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

Herbert Egger, Matthias Schlottbom

Research output: Contribution to journalArticleAcademicpeer-review

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

Fingerprint

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.",
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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

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