Astron. Astrophys. 324, 161-176 (1997)

6. Concluding remarks

The non-linear multigrid methods developed in this paper for multilevel RT applications are characterized by a very high convergence speed that does not deteriorate when the grid resolution is refined. For this reason they are particularly suitable for solving complicated 1D, 2D or 3D multilevel problems, where grid-sizes smaller than about 10 km are required. The computational work one can save by using the non-linear multigrid RT method instead of the operator splitting methods currently in use (like, e.g., the MALI scheme), is the larger the smaller the grid-size. This saving is larger for 3D applications than for 2D, which, in turn, is larger than for 1D.

Our nested multigrid RT code is even more effective. Its storage requirements are only a factor larger than those of the MALI method of Paper I (with for 1D, for 2D, and for 3D applications), and it is a factor 2 faster than the standard multigrid RT technique. It, thus, allows the accurate solution of RT problems with a computational effort equal to the cost of only very few formal solutions of the RT equation. For instance, if n indicates the number of grid-points per decade, the computational work of the Jacobi-type MALI scheme of Paper I scales approximately as , the MUGA method of Trujillo Bueno & Fabiani Bendicho (1995; 1996) (which is based on Gauss-Seidel iterations) as , their SOR-based technique as , while the multilevel RT multigrid methods presented here scale as n, because the very high convergence rate of the multigrid iteration is insensitive to the grid-size, while the cost per MG iteration scales as n. To our knowledge there is presently no other multilevel transfer code capable of offering this very high performance in scalar-class computers. Preliminary results indicate that a similarly good relative improvement is obtained in calculations for realistic atomic models and atmospheres, whose presentation we leave for future publications.

© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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