A WENO algorithm for radiative transfer with resonant scattering: the
  time scale of the Wouthuysen-Field Coupling
  We develop a numerical solver for the integral-differential equations, which
describes the radiative transfer of photon distribution in the frequency space
with resonant scattering of Lyalpha photons by hydrogen gas in the early
universe. The time-dependent solutions of this equation is crucial to the
estimation of the effect of the Wouthuysen-Field (WF) coupling in relation to
the 21 cm emission and absorption at the epoch of reionization. The resonant
scattering leads to the photon distribution in the frequency space to be
piecewise smooth containing sharp changes. The weighted essentially
nonoscillatory (WENO) scheme is suitable to handle this problem, as this
algorithm has been found to be highly stable and robust for solving Boltzmann
equation. We test this numerical solver against analytic solutions of the
evolution of the photon distribution in rest background, analytic solution in
expanding background without resonant scattering and formation of local
Boltzmann distribution around the resonant frequency with the temperature same
as that of atom for recoil. We find that evolution of photon distribution
undergoes three phases; profile is similar to the initial one, a flat plateau
(without recoil) or local Boltzmann distribution (with recoil) forms around the
resonant frequency, and finally the distribution around the resonant frequency
is saturated when the photons from the source is balanced by the redshift of
the expansion. This result indicates that the onset of the W-F coupling should
not be determined by the third phase, but by the time scale of the second
phase. We found that the time scale of the W-F coupling is equal to about a few
hundreds of the mean free flight time of photons with resonant frequency, and
is independent of the Sobolev parameter if this parameter is much less than 1.
