Numerical method for the Compton scattering operator.

In this paper, we present a new discretization of the Kompaneets equation. This equation can be derived from the Quantum Boltzmann equation for cross sections localized around small changes in the energy. The two collision operators share the same properties (conservation, entropy and steady states). The numerical methods are designed in order to preserve these properties which insures the correct long time behaviour. A supplementary difficulty arises for initial data with a density larger than a critical one, associated with a Planck distribution. In this case, the equilibrium state are the sum of a smooth Planck distribution plus a Delta measure on zero energy. The scheme is able to deal with this singularity