Communications in Computational Physics. 24 (4). 1021-1048. doi:10.4208/cicp.2018.hh80.06
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