Extending the discrete maximum principle for the IMC equations Public Deposited



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  • The implicit Monte Carlo (IMC) method [16] for radiative transfer, developed in 1971, provides numerical solutions to the tightly-coupled, highly-nonlinear radiative heat transfer equations in many physical situations. Despite its popularity, there are instances of overheating in the solution for particular choices of time steps and spatial grid sizes. To prevent overheating, conditions on teh time step size Δt have been sought to ensure that the implicit Monte Carlo (IMC) equations satisfy a maximum principle. Most recently, a discrete maximum principle (DMP) for teh IMC equations has been developed [32] that predicts the necessary time step size for boundedness given the spatial grid size. Predictions given by this DMP assumed equilibrium thermal initial conditions, was developed using pseudo-analytic and symbolic algebra tools that are computationally expensive, has only been applied to one-dimensional Marshak wave problems, and has not considered the evolution of the DMP predictions over multiple time steps. These limitations restrict the utility of the DMP predictions. We extend the DMP derivation to overcome these limitations and provide an algorithm that can be introduced into IMC codes with minimal impact on simulation CPU time. This extended DMP effectively treats non-equilibrium thermal initial conditions, decreases calculation time by using multigroup approximations in frequency, considers multiple spatial dimensions with an arbitrary number of neighboring sources, and overcomes inherent difficulties for the DMP in time-dependent problems. Disequilibrium in the initial conditions is introduced through a redefinition of existing terms from [32] to different radiation and material temperatures on the first time step. This results in a limiting DMP inequality similar in form to the original. Multifrequency approximations are then applied by assuming separation of variables. Energy deposition from multiple sources is assumed to follow linear superposition and the DMP from [32] is re-derived to incorporate multiple incident sources of energy in multiple dimensions. Lastly, an inherent flaw in the DMP resulting in poor predictions when temperature varies slowly over a region is overcome by developing a threshold temperature difference, above which the DMP operates. We have numerically implemented these improvements and validated the results against IMC solutions, showing the predictive capacity of the more general DMP algorithm. We find the disequlibrium conditions to be properly incorporated into the DMP, and multifrequency approximations to be accurate over a large range of time step and spatial grid sizes. The linear superposition assumption is generally very accurate, but infrequently leads to DMP predictions which are not conservative. We also demonstrate that the temperature difference threshold prevents inaccurate predictions by the DMP while preserving its functionality.
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