A Discrete Maximum Principle for the Implicit Monte Carlo Equations

摘要

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 △_t, that is sufficient to eliminate these violations have recommended a △_t that is typically too small to be used in practice and that appeared to be much too conservative when compared to the actual △_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 △_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 △_x is to reduce the limitation on A,. 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.