Convergence analysis of the Random ordinate method for mitigating the ray effect

The Discrete Ordinates Method (DOM) is widely used for velocity discretization in radiative transport simulations. However, DOM tends to exhibit the ray effect when the velocity discretization is not sufficiently refined, a limitation that is well documented. To counter this, we have developed the Random Ordinates Method (ROM) by integrating randomness into the velocity discretization, which mitigates the ray effect without incurring additional computational costs. ROM partitions the velocity space into $ n $ cells, selects a random ordinate from each cell, and solves a DOM system with these ordinates. It leverages the average of multiple samples to achieve a higher convergence order, especially for solutions with low regularity in the velocity variable. In this work, we provide a detailed convergence analysis for ROM, focusing on bias and single-run errors. This analysis is crucial for determining the necessary mesh size and the optimal number of samples required to attain a specified level of accuracy.