The global optimal reference field for the difference formulation in the implicit Monte Carlo radiation transport

Abstract

Statistical noise has always been one of the most important problems hindering the wide application of the Monte Carlo method in thermal radiative transfer. The difference formulation which has been introduced in recent years is a source manipulation variance reduction technique, in which the radiation intensity is defined relative to a reference field. In this formulation, the transport operator remains unchanged, but the solution and source functions are changed. Numerical results show that in this formulation the symbolic implicit Monte Carlo (SIMC) method can significantly reduce statistical noise in some cases. The standard formulation can be considered as a special formulation where the reference field is zero. In general, the reference field is chosen as the thermal equilibrium radiation field at the beginning or end of the time step. However, we found that this choice can lead to a larger statistical noise than in the standard formulation in some cases, and even non-physical oscillation of the calculation results, which severely restrict the application of this method.In this paper, we develop the difference formulation based on generalized reference fields, and the Fleck-Cummings implicit Monte Carlo technique is employed to solve the functions. Through analysis, we find that the choice of the reference field is the key to reduce statistical noise. We define the global optimal reference field, which satisfies the minimum for the L1 norm of energy weights of the particles emitted during a time step. Although it is difficult to find the global optimal reference field, we propose a scheme to select a reference field which satisfies that the statistical noise is lower than the zero reference field in most cases. Numerical results show that the new method can avoid the nonphysical oscillation that occurs in regions of the problem with sharp spatial gradients in the solution in the classical difference formulation. The method also leads to a remarkable reduction in noise comparing to the Fleck-Cummings implicit Monte Carlo method in the standard formulation.