An Adaptive Nested Source Term Iteration for Radiative Transfer Equations

TL;DR

This paper introduces an adaptive iterative method with certified error bounds for solving radiative transfer equations, utilizing Petrov-Galerkin formulations, a posteriori error estimates, and low-rank approximations to improve efficiency and accuracy.

Contribution

It develops a novel adaptive nested iteration scheme for radiative transfer equations that guarantees convergence with controlled error reduction and employs advanced error estimation and matrix compression techniques.

Findings

Guaranteed convergence with fixed error reduction per iteration

Significant reduction in computational complexity through adaptive meshing

Effective use of low-rank approximation for the scattering operator

Abstract

We propose a new approach to the numerical solution of radiative transfer equations with certified a posteriori error bounds. A key role is played by stable Petrov--Galerkin type variational formulations of parametric transport equations and corresponding radiative transfer equations. This allows us to formulate an iteration in a suitable, infinite dimensional function space that is guaranteed to converge with a fixed error reduction per step. The numerical scheme is then based on approximately realizing this iteration within dynamically updated accuracy tolerances that still ensure convergence to the exact solution. To advance this iteration two operations need to be performed within suitably tightened accuracy tolerances. First, the global scattering operator needs to be approximately applied to the current iterate within a tolerance comparable to the current accuracy level. Second, parameter dependent linear transport equations need to be solved, again at the required accuracy of the iteration. To ensure that the stage dependent error tolerances are met, one has to employ rigorous a posteriori error bounds which, in our case, rest on a Discontinuous Petrov--Galerkin (DPG) scheme. These a posteriori bounds are not only crucial for guaranteeing the convergence of the perturbed iteration but are also used to generate adapted parameter dependent spatial meshes. This turns out to significantly reduce overall computational complexity. Since the global operator is only applied, we avoid the need to solve linear systems with densely populated matrices. Moreover, the approximate application of the global scatterer accelerated through low-rank approximation and matrix compression techniques. The theoretical findings are illustrated and complemented by numerical experiments with non-trivial scattering kernels.