Robust High-Order Implicit Shock Tracking Solvers for Shock Dominated Flows

This dissertation enhances the High-Order Implicit Shock Tracking (HOIST) method, a computational technique that addresses the challenges posed by shocks and discontinuities in high-speed flow simulations. Traditional high-order numerical methods, while accurate, struggle in the presence of shocks, often requiring complex parameter tuning or losing accuracy due to shock misalignment. HOIST represents an advancement by implicitly aligning mesh elements with discontinuities through an optimization process, thus eliminating the need for explicitly generating shock-aligned meshes. Significant improvements are introduced that include the development of a specialized sequential quadratic programming optimization solver which incorporates robustness measures to effectively handle complex shock interactions, such as shock reflections. This solver enhances the adaptability and accuracy of the HOIST method across various flow scenarios without extensive user intervention. A notable innovation is the creation of a specialized HOIST framework designed for "many-query" analyses involving parametrized shocks. This framework leverages the inherent capabilities of the HOIST method to efficiently explore parameter spaces, facilitating optimization, uncertainty quantification, and extensive parameter sweeps, providing capabilities not available in traditional nonlinear stabilization methods. The dissertation also extends the applicability of the HOIST method to viscous flow scenarios. It introduces strategies such as residual scaling and viscosity continuation to manage viscous shocks and avoid extensive mesh distortion. Additionally, it tackles the challenge of curved shocks with a novel mesh parametrization technique that controls element curvature, enhancing the method's overall robustness and solution quality. In summary, these advancements make the HOIST method a highly effective and robust tool for analyzing complex fluid dynamics problems involving shocks, significantly enhancing its efficiency, applicability, and accuracy.