Analysis of eigenvalue clustering leads to optimal scaling in numerical radiative transfer

TL;DR

This paper analyzes the spectral properties of matrices from multidimensional radiative transfer problems, demonstrating that Krylov methods converge robustly due to eigenvalue clustering, leading to optimal scaling in simulations of stellar atmospheres.

Contribution

It provides a theoretical spectral analysis explaining the robustness of Krylov methods for radiative transfer matrices, highlighting eigenvalue clustering as the key factor.

Findings

Krylov methods are effective even with dense matrices.

Eigenvalues cluster at unity, ensuring robust convergence.

Numerical experiments confirm theoretical spectral analysis.

Abstract

We consider a multidimensional polychromatic radiative transfer (RT) problem, accounting for scattering processes in a general form, i.e. anisotropic (dipole) scattering with partial frequency redistribution. Given a discrete ordinates discretization, we report the corresponding matrix structures, depending on model and discretization parameters. Despite the possibly dense nature of these matrices, the use of Krylov methods is effective (especially in the matrix-free context) and robust. We propose a theoretical analysis, using the spectral tools of the symbol theory, explaining why Krylov convergence is robust w.r.t. all the discretization parameters, even in the unpreconditioned case. In fact, the compactness of the continuous operators used in the modeling leads to zero-clustered dense matrix sequences plus identity, so that the clustering at the unity of the spectra is deduced. Numerical experiments confirm the theoretical results, which have a direct application, for example, in the simulation of radiative transfer in stellar atmospheres, a key problem in astrophysical research. In general, we demonstrate that optimal scaling with respect to RT discretization parameters is expected for Krylov solution strategies.