Testing environment

We use the framework discussed in Paper I and II as the baseline for the line transfer problems discussed in this paper. Our basic setup is similar to that discussed in Paper II. Periodic boundary conditions (PBCs) are realized in a plane parallel slab. We use PBCs on the $x$ and $y$ axes, $z_{\rm max}$ is at the outside boundary, $z_{\rm min}$ the inside boundary. The slab has a finite optical depth in the $z$ axis. The grey continuum opacity is parameterized by a power law in the continuum optical depth $\ifmmode{\tau_{\rm std}}\else\hbox{$\tau_{\rm std}$}\fi$ in the $z$ axis. The basic model parameters are

1. thickness of the slab, $z_{\rm max}- z_{\rm min} = 10^7\,$cm
2. Minimum optical depth in the continuum, $\tau _{\rm std}^{\rm min} = 10^{-8}$ and maximum optical depth in the continuum, $\tau _{\rm std}^{\rm max} = 10$.
3. Constant temperatures (in all axes), $T=10^4$ K
4. Outer boundary condition, $I_{\rm bc}^{-} \equiv 0$ and diffusion inner boundary condition for all wavelengths.
5. Parameterized coherent & isotropic continuum scattering by defining
   
   $$\chi_c = \epsilon_c \kappa_c + (1-\epsilon_c) \sigma_c$$
   
   
   with $0\le \epsilon_c \le 1$. $\kappa_c$ and $\sigma_c$ are the continuum absorption and scattering coefficients.

The line of the simple 2-level model atom is parameterized by the ratio of the profile averaged line opacity $\chi_l$ to the continuum opacity $\chi_c$ and the line thermalization parameter $\epsilon_l$. For the test cases presented below, we have used $\epsilon _c=1$ and a constant temperature and thus a constant thermal part of the source function for simplicity (and to save computing time) and set $\chi_l/\chi_c = 10^6$ to simulate a strong line, with varying $\epsilon_l$ (see below). With this setup, the optical depths as seen in the line range from $10^{-2}$ to $10^6$. We use 32 wavelength points to model the full line profile, including wavelengths outside the line for the continuum. We did not require the line to thermalize at the center of the test configurations, this is a typical situation one encounters in a full 3D configurations as the location (or even existence) of the thermalization depths becomes more ambiguous than in the 1D case.

The slab is mapped onto a Cartesian grid. For the test calculations we use voxel grids with the same number of spatial points in each direction (see below). The solid angle space was discretized in $(\theta,\phi)$ with $n_\theta=n_\phi$ if not stated otherwise. In the following we discuss the results of various tests. In all tests we use the full characteristic method for the 3D RT solution as described above. Unless otherwise stated, the tests were run on parallel computers using 128 CPUs. For the 3D solver we use $n_x=n_y=n_z=2*32+1=65$ points along each axis. The solid angle space discretization uses $n_\theta=n_\phi=64$ points.