Symbolic implicit Monte Carlo radiation transport in the difference formulation: a piecewise constant discretization

The equations of radiation transport for thermal photons are notoriously difficult to solve in thick media without resorting to asymptotic approximations such as the diffusion limit. One source of this difficulty is that in thick, absorbing media, thermal emission and absorption are almost completely balanced. A new formulation for thermal radiation transport, called the difference formulation, was recently introduced in order to remove the stiff balance between emission and absorption. In the new formulation, thermal emission is replaced by derivative terms that become small in thick media. It was proposed that the difficulties of solving the transport equation in thick media would be ameliorated by the difference formulation, while preserving full rigor and accuracy of the transport solution in the streaming limit. In this paper, the transport equation is solved by the symbolic implicit Monte Carlo method and comparisons are made between the standard formulation and the difference formulation. The method is easily adapted to the derivative source terms of the difference formulation, and a remarkable reduction in noise is obtained when the difference formulation is applied to problems involving thick media.