A predictor-corrector Monte Carlo method for thermal radiative transfer equations

In this paper, we develop a predictor-corrector Monte Carlo method for solving the thermal radiative transfer equations. A two-step numerical method is employed when the material temperature violates the maximum principle. It utilizes the effective scattering technique in the standard implicit Monte Carlo (IMC) method in each step. Then a linear combination of the prediction and correction solutions is obtained as the final solution, where the coefficient is determined by solving an optimization problem. The new method is more implicit than the IMC method in that it uses a more implicit estimate for the material temperature dependent quantities. Moreover, we note that the last modification step is necessary because using the iterative solution directly may overestimate the material temperature when large time steps are used. Numerical simulations show that the new method can significantly mitigate the violation of maximum principle in comparison with the standard IMC method, and results in more accurate solutions than the original predictor-corrector method when the time step is large.