Truncation analysis and numerical method improvements for the Thermal Radiative Transfer Equations.

摘要

The Implicit Monte Carlo (IMC) method has been the standard Monte Carlo approach to solving the Thermal Radiative Transfer Equations for the last 38 years. While this method has proven itself to be robust at reaching the equilibrium solution, there has been no work published on the detailed sources of truncation error associated with the method. In this thesis, we explore the sources of error in the IMC method and compare them to another method proposed by Carter and Forrest (CF) in 1973. The CF method is exact in zero-D linear problems and was used to quantify the bias in the IMC approximations.;A detailed time truncation analysis leads to the identification of the leading source of truncation error for both the IMC and CF methods. This analysis suggests that by applying a predictor-corrector to estimate the opacity at the middle of the time step, the CF method can be made a second order accurate method in a nonlinear zero-D problem. However, even with this better opacity estimate the IMC method generally remains first order accurate. Using our knowledge of the IMC bias however, we can create a second order accurate predictor-corrector IMC method when using alpha = 0.5 although alpha = 1 in practice for stability concerns. We also create a Variable Weight Predictor-Corrector, which uses fewer particles with higher energy-weight in the predictor step than the corrector step. This greatly reduces computation time of the predictor-corrector methods while preserving their accuracy.;We also examine the spatial discretization error known as photon teleportation. Photon teleportation is due to the difference between the distribution of absorption and emission locations. The current method used to reduce photon teleportation is called source tilting and is compared to our implementation of Functional Expansion Tallies (FET's). FET's are a significant improvement at reducing photon teleportation over the current source tilting techniques.;Finally, we examine the use of time step controllers. Based on our detailed truncation analysis, we propose a time step controller that will control the leading source of error in a zero-D problem. We also implement a predictor-corrector scheme with this approach to aid in the selection of a time step size while also gaining information to be used in the corrector step. The time step controller with predictor-corrector method proved to be more accurate and efficient than the currently used time step controllers.