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An Adaptive Angular Domain Compression Scheme For Solving Multiscale Radiative Transfer Equation

Qinchen Song, Jingyi Fu, Min Tang, Lei Zhang

Abstract

When dealing with the steady-state multiscale radiative transfer equation (RTE) with heterogeneous coefficients, spatially localized low-rank structures are present in the angular space. This paper introduces an adaptive tailored finite point scheme (TFPS) for RTEs in heterogeneous media, which can adaptively compress the angular space. It does so by selecting reduced TFPS basis functions based on the local optical properties of the background media. These reduced basis functions capture the important local modes in the velocity domain. A detailed a posteriori error analysis is performed to quantify the discrepancy between the reduced and full TFPS solutions. Additionally, numerical experiments demonstrate the efficiency and accuracy of the adaptive TFPS in solving multiscale RTEs, especially in scenarios involving boundary and interface layers.

An Adaptive Angular Domain Compression Scheme For Solving Multiscale Radiative Transfer Equation

Abstract

When dealing with the steady-state multiscale radiative transfer equation (RTE) with heterogeneous coefficients, spatially localized low-rank structures are present in the angular space. This paper introduces an adaptive tailored finite point scheme (TFPS) for RTEs in heterogeneous media, which can adaptively compress the angular space. It does so by selecting reduced TFPS basis functions based on the local optical properties of the background media. These reduced basis functions capture the important local modes in the velocity domain. A detailed a posteriori error analysis is performed to quantify the discrepancy between the reduced and full TFPS solutions. Additionally, numerical experiments demonstrate the efficiency and accuracy of the adaptive TFPS in solving multiscale RTEs, especially in scenarios involving boundary and interface layers.
Paper Structure (25 sections, 6 theorems, 95 equations, 13 figures, 1 table)

This paper contains 25 sections, 6 theorems, 95 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Outline
  3. Low rank structure in velocity domain: 1D illustration
  4. DOM for 1D RTE
  5. Basis functions as analytical solutions of discrete ordinate RTE
  6. Low rank structure indicated by basis functions:
  7. Low rank structure in 2D case and TFPS
  8. DOM for RTE in x-y geometry
  9. Construction of exponential decaying basis functions and TFPS
  10. Approximate Solution by TFPS
  11. Low rank structure in the angular domain
  12. Adaptive TFPS: An adaptive angular domain compression scheme
  13. Interface condition and boundary condition
  14. Interface condition
  15. Boundary condition
  16. ...and 10 more sections

Key Result

Lemma 5.3

\newlabellemma:Cinf0 Define $C_{\gamma,g,M,\delta,\infty}$ as follows: Then $\langle l,\chi_{\mathfrak{i}}^{(k)}\rangle_{\mathfrak{i}}\leq C_{\gamma,g,M,\delta,\infty}\Vert l\Vert_{\infty}$ for any $\mathfrak{i}\in\mathcal{I}$, $l\in U_{\mathfrak{i}}$, and $k\in\mathcal{V}^{\mathfrak{i}}$. Additionally, $1\leq C_{\gamma,g,M,\delta,\infty}\leq \sqrt{4M}C_{\gamma,g,M,\de

Figures (13)

  • Figure 1: The figure illustrates the values of the eigenvalue $\lambda^{(k)}$ for different choice of $\frac{\sigma_{s}}{\sigma_{T}}$. The eigenvalues with the smallest magnitude ($\lambda^{(10)}$, $\lambda^{(11)}$) are also highlighted in the figure, corresponding to $k=10$ and 11.
  • Figure 1: DOM in x-y geometry. Left figure: a quadrature point in three spatial dimensional case. Right figure: example of DOM ($S_{N}$) in x-y geometry with $N=2$.
  • Figure 1: The layout for the lattice problem is as follows: blue rectangles represent the diffusive regions, while yellow rectangles represent the transport regions.
  • Figure 2: Left figure: the profile of $\phi$ when $\sigma_{s}=5$, $\sigma_{T}=10$. Middle and right figure: the difference between $\phi$ and $\phi_{\delta}$ for $\delta=10^{-2}$, $10^{-3}$, $10^{-4}$, $10^{-5}$ when $\sigma_{s}=5$, $\sigma_{T}=10$. Here, the blue dot-dash line represents the case when $\delta=10^{-2}$, the red dotted line represents the case when $\delta=10^{-3}$, the dashed orange line represents the case when $\delta=10^{-4}$ and the purple solid line represent the case when $\delta=10^{-5}$.
  • Figure 2: Spatial mesh
  • ...and 8 more figures

Theorems & Definitions (20)

  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 4.2
  • Remark 4.3
  • Remark 4.5
  • Remark 4.6
  • Remark 4.7
  • Remark 5.2
  • Lemma 5.3
  • ...and 10 more