Numerical computation of isotropic Compton scattering
  Compton scattering is involved in many astrophysical situations. It is well
known and has been studied in detail for the past fifty years. Exact formulae
for the different cross sections are often complex, and essentially asymptotic
expressions have been used in the past. Numerical capabilities have now
developed to a point where they enable the direct use of exact formulae in
sophisticated codes that deal with all kinds of interactions in plasmas.
Although the numerical computation of the Compton cross section is simple in
principle, its practical evaluation is often prone to accuracy issues. These
can be severe in some astrophysical situations but are often not addressed
properly. In this paper we investigate numerical issues related to the
computation of the Compton scattering contribution to the time evolution of
interacting photon and particle populations. An exact form of the isotropic
Compton cross section free of numerical cancellations is derived. Its accuracy
is investigated and compared to other formulae. Then, several methods to solve
the kinetic equations using this cross section are studied. The regimes where
existing cross sections can be evaluated numerically are given. We find that
the cross section derived here allows for accurate and fast numerical
evaluation for any photon and electron energy. The most efficient way to solve
the kinetic equations is a method combining a direct integration of the cross
section over the photon and particle distributions and a Fokker-Planck
approximation. Expressions describing this combination are given.
