The method of characteristics (MOC) is a deterministic transport method that has been applied to large-scale problems including those in reactor physics and radiative transfer. Long characteristics, (LC) methods, have been used extensively to discretize and solve transport problems in the spatial domain. There is a need for an equally adequate time-dependent discretization for these transport problems. The new contributions from this research include the development of a space-time long characteristic (STLC) method with various source approximations including several that employ a piece-wise linear (PWL) approximation spatially. In the prism-PWL (PPWL) method the coefficient of each PWL spatial function is linear in time in each space-time cell. Along with STLC, a PWL-LC method is developed for steady-state problems in (x, y) and (x, y, z). The methods developed in this work use least-squares projections to determine the coefficients of their source approximations. This work presents a detailed asymptotic analysis of the PWL-LC and STLC methods in the thick diffusion limit, which is of special interest in radiative transfer problems. This is the first such analysis reported for LC methods and these new methods are the first that are expected to perform well in this limit. Results from test problems executed with a modified version of the Parallel Deterministic Transport code, PDT, show the PWL-LC and STLC methods are more accurate than current methods for streaming problems. From asymptotic analysis and test problems, it is found that the steady-state PWL-LC method is accurate in the thick diffusion limit with solutions similar to those of analogous discontinuous finite element method, DFEM, solutions. Similarly, the PPWL-STLC method is found to be accurate in time-dependent thick diffusive problems. STLC is also a promising method for massively parallel applications because it permits the use of track-based sweeping, which appears to have significant advantages over cell-based sweeping. This is a key topic recommended for further research.

Discretization and solving of the transport equation has been an area of great research where many methods have been developed. Under the deterministic transport methods, the method of characteristics, MOC, is one such discretization and solution method that has been applied to large-scale problems. Although these MOC, specifically long characteristics, LC, have been thoroughly applied to discretize and solve transport problems in the spatial domain, there is a need for an equally adequate time-dependent discretization. A method has been developed that uses LC discretization of the time and space variables in solving the transport equation. This space-time long characteristic, STLC, method is a discrete ordinates method that applies LC discretization in space and time and employs a least-squares approximation of sources such as the scattering source in each cell. This method encounters the same problems that previous spatial LC methods have dealt with concerning achieving all of the following: particle conservation, exact solution along a ray, and smooth variation in reaction rate for specific problems. However, quantities that preserve conservation in each cell can also be produced with this method and compared to the non-conservative results from this method to determine the extent to which this STLC method addresses the previous problems. Results from several test problems show that this STLC method produces conservative and non-conservative solutions that are very similar for most cases and the difference between them vanishes as track spacing is refined. These quantities are also compared to the results produced from a traditional linear discontinuous spatial discretization with finite difference time discretization. It is found that this STLC method is more accurate for streaming-dominate and scattering-dominate test problems. Also, the solution from this STLC method approaches the steady-state diffusion limit solution from a traditional LD method. Through asymptotic analysis and test problems, this STLC method produces a time-dependent diffusion solution in the thick diffusive limit that is accurate to O(E) and is similar to a continuous linear FEM discretization method in space with time differencing. Application of this method in parallel looks promising, mostly due to the ray independence along which the solution is computed in this method.