Runge-kutta ray-tracing technique for radiative transfer in a three-dimensional graded-index medium

The Runge-Kutta ray-tracing method is combined with the Monte Carlo method to analyze the radiative transfer in a three-dimensional graded-index media. The analytical solution of the trajectory of a ray in a refractive index distribution of a Maxwell fisheye lens is obtained. The temperature field, heat flux, and apparent emissivity calculated by the analytical solution of the trajectory of a ray are treated as the standard solution to verify the results of the Runge-Kutta ray-tracing methods. The third-order fixed-step, fourth-order fixed-step, and third-order variable-step Runge-Kutta ray-tracing methods are discussed here. More accurate results can be obtained by using a higher-order Runge-Kutta algorithm; however, the third-order variable-step-size Runge-Kutta ray-tracing method has better precision than the fourth-order fixed-step Runge-Kutta formula for the temperature field. The maximum relative error of the temperature as compared to the standard solution is less than 0.19%, the integratedmean relative error is less than 0.06%, and the maximum relative error of the emissivity is less than 0.94% in the case of Δt = 10^-3. The Runge-Kutta ray-tracing method can effectively solve the radiative transfer in a three-dimensional graded-index media.