LTE tests

In this test we have set $\epsilon=1$ to test the accuracy of the formal solution by comparing to the results of the 1D code. The 1D solver uses 50 radial points, distributed logarithmically in optical depth. For the 3D solver we tested `small' grids with $n_x=n_y=n_z=2*32+1$ points along each axis, for a total of $65^3 \approx 2.7\times 10^{5}$ voxels, as well as `medium' ( $n_x=n_y=n_z=2*64+1$ with a total of $129^3\approx 2.1\times 10^{6}$ voxels) and 'large ( $n_x=n_y=n_z=2*96+1$ with a total of $193^3\approx 7.2\times 10^{6}$ voxels). The large grid was limited by available memory for the storage of the non-local $\ifmmode{\Lambda^*}\else\hbox{$\Lambda^*$}\fi$ operator. The solid angle space discretization uses, in general, $n_\theta=n_\phi=64$ points. In Fig. 2 we show the mean intensities as function of distance from the center for both the 1D ($+$ symbols) and the 3D solver. The results show excellent agreement between the two solutions, thus the 3D RT formal solution is comparable in accuracy with the 1D formal solution. We demonstrate below that for the conditions used in these tests a larger number of solid angle points significantly improves the accuracy of the mean intensities.