# On a convergent DSA preconditioned source iteration for a DGFEM method for radiative transfer

## TL;DR

This paper introduces a convergent DSA preconditioned source iteration method for DGFEM discretizations of the radiative transfer equation, demonstrating robustness and convergence properties both theoretically and numerically.

## Contribution

The paper develops a new DSA preconditioned source iteration for DGFEM discretizations, with a convergence analysis and numerical validation in radiative transfer problems.

## Key findings

- Convergence proven for the infinite-dimensional iteration.
- Numerical results show robustness in the diffusion limit.
- Discretization avoids the ray effect in lattice problems.

## Abstract

We consider the numerical approximation of the radiative transfer equation using discontinuous angular and continuous spatial approximations for the even parts of the solution. The even-parity equations are solved using a diffusion synthetic accelerated source iteration. We provide a convergence analysis for the infinite-dimensional iteration as well as for its discretized counterpart. The diffusion correction is computed by a subspace correction, which leads to a convergence behavior that is robust with respect to the discretization. The proven theoretical contraction rate deteriorates for scattering dominated problems. We show numerically that the preconditioned iteration is in practice robust in the diffusion limit. Moreover, computations for the lattice problem indicate that the presented discretization does not suffer from the ray effect. The theoretical methodology is presented for plane-parallel geometries with isotropic scattering, but the approach and proofs generalize to multi-dimensional problems and more general scattering operators verbatim.