"This thesis provides roughly three contributions to the study of defect correction methods and hybrid methods for radiation transport. First, a modification of traditional integral deferred correction (IDC) time-integration schemes is proposed that significantly reduces the storage requirements of the methods. These methods, which we call low-storage IDC or LS-IDC methods, require storing only one copy of each stage vector throughout the iteration process, whereas traditional IDC methods require two copies of each vector. The convergence and stability properties of the methods are examined in a variety of settings, and both analytical and experimental results are provided. A nonlinear ODE and a linear transport equation are used to compare the accuracy and storage requirements of LS-IDC integrators with other fully implicit schemes including diagonally implicit Runge-Kutta (DIRK) and space-time discontinuous Galerkin (STDG) methods. The results demonstrate that LS-IDC methods have similar accuracy but significantly reduced memory requirements compared to other fully implicit methods. Second, extensions of a collision-based hybrid method for time-dependent radiation transport simulations are discussed. The hybrid methods are constructed by splitting the radiation flux into collided and uncollided components to which low- and high-resolution discrete ordinates approximations are applied, respectively. The use of arbitrarily high-order numerical approximations is emphasized, with particular attention paid to high-order time-integration schemes. A range of time-integrators are considered including DIRK, STDG, IDC, and LS-IDC methods of up to fifth-order accuracy. Convergence results in one spatial dimension are provided, and it is found that while the hybrid methods exhibit convergence stagnation and order reduction in certain scenarios, the overall accuracy of the hybrid approximations is comparable to standard discrete ordinates approximations in many cases. A test problem in two-dimensional x y-geometry consisting of a mock-up of a standard hohlraum configuration is used to compare the computational efficiency of the hybrid methods with standard discrete ordinates methods. It is observed that replacing a standard discrete ordinates method using an angular quadrature of order N with a hybrid discrete ordinates method using angular quadratures of order 2N and N/2 for the uncollided and collided fluxes, respectively, usually reduces solve time by a factor of 2 or more and in many cases also yields a reduction in solution error by a factor of up to 2. However, it is noted that the specified hybrid methods require approximately 4.25 times the storage of the corresponding standard discrete ordinates methods. Finally, two mechanisms for increasing the effectiveness of the hybrid methods are presented. The first is a reconstruction procedure for mapping between arbitrary discrete ordinates quadratures within the context of these hybrid methods. This approach, called Nyström reconstruction, is shown to be significantly more accurate than previous reconstruction methods. When applied to the two-dimensional hohlraum problem, it is observed that replacing a standard discrete ordinates method using an angular quadrature of order N with a hybrid discrete ordinates method using a Nyström reconstruction procedure and angular quadratures of order N and N/4 for the uncollided and collided fluxes, respectively, consistently reduces solve time by a factor of between 4 and 8 while increasing memory requirements by only 6% and producing little to no increase in solution error. Lastly, variations of hybrid methods using IDC integrators are described in which the hybrid approach is written as a two-grid iterative method in angle that is combined with an IDC time-integration scheme. It is demonstrated that the resulting methods are able to iteratively reduce the error due to the application of discrete ordinates quadratures of different resolutions to the collided and uncollided components."--Pages ii-iii.
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