Adaptive source term iteration : a stable formulation for radiative transfer

The radiative transfer problem is a model used to describe particles moving in a medium with which the particles might interact. It is used in a broad variety of fields including nuclear physics, medical imaging and astrophysics. From a numerical perspective, it is a challenging problem, due to its transport character and relatively high dimensionality with a 2d−1 dimensional solution (d spatial and d−1 directional dimensions). An integral operator over the directional domain introduces a global coupling of all directions that further complicates the high dimensionality. Solving the radiative transfer problem is traditionally done either using the non-deterministic Monte Carlo method or with deterministic solvers like the method of moments and the discrete ordinates method. Those deterministic methods usually use rather strong assumptions to obtain a priori estimates on the discretization error that might not hold in realistic physical settings. In this thesis, we propose a new deterministic method for solving the radiative transfer problem that gives rigorous a posteriori error estimates on the discrete solution. This method is based on an ideal fixed-point iteration in an infinite-dimensional setting that is solved approximately with dynamically updated accuracy. Thus, we call this new method Adaptive Source Term Iteration or ASTI for short. The use of a posteriori error estimates allows us to solve problems with less regular solutions and also reduces the computational costs by using adaptively chosen grids. The main difference with regard to existing Source Term Iteration methods, which iterate in fixed discrete spaces, is that ASTI adapts the spaces, in both the spatial and directional domain, during the iteration. This way, we can control the error of our iteration to guarantee convergence towards the exact solution. For the transport solver, we use a Discontinuous Petrov-Galerkin (DPG) method from Broersen, Dahmen and Stevenson. It is well suited for the kind of linear trans-port problems we obtain from the Adaptive Source Term Iteration and gives reliable a posteriori error estimates. This is based on Banach-Nečas-Babuška stability theory which centers around the existence of inf-sup estimates. All this adaptivity theory based upon a posteriori error estimators is new in the context of radiative transfer problems. As the analysis gets more involved, we also have to solve new implementational challenges. This can especially be seen in the grid management which involves combining transport solutions living on different adaptively refined grids. Our implementation of the Adaptive Source Term Iteration is built upon the general-purpose Dune-DPG library which was extended by code for an adaptive scattering approximation and for combining solutions living on differently adapted grids. Finally, we give two example problems computed with our ASTI implementation which illustrate how the adaptivity keeps the size of the discretized formulation in a feasible range while guaranteeing certified error bounds