Accurate and efficient computation of the 3D radiative transfer equation in highly forward-peaked scattering media using a renormalization approach

This paper considers an accurate and efficient numerical scheme for solving the radiative transfer equation (RTE) based on the discrete ordinates method (DOM) in highly forward-peaked scattering media. The DOM approximates the scattering integral of the RTE including the phase function into a quadrature sum with the total number of discrete angular directions (TND) by a quadrature set. Due to large numerical errors of the scattering integral based on the DOM in highly forward-peaked scattering, the phase function is renormalized to satisfy its normalization conditions. Although the renormalization approaches of the phase function improve the accuracy of the numerical results of the RTE, the computational efficiency of the RTE is still required. This paper develops the first order renormalization approach using the double exponential formula for three quadrature sets: level symmetric even, even and odd, and Lebedev sets in a wide range of the TND from 48 to 1454. Numerical errors of the three-dimensional time-dependent RTE are investigated by the analytical solutions of the RTE. The investigation shows that the level symmetric even set with the TND of 48 using the developed approach provides the most accurate results of the RTE among the quadrature sets in the range of the TND, while to obtain the same accuracy by the conventional zeroth order renormalization approach, the TND needs to be larger than 360. The results suggest the large reduction of computational loads by the developed approach to less than 10 percent from those in the conventional approach.