Framework

The 3D RT equations are easiest solved in a Cartesian coordinate system (e.g., Fabiani Bendicho et al., 1997), therefore we use a Cartesian grid of volume cells (voxels). Basically, the voxels are allowed to have different sizes but for the tests presented later in this paper we use fixed size voxels (as the simplest option). The values of physical quantities, such as temperatures, opacities and mean intensities, are averages over a voxel, which, therefore, also fixes the local physical resolution of the grid. In the following we will specify the size of the voxel grid by the number of voxels along each positive axis, e.g., $n_x = n_y = n_z = 32$ specifies a voxel grid from voxel coordinates $(-32,-32,-32)$ to $(32,32,32)$ for a total of $(2*32+1)^3 = 274625$ voxels, $65$ along each axis. The voxel $(0,0,0)$ is at the center of the voxel grid. The voxel centers are the grid points. The voxel coordinates are related by grid scaling factors to physical space, depending on the problem. The framework does not require $n_x = n_y = n_z$, we use this for the tests presented in this paper for convenience.

The applications that we intend to solve with the 3DRT framework will involve optically thick environments with a significant scattering contribution, e.g., modeling the light reflected by an extrasolar giant planet close to its parent star. Therefore, not only a formal solution is required but the full solution of the 3D radiative transfer equation with scattering. In this paper, we describe a method based on the operator splitting approach. Operator splitting works best if a non-local $\ifmmode{\Lambda^*}\else\hbox{$\Lambda^*$}\fi$ operator is used in the calculations (e.g., Hauschildt et al., 1994), therefore we describe a non-local $\ifmmode{\Lambda^*}\else\hbox{$\Lambda^*$}\fi$ method here. The operator splitting method can be combined with other methods, like multigrids, to allow for greater flexibility and better convergence, which we will discuss in a later paper.