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

1. Introduction

In astrophysics and other branches of physics the problems needing investigation often exceed the capacity of even the most powerful computers using the available methods. One is, thus, confronted with the need to develop and implement new numerical methods capable of offering much higher efficiency. That is the goal of this paper for the solution of complicated non-LTE transfer problems. We show how multidimensional, multilevel problems can be accurately and rapidly solved by means of non-linear multigrid iteration. By "accurately" we mean a true accuracy better than, say, 0.1% at all points, while by "rapidly" we mean that the CPU-time required to obtain the converged solution (either for 1D, 2D or 3D multilevel problems) is roughly the time needed to perform very few () formal solutions of the radiative transfer equation.

The development of powerful numerical methods for solving multilevel transfer problems is relevant not only in the stellar atmospheres context where there is strong interest in investigating 2D and 3D non-LTE effects using complex atomic models, but also in many other astrophysical research fields where the radiation-matter interaction must be considered in detail. Such problems are found, for instance, in the subjects of astronomical masers, spectral line formation in accretion disks, and the interpretation of spectro-polarimetric observations to determine cosmic magnetic fields. Further, there are a variety of astrophysical problems which require extreme grid-fineness for their correct solution. For example, magneto-hydrodynamic simulations of stellar atmospheric processes show the existence of abrupt changes in the physical state of the stellar plasma because of wave propagation and shock formation. In order to obtain the spectral signatures corresponding to selected snapshots of the time-dependent simulations, it is often necessary to use very fine spatial grids to resolve fully the abrupt changes in the excitation and ionization balance of the chemical species. The combination of multigrid and Gauss-Seidel iteration which we present here for the first time, is a remarkably effective general tool for meeting these needs.

Multigrid methods have their roots in the following observation about the convergence properties of "classical" operator-splitting methods like the Jacobi scheme: the coarser the grid the faster the convergence of the low-frequency spatial components of the error, while the finer the grid the faster the convergence of the high-frequency components (see, e.g., Hackbush, 1985). Therefore, in order to maximize the overall convergence, the multigrid iteration should be composed of two essential parts: a smoothing one where a small number of iterations on the desired fine grid damp the high-frequency spatial components of the error in the current estimate, and a coarser grid correction to suppress the low frequency components. The advantages of such multigrid iteration are its very high convergence rate and that this high convergence speed does not deteriorate when the discretization is refined. This is in sharp contrast with the fastest iterative methods currently in use for the solution of radiative transfer problems (see, e.g., Rybicki & Hummer, 1991, 1992; Auer, Fabiani Bendicho & Trujillo Bueno, 1994), for which the convergence speed decreases for increasing grid resolution.

The convergence speed of the recently-developed RT methods based on Gauss-Seidel (GS) and successive over-relaxation (SOR) iteration (see Trujillo Bueno & Fabiani Bendicho, 1995; hereafter TF-1), which have dramatically better convergence properties and are much faster than the above-mentioned ones, is also sensitive to the spatial resolution of the grid. Although these GS-based iterative schemes can rapidly solve complicated non-LTE problems in very fine grids, the critical point is that they provide extremely good smoothing properties and, therefore, make a multigrid approach exceptionally effective. The combination can treat fine spatial discretization with great effectiveness, as the time for solution scales only linearly with the number of grid points.

The mathematical foundations of multigrid methods are described in the monograph by Hackbusch (1985), where references to original work are also given, while numerical recipes for the application of multigrid methods can be found in Press et. al. (1994). The first application of the linear multigrid method to RT was done by Steiner (1991) for the particular case of coherent scattering. For the smoothing part he tried the local and non-local operator splitting methods described, respectively, by Olson, Auer and Buchler (1986; hereafter OAB) and by Olson & Kunasz (1987), with the result that the local approximate operator method has a much worse smoothing capability than a non-local one, often leading to failure of the multigrid iteration. For one-dimensional applications there is no problem in using a non-local operator splitting method which takes nearest depth couplings into account; however, in 2D and 3D applications the currently-used non-local approximate operator methods are not suitable because they require prohibitively large computing time and memory storage (see the discussion on this point in TF-1). The method of choice for the smoothing part of multigrid RT codes is the new RT method developed by TF-1, which is based on Gauss-Seidel (GS) iterations. With this new radiative transfer method there is no need for constructing and inverting any non-local operator and the computing time per iteration is minimal (i.e. virtually identical to that of a local operator splitting method). Further, the method has excellent smoothing capabilities. As we shall show, its efficiency in smoothing the high-frequency error components is so great, that only one or two smoothing iterations normally suffice to obtain the very high convergence rate which is obtainable with the multigrid method.

The above remarks also imply that 2D and 3D two-level atom RT codes based on the same linear multigrid scheme applied by Steiner (1991) and Väth (1994) should be implemented by making use of the GS-based method of TF-1 for their smoothing part. In this paper, however, we shall concentrate on the much more complicated problem of applying the non-linear multigrid method of Brandt (1977) to multilevel radiative transfer. This non-linear multigrid method is characterized by two essential parts: a smoothing one and a coarse-grid correction. For the smoothing part one needs a non-linear iterative relaxation method suitable for the multilevel atom problem, which, of course, should also be extremely efficient in its smoothing capability. For this smoothing purpose we shall correspondingly apply the method described by Trujillo Bueno & Fabiani Bendicho (1996; hereafter TF-2), which is nothing but the generalization to the multilevel non-linear case of the GS-based method of TF-1.

The outline of this paper is as follows. Sect. 2 discusses the application to multilevel RT of the non-linear multigrid method of Brandt (1977), showing examples that illustrate the complementary action of the coarse grid correction (CGC) and the smoothing steps. It also quantifies and compares the smoothing ability of two non-linear iterative schemes for multilevel RT calculations (one based on Jacobi's method and the other on Gauss-Seidel iterations), showing that the method of choice for the smoothing part of a non-linear RT multigrid code is the GS-based method of TF-2. Sect. 3 studies in detail the convergence rate of the multigrid iteration. Sect. 4 derives and discusses analytical formulae concerning the computational work and efficiency of our non-linear multigrid RT code. Sect. 5 introduces an improved multigrid RT technique (called the "nested multigrid method"), which is nothing but the combination of our grid-doubling strategy of Paper I (cf. Auer, Fabiani Bendicho and Trujillo Bueno, 1994) with the standard multigrid method. Here we show examples of 2D line-transfer calculations, comparing the efficiency of this nested multigrid RT method with the one of the grid-doubling strategy of Paper I. Finally, in Sect. 6 we present our conclusions.

Before entering into details we would like to clarify from the outset the two following points. First, in this paper we put the emphasis on describing in great detail the basic ingredients of the new RT method and the properties of our non-linear multigrid RT code. To this end, we derive some general analytical formulae, and concentrate on showing demonstrative numerical results for the same 5-level Ca II 1D and 2D transfer problems described in Paper I. Future papers will show results for other atoms (like, e.g., K, Fe and H) and inhomogeneous model atmospheres, demonstrating that a similarly good behaviour is obtained in these cases. Second, we should stress that the methods of this paper require only scalar-class computers, and that our nested multigrid RT method permits the modeling of 2D and 3D multilevel problems with no more than a workstation.

© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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