Improved integration strategies for the singularity subtraction method to solve radiative integral transfer equations with specified temperature field

Abstract

Solutions of the radiative integral transfer equations, which can avoid the ray effect, are usually adopted as benckmarks in the developing of discrete-ordinates-type methods for the radiative transfer equation. The singularity subtraction method is a popular method to deal with the singularity in radiative integral transfer equations. After singularity is removed, the remaining integral equations can be approximated by numerical quadratures. However, it was often ignored in previous researches that the integrands for the specified temperature problem are non-smooth in the whole domain, which would lead to reduced accuracy for the high order quadratures. In this paper, this issue is clarified, and also special treatments are proposed for the composite Boole's quadrature and the Chebyshev quadrature respectively. For the former, the quadrature weights are modified near the point on which the variables are to be solved, whereas for the latter, segments integration with interpolation is applied. The results for the plane-parallel problem show that, with such treatments, the accuracy of composite Boole's quadrature can be improved from the second order to the second/third order, and the accuracy of Chebyshev quadrature can be improved from the second order to the fourth order. The improved integration methods can be extended for the two-dimensional problem straightly, and the improvement on the accuracy is still observed. Further, since the improved Chebyshev quadrature can provide higher order accuracy than the improved composite Boole's quadrature, it is more suggested to adopt the improved Chebyshev quadrature to obtain the results with desired accuracy. Besides, the effects of parameters on the numerical errors are investigated. The errors increase with the optical thickness and the scattering albedo in the quadratic and linear relations, respectively.