A Discrete Maximum Principle for the Implicit Monte Carlo Equations

It is well known that temperature solutions of the Implicit Monte Carlo (IMC) equations can exceed the external boundary temperatures, a violation of the “maximum principle.” Previous attempts to prescribe a maximum value of the time-step size Δ_t that is sufficient to eliminate these violations have recommended a Δ_t that is typically too small to be used in practice and that appeared to be much too conservative when compared to the actual Δ_t required to prevent maximum principle violations in numerical solutions of the IMC equations. In this paper we derive a new, approximate estimator for the maximum time-step size that includes the spatial-grid size Δ_x of the temperature field. We also provide exact necessary and sufficient conditions on the maximum time-step size that are easier to calculate. These explicitly demonstrate that the effect of coarsening Δ_x is to reduce the limitation on Δ_t. This helps explain the overly conservative nature of the earlier, grid-independent results. We demonstrate that the new time-step restriction is a much more accurate predictor of violations of the maximum principle. We discuss how the implications of the new, grid-dependent time-step restriction can affect IMC solution algorithms.