A Space-Time High-Order Implicit Shock Tracking Method for Shock-Dominated Unsteady Flows

Shock-dominated flows commonly arise in a wide variety of science and engineering disciplines. However, despite their prevalence, accurate and robust simulation of shock-dominated flows remains a significant challenge for modern computational fluid dynamics methods. High-order numerical methods are highly accurate per degree of freedom, introduce minimal dissipation, offer geometric flexibility, and have a high degree of parallel scalability but lack robustness for shocked flows because high-order approximation of discontinuities leads to spurious oscillations that cause failure of solvers. Popular methods to suppress these oscillations, known as shock capturing, are effective but often suffer from limitations in accuracy and are limited to global first order accuracy. Popular methods known as explicit shock fitting can provide high order accurate solutions but require significant effort in generating a shock-aligned mesh.
High-order implicit shock tracking (fitting) (HOIST) is a class of high-order, optimization-based numerical methods to approximate solutions of conservation laws with non-smooth features by aligning elements of the computational mesh with non-smooth features. This ensures the non-smooth features are perfectly represented by inter-element jumps and high-order basis functions approximate smooth regions of the solution without nonlinear stabilization, which leads to accurate approximations on traditionally coarse meshes.
This dissertation presents several advancements in the HOIST method, aimed at developing a robust approach for time-dependent problems and using the HOIST method to address multi-material flow problems and real gas flows. Key features of this work are the development of a slab-based space-time approach for implicit shock tracking, an implicit shock tracking approach using hyper-cube geometries for higher dimensional problems, and a high-order sharp-interface method for real gas single and two-phase flow simulations based on the HOIST method. The methods present in this work are tested on a sweep of problems of varying difficulty in 1-, 2-, and 3-dimensions of space and time.