A multigrid Newton-Krylov method for multimaterial equilibrium radiation diffusion

We focus on a fully implicit, nonlinearly converged, solution of multimaterial equilibrium radiation diffusion problems. The nonlinear method of solution is a Newton-Krylov (generalized minimum residual, GMRES) method preconditioned by a multigrid method. The multigrid iteration matrix results from a Picard-type linearization of the governing equations. The governing equation is highly nonlinear with the principal forms of nonlinearity found in the fourth-order dependence of the radiation energy on temperature, the temperature dependence of the opacity, and flux limiting. The efficiency of both the linear and nonlinear iterative techniques is investigated. With the realistic time step control the solution of the linear system does not scale linearly with multigrid as might be expected from theory. In contrast, we find that the use of multigrid to precondition a Newton-Krylov (GMRES) method provides a robust, scalable solution for the nonlinear system. Also only through converging the nonlinearities within a time step does the solution method achieve its design accuracy.