Solution of Equilibrium Radiation Diffusion Problems Using Implicit Adaptive Mesh Refinement

Abstract

Diffusion approximations to radiation transport feature a nonlinear conduction coefficient that leads to formation of a sharp front, or Marshak wave, under suitable initial and boundary conditions. The front can vary several orders of magnitude over a very short distance. Resolving the shape of the Marshak wave is essential, but using a global fine mesh can be prohibitively expensive. In such circumstances it is natural to consider using adaptive mesh refinement (AMR) to place a fine mesh only in the vicinity of the propagating front. In addition, to avoid any loss of accuracy due to linearization, implicit time integration should be used to solve the equilibrium radiation diffusion equation. Implicit time integration on AMR grids introduces a new challenge, as algorithmic complexity must be controlled to fully realize the performance benefits of AMR\@. A Newton--Krylov method together with a multigrid preconditioner addresses this latter issue on a uniform grid. A straightforward generalization is to use a multilevel preconditioner that is tuned to the structure of the AMR grid, such as the fast adaptive composite grid (FAC) method. We describe the resulting Newton--Krylov-FAC method and demonstrate its performance on simple equilibrium radiation diffusion problems.