A perfectly matched layer approach for radiative transfer in highly scattering regimes

TL;DR

This paper introduces a perfectly matched layer method for radiative transfer problems that improves boundary condition approximation, reduces computational complexity, and enhances efficiency in highly scattering regimes.

Contribution

It develops a PML-based boundary condition approach combined with a mixed variational formulation for better numerical approximation in radiative transfer.

Findings

The method achieves arbitrarily small consistency errors with larger absorbing layers.

It simplifies the variational treatment by avoiding half-space integrals.

Numerical tests confirm high accuracy and efficiency in scattering regimes.

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 PN-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.