Asymptotically Derived Discretization and Acceleration Schemes for Radiation Transport

Heath Hanshaw, University of Michigan

Radiation transport theory is the science of calculating radiation fields in nuclear power, medical diagnostics and radiation therapy planning, non-destructive assay, thermal radiative transfer, and high energy physics experiments. Realistic problems are computationally large, even in the linear case, because typical media are extremely heterogeneous in phase space. Consequently, trade offs must be made between expense and various levels of approximation. Transport calculations are typically based either on deterministic solution of the Boltzmann equation or on Monte Carlo simulation of large numbers of individual particle tracks. Deterministic methods are generally much faster than Monte Carlo and yield solutions for all points of discretized phase space, not just a tally region of interest. However, the discretization typically results in an algorithm with non-uniform error and convergence properties over different medium properties and different initial and boundary conditions.

I am developing new discretization and acceleration/preconditioning schemes based on an asymptotic approach. Asymptotic methods applied to discretized differential equations can yield more stable numerical methods or reveal whether or not a numerical technique obeys the correct physics under scaling limits. One scaling limit is the thick diffusion limit, in which radiation particles undergo a very large number of collisions in a region of phase space small relative to the dimensions of interest. Problems with this type of behavior over significant regions of phase space are extremely computationally intensive if not accelerated, and many existing acceleration techniques are prone to instability or poor performance outside of idealized problems. My current work aims to build discretization schemes that have the correct asymptotic diffusion limit and to use that limit to establish stable acceleration and better preconditioning techniques for solving large scale radiation transport problems.

Abstract Author(s): Heath L. Hanshaw