Hybrid Numerical Methods and Solution Verification for the Neutron Transport Equation

The neutron transport equation approximates how neutrons interact with different materials in various systems. This work focuses on applying hybrid numerical methods to improve the current methods used to approximate the neutron flux in addition to further developing a single spatial grid error estimator. One hybrid method employs machine learning and neural networks to predict the matrix-vector multiplication product in k-eigenvalue problems. This alleviates the use of large scattering and fission matrices where these machine learning models require less than 3% of the original memory storage. These models are able to remain accurate to the reference solution and improve convergence times while being used in a plurality of problems not trained on. The training process is also improved by using random forests to initialize neural networks and encoders to limit the model input size. A collision-based hybrid method separates the time-dependent neutron transport equation into uncollided and collided equations. High fidelity angular and energy grids are used with the uncollided equation to increase accuracy while low fidelity grids are used with the collided equation to improve convergence times. This technique is applied to one- and two-dimensional problems which demonstrate better convergence times than similar high fidelity grid coarsening schemes and more accurate results when compared to low fidelity grid coarsening. In certain instances, the hybrid method converges in half the time it takes source iteration. Solution verification is important to ensure an approximate solution is sufficiently close to the true solution of the governing equation. The method of nearby problems is a spatial grid error estimator that can identify locations of high spatial error without requiring multiple spatial grids. This is shown to be effective with one- and two-dimensional problems for fixed source and k-eigenvalue problems using both discrete ordinates and Monte Carlo neutron transport solvers. The method of nearby problems identifies areas of high spatial error with similar accuracy to methods that use multiple spatial grids.