Exact and numerical solutions of the kompaneets equation: Evolution of the spectrum and average frequencies

The evolution of the spectrum of isotropic uniform radiation in an infinite space filled with a homogeneous, nonrelativistic electron gas is calculated by solving the Kompaneets equation. For an infinitely narrow initial spectrum, the time dependence of the average frequency and frequency dispersion is determined in a linear approximation of the equation. Characteristic times corresponding to changes in the character of this dependence are introduced. Two schemes are proposed for the numerical solution of the nonlinear equation: a nonconservative scheme with a grid that is uniform in frequency and a conservative scheme with automatic selection of an adaptive grid in frequency and time. For the linear equation the method yields results consistent with calculations of its solutions in terms of an eigenfunction expansion of the Kompaneets operator calculated in [D. I. Nagirner and V. M. Loskutov, Astrofizika, 40, 97 (1977)]. The influence of nonlinearity on the evolution of the spectrum of initially monochromatic radiation of various intensities is traced as an example of the application of the method.