Anisotropic Diffusion Approximations for Time-dependent Particle Transport.

Abstract

In this thesis, we develop and numerically test new approximations to time-dependent radiation transport with the goal of obtaining more accurate solutions than the diffusion approximation can generate, yet requiring less computational effort than full transport. The first method is the nascent anisotropic diffusion (AD) approximation, which we extend to time-dependent problems in finite domains; the second is a novel anisotropic P_1-like (AP_1) approximation. These methods are ``anisotropic'' in that, rather than operating under the assumption of linearly anisotropic radiation, they incorporate an arbitrary amount of anisotropy via a transport-calculated diffusion coefficient. This anisotropic diffusion tensor is the second angular moment of a simple, purely absorbing transport problem.

In this thesis, much of the computational testing of the new methods is performed in ``flatland'' geometry, a fictional two-dimensional universe that provides a realistic but computationally inexpensive testbed. As work ancillary to anisotropic diffusion and the numerical experiments, a complete description of flatland diffusion, including boundary conditions, is developed. Also, implementation details for both Monte Carlo and S_N transport in flatland are provided.

The two new anisotropic methods, along with a ``flux limited'' modification to anisotropic diffusion, are tested in a variety of problems. Some aspects of the theory, including the newly formulated boundary conditions, are tested first with diffusive, steady-state problems. The new methods are compared against existing ones in linear, time-dependent radiation transport problems. Finally, the efficacy and performance of the anisotropic methods are investigated in several thermal radiative transfer (TRT) computational experiments.

Our results demonstrate that for many multi-dimensional problems, the new anisotropic methods perform much better than their conventional counterparts. In every time-dependent test, the flux-limited anisotropic diffusion approach produced the most accurate solutions of the new methods. Based on our numerical testing, we believe this method to be a strong contender for accurate, inexpensive simulations of time-dependent transport and thermal radiative transfer problems.