Compton and inverse Compton scattering from relativistic Maxwellian electrons both have an important feature, i.e. calculating the radiation transport in high-temperature and full-ionized plasma. Description and evaluation of relativistic photon-Maxwellian electron scattering are numerically complex and computationally time consuming. A Monte Carlo method is proposed to simulate photon scattering with relativistic Maxwellian electron and compute the scattering cross-sections. To compute the total cross-section of a photon of energy hν interacting with electrons at temperature Te in the laboratory coordinate, the calculation steps of Monte Carlo scheme are described as follows. The first step is to sample the velocity of an electron, the directions are isotropically sampled, and the speed is sampled from the relativistic Maxwellian distribution at temperature Te. The second step is to transform the photon energy hν into the photon energy in the coordinate in which the electron is at rest. The third step is to use the exact Klein-Nishina formula to compute the cross-sections. The fourth step is go back to the first step, and cycle this many times. The last step is to summarize all computed cross-sections and averaged them, and the average value is what we need. The operation and corresponding formula for each step are described in this paper. A better method of sampling the speed of a relativistic electron is expected to be found.A Monte Carlo processor is developed to compute the scattering cross-section of a photon of any energy, interacting with electrons at any temperature. To check this method, scatterings of the photons of various energies with electrons with various temperatures are simulated, and the results are compared with those from the numerical integration method. The comparison indicates that the simulated cross-sections are in pretty good agreement with those from the multiple integration method for the cases of electron temperature less than 25 keV. But unfortunately, the difference is obvious for the case of temperature more than 25 keV, and the error increases with temperature increasing. Why so? When the temperature is more than 25 keV, the sampling of electron speed is inaccurate when using the present method, which maybe results in this difference. So, we need to find a more accurate method of sampling relativistic electron speed to solve this problem in the future.
