An Essentially Implicit Monte Carlo Method for Radiative Transfer Equations

The implicit Monte Carlo (IMC) method has been widely used for solving the nonlinear thermal radiative transport equations for over 40 years. It is well known that the solutions of IMC method may non-physically violate the maximum principle for large time steps. In this article, we propose a variant of the IMC method called essentially implicit Monte Carlo (EIMC) method, which can eliminate the violations of the maximum principle. The EIMC method involves nonlinear iterations for the material temperature based on the Newton's method. This method is more implicit than the original IMC method in that it uses a better estimate for the end of time step material temperature, which allows for a more implicit estimate of the temperature-dependent quantities. Numerical simulations show that the new method can eliminate the violations of maximum principle that occurs in standard IMC method when the time step is large.