The gray approximation for thermal non-equilibrium radiation diffusion problems yields a time dependent non-linear system of equations coupling the radiation energy and the material temperature. A flux-limiting term is added to the diffusion coefficient for the radiation energy. Implicit time stepping schemes lead to the solution of non-linear systems of equations.

A Newton-Krylov-method is employed in the solution of arising non-linear problems. For GMRES iterations, we study preconditioners based on multigrids methods. The first approach is to apply a geometric multigrid method directly to the coupled linearized problems. Another approach is to construct an operator splitting which treats the transport phenomenon and the equilibration coupling in separate substeps. A similar approach was proposed by Mousseau, Knoll and Rider, 2000.

We demonstrate the effectiveness of the proposed approaches by solving one-dimensional and two-dimensional model problems.