LINEARIZATION VERSUS PRECONDITIONING - WHICH APPROACH IS BEST FOR SOLVING MULTILEVEL TRANSFER PROBLEMS

We present a critical analysis of linearization and preconditioning, t he two most used approaches proposed for achieving the required linear ity in the iterative solution of the multilevel transfer problem. By d istinguishing from the outset between the response of the radiation he ld to the source function and opacity perturbations, we are able to de monstrate that if the linearization strategy, on which the local appro ximate Lambda-operator option of the multilevel transfer code MULTI is based, is applied neglecting the terms coming from the response of th e radiation held to the opacity perturbations, one then recovers the s ame equations obtained using the preconditioning technique of Rybicki & Hummer. It is also shown that if this preconditioning technique is a pplied taking into account the response of the radiation field to both the source function and opacity variations, one then ends up with the same equations found via the linearization method. Thus these two app roaches to the numerical solution of the multilevel transfer problem t urn out to be essentially the same, because similar equations are obta ined if the same information is taken into account. Finally, it is poi nted out that, if one wishes to guarantee positivity for the atomic le vel populations, it is necessary to neglect the terms associated with the response of the radiation field to the opacity perturbations. Negl ecting such terms does not deteriorate the convergence rate of multile vel transfer methods that make use of a local approximate operator.