# Rapidly-Converging Methods for Solving Multilevel Transfer Problems

## Summary

It is well known that lambda iterations can be used to solve multilevel non-LTE transfer equations in a reasonable number of iterations when the lambda operator is preconditioned, e.g., when the diagonal part of the operator is combined with other terms analytically. This approach is currently used successfully for the solution of model atoms with many line transitions, but sometimes a very large number of iterations is needed.

Lambda iteration consists of alternate solutions of the separate transfer and rate equations. For any given line transition the transfer and rate equations can be combined so that a solution can be obtained directly for that transition with no iterations needed between the transfer and rate equations. However, iterations are needed to determine the coupling between transitions. This can be time-consuming for model atoms with a large number of transitions that are treated in this way.

Here we show that 1) a hybrid approach involving such a direct solution for a few of the strongest transitions, and lambda iterations for the rest, gives rapid convergence, often with oscillations that need to be damped, and 2) this approach should include preconditioning of the lambda operator that occurs in the radiative coupling terms.

We illustrate these results with a simple three-level hydrogen atom and a finite, plane-parallel, symmetric atmosphere resembling a solar prominence, with a temperature of 8,000 K at the center, rising to very large values at each boundary (so that hydrogen is only partly ionized at the center and fully ionized at each boundary). Lambda iterations essentially fail to give a solution for this problem, while the hydrid solution converges in 5 to 10 iterations.

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## References

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