# Measure of Diffusion Model Error for Thermal Radiation Transport

## Abstract

The diffusion approximation to the equation of transfer (Boltzmann transport equation) is usually applied to media where scattering dominates the interactions. Diffusion approximation helps in significant savings in terms of code complexity and computational time. However, this approximation often has significant error. Error due to the inherent nature of a physics model is called model error. Information about the model error associated with the diffusion approximation is clearly desirable. An indirect measure of model error is a quantity that is related in some way to the error but not equal to the error. In general, indirect measures of error are expected to be less costly than direct measures. Perhaps the most well-known indirect measure of the diffusion model error is the variable-Eddington tensor. This tensor provides a great deal of information about the angular dependence of the angular intensity solution, but it is not always simple to interpret.

We define a new indirect measure of the diffusion model error called the diffusion model error source (DME source). When this DME source is added to the diffusion equation, the transport solution for the angular-integrated intensity is obtained. In contrast to the variable-Eddington tensor, our DME source is a scalar that is conceptually easy to interpret. In addition to defining the DME source analytically, we show how to generate this source numerically relative to the Sn radiative transfer equations with linear-discontinuous spatial discretization. This numerical source is computationally tested and shown to reproduce the Sn solution for a number of problems. Our radiative transfer model solves a coupled, time dependent, multi-frequency, 1-D slab equation and material heat transfer equation. We then use diffusion approximation to solve the same problem. The difference due to this approximation can be modelled by a ?diffusion source?. The diffusion source is defined as an amount of inhomogeneous source that, when added to a diffusion calculation, gives a solution for the angle-integrated intensity that is equal to the transport solution.