A Forth-and-Back Implicit Lambda -Iteration

Atanackovic-Vukmanovic, O.; Crivellari, L.; Simonneau, E.
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Astrophysical Journal v.487, p.735

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In many radiative transfer (RT) problems, the sources contain a scattering term that couples all the specific RT equations, one for each frequency and direction, so that solving the problem means solving the system formed by these equations. Each of them is a first-order linear differential equation with its own initial condition assigned at a different point of the medium, which makes the solution of the system extraordinarily difficult. One simple way to achieve a solution is with the so-called Lamda -iteration: sources and sinks given as a first approximation --> computation of the specific intensities from their own RT equations --> computation of the scattering terms --> recomputation of the sources and sinks. This scheme is straightforward, but unfortunately in practice its convergence rate is too slow to be of value in the case of optically thick systems. The aim of this paper is to show that a forth-and-back approach (the natural approach to describing sequentially the two intensities propagating along the two directions of a straight line), together with an implicit representation of the source function in the computation of the intensities within the above iterative scheme, can dramatically accelerate the convergence of the iterative process while retaining the straightforwardness of ordinary Lamda -iteration.