The composite Chebyshev integration method for radiative integral transfer equations in rectangular enclosures

Recently, the Chebyshev integration method was developed to solve the radiative integral transfer equations (RITEs) after singularity removing (IJTS, 149 (2020) 106,158), and shown fourth order convergence accuracy. This superior property makes the Chebyshev integration method attractive to produce results with quite high accuracy to be benchmark solutions. However, the Chebyshev integration method, which is a global method, is only suitable for problems with smooth parameters and temperature distribution. To overcome this drawback, in this paper, the composite Chebyshev integration method is proposed. The computational domain is divided into several subdomains, and then the Chebyshev quadrature is applied in each subdomain. Several benchmark problems in the rectangular medium with continuous/stepwise-change scattering albedo and emissions are solved. Increasing the grid number with fixed subdomains, one can observe that the composite Chebyshev integration method still has the high order convergence accuracy. Besides, the solutions rounded to seven significant digits are given in tabular form for convenience. They are also used as benchmark solutions to assess the collocation spectral method (CSM) and its modified version for radiative transfer equation (RTE). The results indicate that the CSM suffers from severe ray effect when the discontinuities appear in the scattering albedo and temperature distribution. Though the modified CSM can avoid the ray effect due to stepwise-change emissions, it requires heavier computational load but produces less accurate results than the composite Chebyshev integration method for RITEs under the same spatial grid system. Compared with the CSM or modified CSM to solve the RTE, the composite Chebyshev integration method for RITEs can achieve much higher accuracy with acceptable computational time, thus could also be a good alternative for thermal radiation calculations in simple geometry.