Towards a multigrid method for the M1 model for radiative transfer

We present a geometric multigrid solver for the M1 model of radiative transfer without source terms. In radiative hydrodynamics applications, the radiative transfer needs to be solved implicitly because of the fast propagation speed of photons relative to the fluid velocity. The M1 model is hyperbolic and can be discretized with an HLL solver, whose time implicit integration can be done using a nonlinear Jacobi method. One can show that this iterative method always preserves the admissible states, such as positive radiative energy and reduced flux less than 1. To decrease the number of iterations required for the solver to converge, and therefore to decrease the computational cost, we propose a geometric multigrid algorithm. Unfortunately, this method is not able to preserve the admissible states. In order to preserve the admissible state states, we introduce a pseudo-time such that the solution of the problem on the coarse grid is the steady state of a differential equation in pseudo-time. We present preliminary results showing the decrease of the number of iterations and computational cost as a function of the number of multigrid levels used in the method. These results suggest that nonlinear multigrid methods can be used as a robust implicit solver for hyperbolic systems such as the M1 model.