Application examples

As a first step we have implemented the method as a MPI parallelized Fortran 95 program. The parallelization of the formal solution is presently implemented over solid angle space as this is the simplest parallelization option and also one of the most efficient (a domain decomposition parallelization method will be discussed in a subsequent paper). In addition, the Jordan solver of the Operator splitting equations is parallelized with MPI (see below for scaling properties of the MPI implementation). The number of parallelization related statements in the code is small, about 320 out of a total of about 7900.

Our basic continuum scattering test problem is similar to that discussed in Hauschildt (1992) and in Hauschildt & Baron (2004). This test problem covers a large dynamic range of about 9 dex in the opacities and overall optical depth steps along the characteristics and, in our experience, constitutes a reasonably challenging setup for the radiative transfer code.The application of the 3D code to 'real' problems is in preparation and requires a substantial amount of development work (in progress). For the 1D code we have found that the test case is actually pretty much a worst case scenario and that it generally works better in real world problems. We use a sphere with a grey continuum opacity parameterized by a power law in the continuum optical depth $\ifmmode{\tau_{\rm std}}\else\hbox{$\tau_{\rm std}$}\fi$. The basic model parameters are

1. Inner radius $r_c=10^{13}\,$cm, outer radius $\hbox{$r_{\rm out}$} = 1.01\times 10^{15}\,$cm.
2. Minimum optical depth in the continuum $\tau _{\rm std}^{\rm min} = 10^{-4}$ and maximum optical depth in the continuum $\tau _{\rm std}^{\rm max} = 10^{4}$.
3. Grey temperature structure with $\hbox{$\,T_{\rm eff}$} =10^4$ K.
4. Outer boundary condition $I_{\rm bc}^{-} \equiv 0$ and diffusion inner boundary condition for all wavelengths.
5. Continuum extinction $\chi_c = C/r^2$, with the constant $C$ fixed by the radius and optical depth grids.
6. Parameterized coherent & isotropic continuum scattering by defining
   

   $$\chi_c = \epsilon_c \kappa_c + (1-\epsilon_c) \sigma_c$$ (28)

   
   
   with $0\le \epsilon_c \le 1$. $\kappa_c$ and $\sigma_c$ are the continuum absorption and scattering coefficients.

The test model is just an optically thick sphere put into the 3D grid. This problem is used because the results can be directly compared with the results obtained with our 1D spherical radiation transport code (Hauschildt, 1992) to assess the accuracy of the method. The sphere is centered at the center of the Cartesian grid, which is in each axis 10% larger than the radius of the sphere. 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 LC method for the 3D RT solution unless stated otherwise.