A WENO Algorithm for the Growth of Ionized Regions at the Reionization Epoch

We investigate the volume growth of ionized regions around UV photon sources with the WENO algorithm, which is an effective solver of photon kinetics in the phase space described by the radiative transfer equation. We show that the volume growth rate, either of isolated ionized regions or of clustered regions in merging, generally consists of three phases: fast or relativistic growth phase at the early stage, slow growth phase at the later stage, and a transition phase between the fast and slow phases. We also show that the volume growth of ionized regions around clustered sources with intensity $\dot{E}_i$ ($i=1, 2, ...$) would have the same behavior as a single source with intensity $\dot{E}=\sum_i\dot{E}_i$, if all the distances between nearest neighbor sources $i$ and $j$ are smaller than $c(t^i_c+t^j_c)$, $t^i_c$ being the time scale $t_c$ of source $i$. Therefore, a tightly clustered UV photon sources would lead to a slow growth of ionized volume. This effect would be important for studying the redshift-dependence of 21cm signals from the reionization epoch.