Basic numerical methods

The co-moving frame radiative transfer equation for spherically symmetric flows can be written as :

We set c=1; $\beta$ is the velocity; and $\gamma = (1-\beta^2)^{-1/2}$ is the usual Lorentz factor. Equation  is a integro-differential equation, since the emissivity $\eta_\nu$ contains J_$J_{\nu}$ , the zeroth angular moment of I_$I_{\nu}$ :

$$\eta_\nu = \kappa_\nu \ifmmode{S_{\nu}}\else{\hbox{$S_{\nu}$} }\fi+ \sigma_\nu \ifmmode{J_{\nu}}\else{\hbox{$J_{\nu}$} }\fi,$$

and

$$\ifmmode{J_{\nu}}\else{\hbox{$J_{\nu}$} }\fi= 1/2 \int_{-1}^{1} d\mu\, \ifmmode{I_{\nu}}\else{\hbox{$I_{\nu}$} }\fi,$$

where $\ifmmode{S_{\nu}}\else{\hbox{$S_{\nu}$} }\fi$ is the source function, $\kappa_\nu$ is the absorption opacity, and $\sigma_\nu$ is the scattering opacity. With the assumption of time-independence $\frac{\partial\ifmmode{I_{\nu}}\else{\hbox{$I_{\nu}$} }\fi}{\partial t} = 0$ and a monotonic velocity field Eq. 1 becomes a boundary-value problem in the spatial coordinate and an initial value problem in the frequency or wavelength coordinate. The equation can be written in operator form as:

where $\Lambda$ is the lambda-operator.