Verification of the Lebesgue averaging method

Abstract

This work is devoted to the study of the accuracy of the Lebesgue averaging method for the spectra of resonance radiation in solving the transport equation. The method is tested for the problem of the thermal radiation transfer in the Earth atmosphere. The results of numerical calculations in two versions of the Lebesgue averaging method are compared with the results of the direct pointwise (lineby-line) spectral calculations. In the first version, the classical Lebesgue integral is used, and the absorption coefficient is taken in it as the independent variable instead of the photons energy. In the second version, we apply the Lebesgue-Stieltjes integral with the measure of Lebesgue’s set as the independent variable. All calculations are performed by using one scheme of spatial and angular discretization of the transport equation. This ensures the purity of the computational experiment. The second variant of averaging by the use of Lebesgue-Stieltjes integration showed the highest accuracy (within 5%) with a significant decrease of the number of arithmetic operations (approximately by a factor of 103–104 ) in relation to the direct spectral calculations.