Truncation Analysis and Numerical Method Improvements for the Thermal Radiative Transfer Equations.

Abstract

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. 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 accuracy.

We also examine the spatial discretization error known as photon

teleportation which is the difference between the distribution

of absorption and emission locations. The current method used to reduce photon

teleportation, called source tilting, is compared to our implementation of Func-

tional Expansion Tallies (FET’s). FET’s are a significant improvement at reducing

photon teleportation over the source tilting techniques.

Finally, we examine the use of time step controllers. Based on our detailed trunca-

tion analysis, we propose a time step controller to 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 current approach.