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

Multidimensional radiative transfer with multilevel atoms

II. The non-linear multigrid method

P. Fabiani Bendicho ^{1, 2}, J. Trujillo Bueno ² and L. Auer <sup>3

1</sup> Kiepenheuer Institut für Sonnenphysik, Schöneckstr. 6, D-79104 Freiburg, Germany
² Instituto de Astrofísica de Canarias, E-38200, La Laguna (Tenerife), Spain
³ Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Received 2 September 1996 / Accepted 2 January 1997

Abstract

A new iterative method for solving non-LTE multilevel radiative transfer (RT) problems in 1D, 2D or 3D geometries is presented. The scheme obtains the self-consistent solution of the kinetic and RT equations at the cost of only a few () formal solutions of the RT equation. It combines, for the first time, non-linear multigrid iteration (Brandt, 1977; Hackbush, 1985), an efficient multilevel RT scheme based on Gauss-Seidel iterations (cf. Trujillo Bueno & Fabiani Bendicho, 1995), and accurate short-characteristics formal solution techniques. By combining a valid stopping criterion with a nested-grid strategy a converged solution with the desired true error is automatically guaranteed. Contrary to the current operator splitting methods the very high convergence speed of the new RT method does not deteriorate when the grid spatial resolution is increased. With this non-linear multigrid method non-LTE problems discretized on N grid points are solved in O(N) operations. The nested multigrid RT method presented here is, thus, particularly attractive in complicated multilevel transfer problems where small grid-sizes are required. The properties of the method are analyzed both analytically and with illustrative multilevel calculations for Ca II in 1D and 2D schematic model atmospheres.

Key words: radiative transfer stars: atmospheres methods: numerical

Send offprint requests to: J. Trujillo Bueno

Contents

• 1. Introduction
• 2. The non-linear multigrid method
  • 2.1. The multilevel transfer problem
  • 2.2. The smoothing ability of the MALI and MUGA schemes
  • 2.3. The coarse-grid equation
  • 2.4. The standard multigrid method
  • 2.5. A stopping criterion for the standard MG method
• 3. The convergence rate
  • 3.1. Measuring the convergence speed
  • 3.2. The influence of the smoothing number
  • 3.3. The influence of the number of coarse grids
• 4. Efficiency
  • 4.1. The MUGA method
  • 4.2. Two-grid method
  • 4.3. Multigrid method
  • 4.4. Results for the multidimensional case.
• 5. A nested multigrid RT method
• 6. Concluding remarks
• Acknowledgements
• References

© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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