Analysis of Eigenvalue Clustering Leads to Optimal Scaling in Numerical Radiative Transfer

Abstract

ABSTRACT We consider a multidimensional polychromatic radiative transfer (RT) problem, accounting for scattering processes in a general form, that is, anisotropic (dipole) scattering with partial frequency redistribution. Given a discrete ordinates () discretization, we report the corresponding matrix structures, depending on the 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. More generally, we provide a new spectral framework to demonstrate optimal scaling with respect to RT discretization parameters for Krylov solution strategies, with implications for preconditioner design.