$2^{3(l_{\rm max} - l)}$ meshes. []{data-label="fig:supermesh"}](fig1.eps){width="8.0cm"}

We first introduce the supermesh approximation. In Fig. \[fig:supermesh\], we show a schematic illustration of the supermesh structure. If a three-dimensional computational domain is discretised by $2^{3 l_{\rm max}}$ meshes, a level $l$ supermesh consists of $2^{3 (l_{\rm max} -l)}$ meshes. We can calculate the mean density of each chemical species for every supermesh by using the meshes contained in it. The meshes can be used as the highest level supermeshes. The supermesh structure is resembling to an adaptive mesh refinement (AMR) structure and thus this method is well-suited to couple with the hydrodynamics by AMR codes.

Let us consider the case in which plane-parallel radiation with the specific intensity $I_0$ enters a supermesh that consists of $N_x \times N_y$ meshes. What we want to know is the mean intensity of the ray emerging from the other side of the supermesh, $\langle I_{\rm out} \rangle$ (see Fig. \[fig:supermeshapprox\]).

![Plane-parallel radiation with specific intensity $I_0$ entering to a supermesh that consists of $N_x \times N_y$ meshes. The $(i, j)$-th mesh has the ${\ensuremath{\mbox{{H$\,${\sc i}}}}}$ number density, $n_{i, j}$. []{data-label="fig:supermeshapprox"}](fig2.eps){width="8.0cm"}

For simplicity, we here only consider the absorption by ${\ensuremath{\mbox{{H$\,${\sc i}}}}}$ atoms and drop the frequency 