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Abstract

The development of advanced discretization methods for the radiation transport equation is of fundamental importance, since the numerical effort of modeling increasingly complex multidimensional problems with increasing accuracy is extremely challenging. Different expressions of this equation arise in several science fields, from nuclear fission and fusion to astrophysics, climatology and combustion.

Mathematically, the radiation intensity is usually a rapidly changing function, causing a considerable loss in accuracy for many discretization methods. Depending on the coefficient ranges, the equation behaves like totally different equation types, making it very difficult to find a discretization method that is efficient in all regimes. Computationally, the huge amount of unknowns involved demands not only extremely powerful computers, but also efficient numerical methods and optimized implementations. Today, solvers covering all the coefficient ranges and still being robust in the diffusion dominated case are very scarce.

In the last 20 years, Discontinous Galerkin (DG) methods have been studied for the monoenergetic problem, unsuccessfully, due to lack of stability for diffusion-dominated cases. Recently, new mathematical developments have fully explained the instability and provided a remedy by using a numerical flux depending on the scattering cross section and the mesh size. The new formulation has proven to be stable and allows the application of multigrid, matrix-free methods, reducing the memory needed for such an amount of unknowns.

We use these numerical methods to address the solution of a energy dependent problem with a multigroup approach. We study the diffusion approximation to the transport problem, obtaining convergence proofs for the symmetric scattering case and advances in the nonsymmetric case, using field of values analysis.

For the full transport case, we discretize by means of an asymptotic preserving, weakly penalized discontinuous Galerkin method that we solve with a multigrid preconditioned GMRES solver, using nonoverlapping Schwarz smoothers for the energy and direction dependent radiative transfer problem.

To address the local thermodynamic equilibrium (LTE) constraint, we use a nonlinear additive Schwarz method to precondition the Newton solver. By solving full local radiative transfer problems for each grid cell, performed in parallel on a matrix-free implementation, we achieve a method capable to address large scale calculations arising from applications such as astrophysics, atmospheric radiation calculations and nuclear applications.

To the best of our knowledge, this is the first time this preconditioner combination has been used in LTE radiation transport and in several tests we show the robustness of the approach for different mesh sizes, cross sections, energy distributions and anisotropic regimes, both in the linear and nonlinear cases.