Fast multilevel radiative transfer

The vast majority of recent advances in the field of numerical radiative transfer relies on approximate operator methods better known in astrophysics as Accelerated Lambda-Iteration (ALI). A superior class of iterative schemes, in term of rates of convergence, such as Gauss-Seidel and successive overrelaxation methods were therefore quite naturally introduced in the field of radiative transfer by Trujillo Bueno and Fabiani Bendicho [A novel iterative scheme for the very fast and accurate solution of non-LTE radiative transfer problems. Astrophys J 1995;455:646]; it was thoroughly described for the non-LTE two-level atom case. We describe hereafter in details how such methods can be generalized when dealing with non-LTE unpolarised radiation transfer with multilevel atomic models, in monodimensional geometry.