Astron. Astrophys. 332, 610-628 (1998)

4. The numerical method of solution

4.1. The iterative procedure

The integral equation for can be written in a symbolic form as

After discretization of the -variable, the operator becomes a matrix, where is the number of points in the optical depth grid . Each element of is a matrix. Using the same kind of iterative method as in Paper I, we write the correction to the current estimate as

Here is the identity matrix and is the approximate operator. To calculate we solve the transfer equation (50) with as source function and then average the resulting solution over frequencies and directions according to Eq. (49). The operator is constructed by keeping only the matrices , , on the diagonal of . Each matrix is calculated by placing a matrix point source (delta function source) at the grid point (see Paper I). Since this calculation has to be repeated at each grid point it turns out to be the most time consuming part of the iterative method.

4.2. Computational details and test problems

We consider isothermal, self-emitting plane parallel slab atmospheres with no incident radiation at the boundaries. These slab models are characterized by a set of input parameters , where T is the optical thickness of the slab, a the Voigt parameter of the line, the photon destruction probability per scattering, and the unpolarized internal thermal source. We consider the case of a pure line with no background continuum absorption. We restrict our attention to a two-level atom model with an atomic depolarization parameter set to unity (). The magnetic field is characterized by a set of 3 parameters . For the optical depth grid, we use a resolution of 8 points per decade in a logarithmic scale, covering the range . A frequency grid with 2 points per decade in the value of the profile function is used. The last frequency point in the grid, , is chosen such that . A 5-point Gaussian quadrature formula with is employed for angular grid. The grid points correspond to the five angles .

The calculations have been performed with two sets of atmospheric parameters :
A first set . It corresponds to a line which has reached thermalization at mid-slab. This model is used to test the convergence of the iterative method.
A second set . This model is used to study the influence of the magnetic field parameters on the polarization. At small optical depths (order of unity or less), the qualitative behavior of the polarization is almost independent of the total optical thickness of the slab. It is of course computationally much faster to consider a slab with than a slab with .

When the atmospheric and magnetic field parameters are uniform, the polarized radiation field is symmetric about the mid-plane at . The transfer problem can be solved on a half-slab, by imposing as boundary condition at the mid-plane that the derivative of the intensity vector with respect to vanishes. The presence of a unidirectional magnetic field does not break the mid-plane symmetry, because of the symmetries of the Hanle phase matrix. For example .

4.3. Convergence properties of the method

As in Paper  I, we have studied the convergence property of the method by following the dependence on the iteration number n of the , the maximum relative corrections of the components of the source vector . The upper panel in Fig. 3 shows the defined by

with . The lower panel shows the with the denominator in Eq. (66) replaced by

The iterative process is stopped when . In the lower panel of Fig. 3 we also show the effect of an Ng acceleration applied only on (see Paper I for details).

Fig. 3. Maximum relative corrections as function of the iteration number n. Slab model with parameters , and is employed. The upper panel shows the computed with the definition (66). The lower panel shows the same quantity computed with the modification given in Eq. (67). The effect of Ng acceleration is also shown in the lower panel. Note that all the curves (without Ng acceleration) have asymptotically the same slope for large values of n

Comparing Fig. 3 of this paper with Figs. 4 and 5 of Paper I, we see that the convergence properties of the iterative method are exactly the same as in the non-magnetic case. This is a direct consequence of the fact that all the polarization components behave as slave modes of the intensity component in the asymptotic regime of large n. One can verify that the speed of convergence as measured by the ratio keeps the same value when a magnetic field is switched on. Thus a nice property of the PALI-H iterative method is that the convergence rate is independent of the strength and direction of the magnetic field.

Fig. 4 shows the convergence history of the six components of . The component almost reaches its saturation value, , at mid-slab because the line is nearly thermalized. As a consequence at and . All the other components go to zero in the interior. Near the two boundaries they vary rapidly and change their sign.

Fig. 4. Convergence history of the six components of (see Eq. (47)). The upper left panel shows and the other panels , . Same model as in Fig. 3. The dotted lines show the initial solutions ( for , and zero for all other components). The effect of Ng acceleration (3-step jump of towards convergence) is clearly seen. Since the slab is symmetric about the mid-plane, the results are shown only for the half-slab

To verify the accuracy and proper convergence of the iterative scheme we have compared its results with those of a non-iterative Feautrier scheme with a 8 points per decade resolution in spatial grid. Once the stopping criterion has been is satisfied, the two solutions are identical up to 6 significant digits.

To estimate the optical depth grid-truncation error we have followed Auer et al. (1994) grid-doubling strategy. It has allowed us to estimate the errors on the intensity component . Employing a 3-level grid doubling procedure with successively 2, 4 and 8 points per decade, we obtain on , true errors of in the second stage, and in the third. For the polarization components, the grid-doubling strategy does not seem to offer a reliable estimation of the accuracy. More sophisticated methods seem to be required when dealing with functions which do not have a constant sign. We have also found that the grid-doubling strategy does not offer a significant gain in computing time for the reason that it is expensive to compute a Hanle approximate operator on début at each level of the grid doubling scheme. For resonance polarization with partial frequency redistribution this grid-doubling strategy appears on the contrary very promising (Paletou & Faurobert-Scholl 1997).

© European Southern Observatory (ESO) 1998

Online publication: March 23, 1998
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