Conclusions

We have described a framework for solving three-dimensional radiative transfer problems in scattering dominated environments. The method uses a non-local operator splitting technique to solve the scattering problem. The formal solution is based on a long characteristic piece-wise parabolic procedure. For strongly scattering dominated test cases (sphere in a box) we find good convergence with non-local $\ifmmode{\Lambda^*}\else\hbox{$\Lambda^*$}\fi$ operators, as well as minimal numerical diffusion with the long characteristics method and adequate resolution. A simple MPI parallelization gives excellent speedups on parallel clusters. In subsequent work we will implement a domain decomposition method to allow much larger spatial grids. Presently, we have implemented the method for static media, it can be used without significant changes to solve problems in the Eulerian-frame for media with low velocity fields. The distribution of matter over the voxels is, in the general 3D case, arbitrary. We chose a spherical test case to be able to compare the results our 1D code.

In Figs. 19 and 20 we compare the results for the $\epsilon =10^{-4}$ test case using a simple implementation of the short characteristics method and the long characteristics method used in this paper. The test grid contains $129^3$ voxel and $64^2$ angular points. The high diffusivity of the SC method is evident. Other authors (Auer et al., 1994; Steiner, 1991; Trujillo Bueno & Fabiani Bendicho, 1995; Vath, 1994; van Noort et al., 2002; Fabiani Bendicho et al., 1997) have used SC methods in multi-dimensional radiative transport problems. Short characteristics techniques are faster but require special considerations to reduce numerical diffusion (Auer, 2003). We may further look into the SC method in later papers.

We have generalized the operator splitting to include larger bandwidth operators. They lead to faster convergence although they do require more memory and ultimately more computing time. Nevertheless, they will be useful for highly complex problems and we have developed a highly flexible approach to the construction of the $\ifmmode{\Lambda^*}\else\hbox{$\Lambda^*$}\fi$ operator so that the bandwidth may be set for each spatial point individually as the problem and computational resources require.

We have designed an especially general and flexible framework for 3D radiative transfer problems with scattering. In future papers of this series we will describe its extension to line transfer problems, multi-level NLTE calculations, and differentially moving flows.