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A new discrete ordinates method designed for use on modern, large memory, vector and/or parallel processing super computers has been developed. The method is similar to conventional SN techniques in that the medium is divided into spatial mesh cells and that discrete directions are used. However, in place of an approximate differencing scheme, a nearly exact matrix representation of the streaming operator is determined. Although extremely large, this matrix can be stored on today's computers for repeated use in the source iteration. Since the source iteration is cast in matrix form it benefits enormously from vector and/or parallel processing, if available. Several test results are presented demonstrating the reduction in numerical diffusion and elimination of ray effects.

Iterative solutions of the Boltzmann transport equation are computationally intensive. Spatial multigrid methods have led to efficient iterative algorithms for solving a variety of partial differential equations; thus, it is natural to explore their application to transport equations. Manteuffel et al. conducted such an exploration in one spatial dimension, using two-cell inversions as the relaxation or smoothing operation, and reported excellent results. In this dissertation we extensively test Manteuffel??s one-dimensional method and our modified versions thereof. We demonstrate that the performance of such spatial multigrid methods can degrade significantly given strong heterogeneities. We also extend Manteuffel??s basic approach to two-dimensional problems, employing four-cell inversions for the relaxation operation. We find that for uniform homogeneous problems the two-dimensional multigrid method is not as rapidly convergent as the one-dimensional method. For strongly heterogeneous problems the performance of the two-dimensional method is much like that of the one-dimensional method, which means it can be slow to converge. We conclude that this approach to spatial multigrid produces a method that converges rapidly for many problems but not for others. That is, this spatial multigrid method is not unconditionally rapidly convergent. However, our analysis of the distribution of eigenvalues of the iteration operators indicates that this spatial multigrid method may work very well as a preconditioner within a Krylov iteration algorithm, because its eigenvalues tend to be relatively well clustered. Further exploration of this promising result appears to be a fruitful area of further research.

In this dissertation, advanced numerical methods for highly forward peaked scattering deterministic calculations are devised, implemented, and assessed. Since electrons interact with the surrounding environment through Coulomb interactions, the scattering kernel is highly forward-peaked. This bears the consequence that, with standard preconditioning, the standard Legendre expansion of the scattering kernel requires too many terms for the discretized equation to be solved efficiently using a deterministic method. The Diffusion Synthetic Acceleration (DSA), usually used to speed up the calculation when the scattering is weakly anisotropic, is inefficient for electron transport. This led Morel and Manteuffel to develop a one-dimensional angular multigrid (ANMG) which has proved to be very effective when the scattering is highly anisotropic. Later, Pautz et al. generalized this scheme to multidimensional geometries, but this method had to be stabilized by a diffusive filter that degrades the overall convergence of the iterative scheme. In this dissertation, we recast the multidimensional angular multigrid method without the filter as a preconditioner for a Krylov solver. This new method is stable independently of the anisotropy of the scattering and is increasingly more effective and efficient as the anisotropy increases compared to DSA preconditioning wrapped inside a Krylov solver. At the coarsest level of ANMG, a DSA step is needed. In this research, we use the Modified Interior Penalty (MIP) DSA. This DSA was shown to be always stable on triangular cells with isotropic scattering. Because this DSA discretization leads to symmetric definite-positive matrices, it is usually solved using a conjugate gradient preconditioned (CG) by SSOR but here, we show that algebraic multigrid methods are vastly superior than more common CG preconditioners such as SSOR. Another important part of this dissertation is dedicated to transport equation and diffusion solves on arbitrary polygonal meshes. The advantages of polygonal cells are that the number of unknowns needed to mesh a domain can be decreased and that adaptive mesh refinement implementation is simplified: rather than handling hanging nodes, the adapted computational mesh includes different types of polygons. Numerical examples are presented for arbitrary quadrilateral and polygonal grids.

Iterative solutions of the Boltzmann transport equation are computationally intensive. Spatial multigrid methods have led to efficient iterative algorithms for solving a variety of partial differential equations; thus, it is natural to explore their application to transport equations. Manteuffel et al. conducted such an exploration in one spatial dimension, using two-cell inversions as the relaxation or smoothing operation, and reported excellent results. In this dissertation we extensively test Manteuffel??s one-dimensional method and our modified versions thereof. We demonstrate that the performance of such spatial multigrid methods can degrade significantly given strong heterogeneities. We also extend Manteuffel??s basic approach to two-dimensional problems, employing four-cell inversions for the relaxation operation. We find that for uniform homogeneous problems the two-dimensional multigrid method is not as rapidly convergent as the one-dimensional method. For strongly heterogeneous problems the performance of the two-dimensional method is much like that of the one-dimensional method, which means it can be slow to converge. We conclude that this approach to spatial multigrid produces a method that converges rapidly for many problems but not for others. That is, this spatial multigrid method is not unconditionally rapidly convergent. However, our analysis of the distribution of eigenvalues of the iteration operators indicates that this spatial multigrid method may work very well as a preconditioner within a Krylov iteration algorithm, because its eigenvalues tend to be relatively well clustered. Further exploration of this promising result appears to be a fruitful area of further research.

A novel collision source method has been developed to solve the Linear Boltzmann Equation (LBE) more efficiently by adaptation of the angular quadrature order. The angular adaptation method is unique in that the flux from each scattering source iteration is obtained, with potentially a different quadrature order used for each. Traditionally, the flux from every iteration is combined, with the same quadrature applied to the combined flux. Since the scattering process tends to distribute the radiation more evenly over angles (i.e., make it more isotropic), the quadrature requirements generally decrease with each iteration. This method allows for an optimal use of processing power, by using a high order quadrature for the first few iterations that need it, before shifting to lower order quadratures for the remaining iterations. This is essentially an extension of the first collision source method, and is referred to as the adaptive collision source (ACS) method. The ACS methodology has been implemented in the 3-D, parallel, multigroup discrete ordinates code TITAN. This code was tested on a variety of test problems including fixed-source and eigenvalue problems. The ACS implementation in TITAN has shown a reduction in computation time by a factor of 1.5-4 on the fixed-source test problems, for the same desired level of accuracy, as compared to the standard TITAN code. / Ph. D.

The objective of this body of work was to produce a code system capable of processing boundary angular flux data from discrete ordinates calculations in 2D and 3D Cartesian and cylindrical geometries into cumulative probability density functions that can be used with a Monte Carlo radiation transport code to define neutron and photon initial positions, directions, and energies. In order to accomplish this goal, the DISCO (DetermInistic-Stochastic Coupling Operation) code was created to interface between the DORT and TORT deterministic radiation transport codes and the MCNP stochastic radiation transport code. DISCO introduces new methods to use the boundary angular flux data, along with information regarding the deterministic quadrature sets and spatial mesh structure, to create cumulative probability density functions that are passed to MCNP for sampling within the source.F90 subroutine that was also generated as part of this work. Operating in concert, DISCO and the MCNP source.F90 subroutine create a source term according to the discrete ordinates angular flux information. In order to validate the work described herein, 24 test cases were created to exercise the different geometries and execution modes available. The results of these test cases confirm that the methodology and corresponding implementation is appropriate and functioning correctly. Furthermore, this work incorporates several novel features such as compatibility with all 2D and 3D Cartesian and cylindrical geometries, an angular and spatial indexing scheme to reduce random sampling operations, a streamlining of process execution, and the ability for the resulting Monte Carlo code to operate in either serial and parallel mode.

The approaches for discretizing the direction variable in particle transport calculations are the discrete-ordinates method and function-expansion methods. Both approaches are limited if the transport solution is not smooth. Angular discretization errors in the discrete-ordinates method arise from the inability of a given quadrature set to accurately perform the needed integrals over the direction ("angular") domain. We propose that an adaptive discrete-ordinate algorithm will be useful in many problems of practical interest. We start with a "base quadrature set" and add quadrature points as needed in order to resolve the angular flux function. We compare an interpolated angular-flux value against a calculated value. If the values are within a user specified tolerance, the point is not added; otherwise it is. Upon the addition of a point we must recalculate weights. Our interpolatory functions map angular-flux values at the quadrature directions to a continuous function that can be evaluated at any direction. We force our quadrature weights to be consistent with these functions in the sense that the quadrature integral of the angular flux is the exact integral of the interpolatory function (a finite-element methodology that determines coefficients by collocation instead of the usual weightedresidual procedure). We demonstrate our approach in two-dimensional Cartesian geometry, focusing on the azimuthal direction The interpolative methods we test are simple linear, linear in sine and cosine, an Abu-Shumays “base” quadrature with a simple linear adaptive and an Abu-Shumays “base” quadrature with a linear in sine and cosine adaptive. In the latter two methods the local refinement does not reduce the ability of the base set to integrate high-order spherical harmonics (important in problems with highly anisotropic scattering). We utilize a variety of one-group test problems to demonstrate that in all cases, angular discretization errors (including "ray effects") can be eliminated to whatever tolerance the user requests. We further demonstrate through detailed quantitative analysis that local refinement does indeed produce a more efficient placement of unknowns. We conclude that this work introduces a very promising approach to a long-standing problem in deterministic transport, and we believe it will lead to fruitful avenues of further investigation.

The approaches for discretizing the direction variable in particle transport calculations are the discrete-ordinates method and function-expansion methods. Both approaches are limited if the transport solution is not smooth. Angular discretization errors in the discrete-ordinates method arise from the inability of a given quadrature set to accurately perform the needed integrals over the direction ("angular") domain. We propose that an adaptive discrete-ordinate algorithm will be useful in many problems of practical interest. We start with a "base quadrature set" and add quadrature points as needed in order to resolve the angular flux function. We compare an interpolated angular-flux value against a calculated value. If the values are within a user specified tolerance, the point is not added; otherwise it is. Upon the addition of a point we must recalculate weights. Our interpolatory functions map angular-flux values at the quadrature directions to a continuous function that can be evaluated at any direction. We force our quadrature weights to be consistent with these functions in the sense that the quadrature integral of the angular flux is the exact integral of the interpolatory function (a finite-element methodology that determines coefficients by collocation instead of the usual weightedresidual procedure). We demonstrate our approach in two-dimensional Cartesian geometry, focusing on the azimuthal direction The interpolative methods we test are simple linear, linear in sine and cosine, an Abu-Shumays â baseâ quadrature with a simple linear adaptive and an Abu-Shumays â baseâ quadrature with a linear in sine and cosine adaptive. In the latter two methods the local refinement does not reduce the ability of the base set to integrate high-order spherical harmonics (important in problems with highly anisotropic scattering). We utilize a variety of one-group test problems to demonstrate that in all cases, angular discretization errors (including "ray effects") can be eliminated to whatever tolerance the user requests. We further demonstrate through detailed quantitative analysis that local refinement does indeed produce a more efficient placement of unknowns. We conclude that this work introduces a very promising approach to a long-standing problem in deterministic transport, and we believe it will lead to fruitful avenues of further investigation.

This dissertation focuses on the development and implementation of a new method to solve the time-dependent form of the linear Boltzmann transport equation for reactor transients. This new method allows for a stable solution to the fully explicit form of the transport equation with delayed neutrons by employing an error-controlled, adaptive Runge-Kutta-Fehlberg (RKF) method to differentiate the time domain. Allowing for the time step size to vary adaptively and as needed to resolve the time-dependent behavior of the angular flux and neutron precursor concentrations. The RKF expansion of the time domain occurs at each point and is coupled with a Source Iteration to resolve the spatial behavior of the angular flux at the specified point in time. The decoupling of the space and time domains requires the application of a quasi-static iteration between solving the time domain using adaptive RKF with error control and resolving the space domain with a Source Iteration sweep. The research culminated with the development of the 1-D Adaptive Runge-Kutta Time-Dependent Transport code (ARKTRAN-TD), which successfully implemented the new method and applied it to a suite of reactor transient benchmarks.