A Perfectly Matched Layer Approach for $P_N$-Approximations in Radiative Transfer

Herbert Egger, Matthias Schlottbom*

*Corresponding author for this work

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    Abstract

    We consider the numerical approximation of boundary conditions in radiative transfer problems by a perfectly matched layer approach. The main idea is to extend the computational domain by an absorbing layer and to use an appropriate reflection boundary condition at the boundary of the extended domain. A careful analysis shows that the consistency error introduced by this approach can be made arbitrarily small by increasing the size of the extension domain or the magnitude of the artificial absorption in the surrounding layer. A particular choice of the reflection boundary condition allows us to circumvent the half-space integrals that arise in the variational treatment of the original vacuum boundary conditions and which destroy the sparse coupling observed in numerical approximation schemes based on truncated spherical harmonics expansions. A combination of the perfectly matched layer approach with a mixed variational formulation and a $P_N$-finite element approximation leads to discretization schemes with optimal sparsity pattern and provable quasi-optimal convergence properties. As demonstrated in numerical tests these methods are accurate and very efficient for radiative transfer in the scattering regime.
    Original languageEnglish
    Pages (from-to)2166-2188
    Number of pages23
    JournalSIAM Journal on Numerical Analysis
    Volume57
    Issue number5
    DOIs
    Publication statusPublished - 2019

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    Radiative Transfer
    Radiative transfer
    Boundary conditions
    Approximation
    Numerical Approximation
    Mixed Formulation
    Discretization Scheme
    Spherical Harmonics
    Variational Formulation
    Approximation Scheme
    Finite Element Approximation
    Sparsity
    Absorbing
    Convergence Properties
    Half-space
    Numerical Scheme
    Vacuum
    Absorption
    Scattering

    Keywords

    • Radiative transfer
    • PN method
    • Galerkin approximation
    • perfectly matched layers

    Cite this

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    title = "A Perfectly Matched Layer Approach for $P_N$-Approximations in Radiative Transfer",
    abstract = "We consider the numerical approximation of boundary conditions in radiative transfer problems by a perfectly matched layer approach. The main idea is to extend the computational domain by an absorbing layer and to use an appropriate reflection boundary condition at the boundary of the extended domain. A careful analysis shows that the consistency error introduced by this approach can be made arbitrarily small by increasing the size of the extension domain or the magnitude of the artificial absorption in the surrounding layer. A particular choice of the reflection boundary condition allows us to circumvent the half-space integrals that arise in the variational treatment of the original vacuum boundary conditions and which destroy the sparse coupling observed in numerical approximation schemes based on truncated spherical harmonics expansions. A combination of the perfectly matched layer approach with a mixed variational formulation and a $P_N$-finite element approximation leads to discretization schemes with optimal sparsity pattern and provable quasi-optimal convergence properties. As demonstrated in numerical tests these methods are accurate and very efficient for radiative transfer in the scattering regime.",
    keywords = "Radiative transfer, PN method, Galerkin approximation, perfectly matched layers",
    author = "Herbert Egger and Matthias Schlottbom",
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    language = "English",
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    A Perfectly Matched Layer Approach for $P_N$-Approximations in Radiative Transfer. / Egger, Herbert; Schlottbom, Matthias .

    In: SIAM Journal on Numerical Analysis, Vol. 57, No. 5, 2019, p. 2166-2188.

    Research output: Contribution to journalArticleAcademicpeer-review

    TY - JOUR

    T1 - A Perfectly Matched Layer Approach for $P_N$-Approximations in Radiative Transfer

    AU - Egger, Herbert

    AU - Schlottbom, Matthias

    PY - 2019

    Y1 - 2019

    N2 - We consider the numerical approximation of boundary conditions in radiative transfer problems by a perfectly matched layer approach. The main idea is to extend the computational domain by an absorbing layer and to use an appropriate reflection boundary condition at the boundary of the extended domain. A careful analysis shows that the consistency error introduced by this approach can be made arbitrarily small by increasing the size of the extension domain or the magnitude of the artificial absorption in the surrounding layer. A particular choice of the reflection boundary condition allows us to circumvent the half-space integrals that arise in the variational treatment of the original vacuum boundary conditions and which destroy the sparse coupling observed in numerical approximation schemes based on truncated spherical harmonics expansions. A combination of the perfectly matched layer approach with a mixed variational formulation and a $P_N$-finite element approximation leads to discretization schemes with optimal sparsity pattern and provable quasi-optimal convergence properties. As demonstrated in numerical tests these methods are accurate and very efficient for radiative transfer in the scattering regime.

    AB - We consider the numerical approximation of boundary conditions in radiative transfer problems by a perfectly matched layer approach. The main idea is to extend the computational domain by an absorbing layer and to use an appropriate reflection boundary condition at the boundary of the extended domain. A careful analysis shows that the consistency error introduced by this approach can be made arbitrarily small by increasing the size of the extension domain or the magnitude of the artificial absorption in the surrounding layer. A particular choice of the reflection boundary condition allows us to circumvent the half-space integrals that arise in the variational treatment of the original vacuum boundary conditions and which destroy the sparse coupling observed in numerical approximation schemes based on truncated spherical harmonics expansions. A combination of the perfectly matched layer approach with a mixed variational formulation and a $P_N$-finite element approximation leads to discretization schemes with optimal sparsity pattern and provable quasi-optimal convergence properties. As demonstrated in numerical tests these methods are accurate and very efficient for radiative transfer in the scattering regime.

    KW - Radiative transfer

    KW - PN method

    KW - Galerkin approximation

    KW - perfectly matched layers

    U2 - 10.1137/18M1172521

    DO - 10.1137/18M1172521

    M3 - Article

    VL - 57

    SP - 2166

    EP - 2188

    JO - SIAM Journal on Numerical Analysis

    JF - SIAM Journal on Numerical Analysis

    SN - 0036-1429

    IS - 5

    ER -