Results

We test the accuracy of the 3D PBC solution by comparing it to the results of the 1D code for several line scattering parameters. The 1D solver uses 64 depth points, distributed logarithmically in optical depth. Figures 1-4 show the mean intensities ${\bar J}$ at $\ifmmode{\tau_{\rm std}}\else\hbox{$\tau_{\rm std}$}\fi =0$ and the $z$ component of the emergent flux $F$ as function of wavelength for both the 1D ($+$ symbols) and the 3D solver. The agreement is excellent for all values of $\epsilon_l$ from unity to $10^{-8}$, indicating that the 3D code produces an accurate solution even for extreme cases of line scattering. In the case with $\epsilon _l=10^{-8}$ the continuum processes lead to earler thermalization than the classical approximation $J\propto \epsilon^{1/2}$ as the line strength is limited compared to the continuum. This behavior is the same as in the 1D plane-parallel comparison case. The convergence rate of the line source function (here used together with Ng acceleration) is the same as discussed in Paper II, in the case of $\epsilon _l=10^{-8}$ the 3D code needed 29 iterations with the nearest-neighbor $\ifmmode{\Lambda^*}\else\hbox{$\Lambda^*$}\fi$ to reach a relative accuracy of $10^{-8}$ using the simple starting guess $S=B$. The nearest-neighbor $\ifmmode{\Lambda^*}\else\hbox{$\Lambda^*$}\fi$ does allow stopping the iterations earlier than a diagonal (local) $\ifmmode{\Lambda^*}\else\hbox{$\Lambda^*$}\fi$ due to the improved convergence rate (see paper I). This can easily cut the number of iterations by factors of two or more, even greater savings are possible if the accuracy limit is relaxed.

In addition to the mean intensities, we checked that the flux vectors $\vec F$ have vanishing components in the $x$ and $y$ directions, typically $\max(\vert F_x\vert,\vert F_y\vert)/\vert F_z\vert \le 10^{-13}$ in all voxels. We stress that this result is the result of the calculations and is not forced by the numerical scheme.