Computer Science – Sound
Scientific paper
Aug 1998
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1998phdt........10b&link_type=abstract
Thesis (PHD). THE UNIVERSITY OF MICHIGAN , Source DAI-B 59/02, p. 851, Aug 1998, 249 pages.
Computer Science
Sound
Spatial Differencing
Scientific paper
A spatial and angular moment analysis of the linear Boltzmann transport equation is used to compute exact flux-weighted average spatial quantities such as the 'center of mass' and 'radius of gyration' of the flux distribution. This moment analysis is valid for multidimensional general-geometry analytic transport problems, posed in an infinite homogeneous medium, with multiple energy groups and anisotropic scattering. The results from the analysis are used in this thesis to assess how accurately approximations to the transport equation compute these flux-weighted average spatial quantities. The first part of this thesis addresses the theoretical analysis of spatial differencing schemes used to discretize the discrete ordinates approximation of the linear Boltzmann transport equation. Discrete ordinates methods have been utilized for many years to obtain numerical solutions of neutron transport problems in which the optical width of the spatial cells is small. The traditional truncation analysis can be used to assess the accuracy of spatial differencing schemes for these problems. The same discrete ordinates methods have in recent years been utilized for radiative transfer problems characterized by optically thick spatial cells and scattering ratios near unity. In this case, an asymptotic diffusion limit analysis has been applied to discretized transport problems in order to assess the accuracy of spatial differencing schemes. At present, theoretical methods for analyzing discretized transport problems with optically intermediate and thick spatial cells and arbitrary scattering ratios are not available. We develop a moment analysis method for theoretically analyzing discrete ordinates spatial differencing schemes that makes no assumptions on the optical thickness of the spatial cells or on the value of the scattering ratio. The second part of this thesis concerns the Simplified PN (SPN) approximation, a multidimensional generalization of the one-dimensional planar geometry PN approximation. The SPN approximation was proposed in the early 1960's but has not gained wide acceptance because of its weak theoretical basis. Our goal is to establish a sound theoretical basis for the Simplified PN approximation, in the hopes that it will find broader application with increased confidence. In particular, we examine the Simplified P3 approximation and show that it more accurately computes higher-order spatial and angular moments than the P1 (diffusion) approximation. Then, we derive the multigroup Simplified P3 approximation using a variational analysis that yields the SP3 equations along with interface and Marshak-like boundary conditions. We also develop an efficient explicit iterative algorithm for solving the multigroup SP3 equations. Finally, we present numerical solutions of transport problems that verify the accuracy of the multigroup SP3 approximation.
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