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 .
The growth rate can be characterized by a time scale t _ { c } of the transition phase , which is approximately proportional to \dot { E } ^ { 1 / 2 } , \dot { E } being the intensity of the ionizing source .
The larger the time scale t _ { c } , the longer the photons to postpone their contribution to the ionization .
For strong sources , like \dot { E } \geq 10 ^ { 56 } erg s ^ { -1 } , t _ { c } can be as large as a few Myrs , which could even be larger than the lifetime of the sources .
Consequently , most photons from these sources contribute to the reionization only when these sources already ceased .
We also show that the volume growth of ionized regions around clustered sources with intensity \dot { E } _ { i } ( i = 1 , 2 , \dots ) 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 .
We also developed , in this paper , the method of using WENO scheme to solve radiative transfer equation beyond one physical dimension .
This method can be used for high dimensional problems in general as well .