  Context: Dynamical studies of irradiated circumstellar disks require an
accurate treatment of radiation transport to, for example, properly determine
cooling and fragmentation properties. The radiation transport algorithm should
be as fast as the (magneto-) hydrodynamics to allow for an efficient usage of
computing resources.
  Methods: We use a setup of a central star and a slightly flared circumstellar
disk. We perform simulations for a wide range of optical depths of the disk's
midplane from tau(550nm) = 0.1 up to tau(810nm) = 1 million. We check the
accuracy of the gray flux-limited diffusion (FLD) approximation and a gray and
frequency-dependent ray-tracing plus FLD approximation.
  Results:
  1. For moderate optical depths, a gray approximation of the stellar
irradiation yields a slightly hotter inner rim and a slightly cooler midplane
of the disk at larger radii, but is otherwise in agreement with the
frequency-dependent treatment.
  2. The gray FLD approximation fails to compute an appropriate temperature
profile in all regimes of optical depth; the maximum deviations to the
comparison runs are 50 percent in the optically thin and up to 280 percent in
the optically thick limit. For low optical depth, the isotropic assumption
within the FLD method yields a too steep decrease of the radial temperature
slope. For higher optical depths, the FLD approximation does not reproduce the
shadow behind the optically thick inner rim of the circumstellar disk, yielding
artificial heating at larger disk radii.
  3. The frequency-dependent RT + gray FLD approximation yields remarkable
accuracy for the whole range of optical depths.
  Conclusions: The high accuracy of the frequency-dependent hybrid radiation
transport algorithm makes this method ideally suited for (magneto-)
hydrodynamical studies of irradiated circumstellar disks.
