Sparse tensor spherical harmonics approximation in radiative transfer
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TLDR
This work solves the stationary monochromatic radiative transfer equation with a multi-level Galerkin FEM in physical space and a spectral discretization with harmonics in solid angle and shows that the benefits of the concept of sparse tensor products, known from the context of sparse grids, can also be leveraged in combination with a spectralDiscretization.About:
This article is published in Journal of Computational Physics.The article was published on 2011-09-01 and is currently open access. It has received 68 citations till now. The article focuses on the topics: Boundary value problem & Spectral method.read more
Figures
Figure 4: Experiment 1: Convergence of numerical solution uL,N to prescribed solution u for L = 0, 1, . . . , 5 (left) or N = 1, 2, 4, . . . , 32 (right). 
Figure 11: Experiment 5: Magnitude of active degrees of freedom (left FT, right ST). 
Figure 8: Experiment 4: Net emission (left FT, right ST). 
Figure 10: Experiment 4: Convergence in incident radiation (Eq. 82) of numerical solution GL,N to reference solution G for L = 0, 1, . . . , 5 (left) or N = 1, 2, 4, . . . , 32 (right). 
Figure 9: Experiment 4: Heat flux (left FT, right ST). ![Figure 1: Full (left) and sparse (right) tensor product space structure for V L=3D = S(1,1)([0, 1]2, T 3D) and V N=3S = P13.](/figures/figure-1-full-left-and-sparse-right-tensor-product-space-1cfb707v.png)
Figure 1: Full (left) and sparse (right) tensor product space structure for V L=3D = S(1,1)([0, 1]2, T 3D) and V N=3S = P13.
![Figure 1: Full (left) and sparse (right) tensor product space structure for V L=3D = S(1,1)([0, 1]2, T 3D) and V N=3S = P13.](/figures/figure-1-full-left-and-sparse-right-tensor-product-space-1cfb707v.webp)
Citations
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An Introduction to Hierarchical (H-) Rank and TT-Rank of Tensors with Examples
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References
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Book
Radiative heat transfer
TL;DR: In this article, the Monte Carlo method for thermal radiation was used to estimate the radiative properties of one-dimensional Gray Media, and the method of Spherical Harmonics (PN-Approximation) was used for the same purpose.
Book
Numerical Solution of Partial Differential Equations by the Finite Element Method
TL;DR: In this article, the authors present an easily accessible introduction to one of the most important methods used to solve partial differential equations, which they call finite element methods for integral equations (FEME).
Journal ArticleDOI
Ray Effects in Discrete Ordinates Equations
TL;DR: In this paper, the nature of anomalous computational effects due to discretization of the angular variable in transport theory discrete ordinates approximations is analyzed and the origin of these effects is discussed.