On robustly convergent and efficient iterative methods for anisotropic radiative transfer

TL;DR

This paper introduces a robust and efficient iterative method for solving linear systems from anisotropic radiative transfer equations, ensuring convergence regardless of discretization and physical parameters, with practical implementation strategies.

Contribution

It develops a preconditioned Richardson iteration with a novel two-step preconditioner that guarantees mesh independence and reduces optical parameter influence, extending diffusion approximations with spherical harmonics.

Findings

Method achieves mesh-independent convergence.

Preconditioner effectively reduces dependence on optical parameters.

Numerical examples demonstrate near-linear complexity and robustness.

Abstract

This paper considers the iterative solution of linear systems arising from discretization of the anisotropic radiative transfer equation with discontinuous elements on the sphere. In order to achieve robust convergence behavior in the discretization parameters and the physical parameters we develop preconditioned Richardson iterations in Hilbert spaces. We prove convergence of the resulting scheme. The preconditioner is constructed in two steps. The first step borrows ideas from matrix splittings and ensures mesh independence. The second step uses a subspace correction technique to reduce the influence of the optical parameters. The correction spaces are build from low-order spherical harmonics approximations generalizing well-known diffusion approximations. We discuss in detail the efficient implementation and application of the discrete operators. In particular, for the considered discontinuous spherical elements, the scattering operator becomes dense and we show that $\mathcal{H}$- or $\mathcal{H}^2$-matrix compression can be applied in a black-box fashion to obtain almost linear or linear complexity when applying the corresponding approximations. The effectiveness of the proposed method is shown in numerical examples.