Astrophysical observations originate from matter that interacts with radiation or transported particles. We develop a pragmatic approximation in order to enable multi-dimensional simulations with basic spectral radiative transfer when the computation
al resources are not sufficient to solve the complete Boltzmann transport equation. The distribution function of the transported particles is decomposed into trapped and streaming particle components. Their separate evolution equations are coupled by a source term that converts trapped particles into streaming particles. We determine this source term by requiring the correct diffusion limit. For a smooth transition to the free streaming regime, this diffusion source is limited by the matter emissivity. The resulting streaming particle emission rates are integrated over space to obtain the streaming particle flux. A geometric estimate of the flux factor is used to convert the particle flux to the streaming particle density. The efficiency of the scheme results from the freedom to use different approximations for each particle component. In supernovae, reactions with trapped particles on fast time scales establish equilibria that reduce the number of primitive variables required to evolve the trapped particle component. On the other hand, a stationary-state approximation facilitates the treatment of the streaming particle component. Different approximations may apply in applications to stellar atmospheres, star formation, or cosmological radiative transfer. We compare the isotropic diffusion source approximation with Boltzmann neutrino transport of electron flavour neutrinos in spherically symmetric supernova models and find good agreement. An extension of the scheme to the multi-dimensional case is also discussed.
