Computational fluid dynamics (CFD) play an essential role in both industry and research for analyzing compressible flows dominated by shocks, enhancing experimental and theoretical studies. Achieving accurate and robust simulations of these shock-dominated flows, such as those encountered around airplanes, poses a significant challenge for contemporary CFD techniques. High-order numerical methods, such as DG methods, have received considerable attention because: they introduce minimal numerical dissipation, are highly accurate per degree of freedom, provide geometric flexibility, and exhibit excellent parallel scalability. However, these methods often struggle with robustness in scenarios involving shocks and contact discontinuities, as the high-order approximation of shocks and discontinuities can induce spurious oscillations, leading to the failure of numerical solvers.

In recent years, a novel category of numerical methods, termed implicit shock tracking (IST) methods, has been developed, utilizing numerical optimization to achieve high-order DG approximations of shock-dominated flow solutions while aligning the computational mesh with non-smooth features. These methods represent these features accurately through inter-element jumps and allow high-order basis functions to approximate smooth areas of the flow without requiring nonlinear stabilization. As a consequence, IST methods achieve accurate approximations of shock-dominated flows even on traditionally coarse meshes. This dissertation introduces two significant advancements for IST.

First, we integrate ideas from IST with extended DG (XDG) methods, introducing the XDGIST method. This innovative method utilizes a level set function to define discontinuity interfaces, which segment but do not deform the computational grid, thereby circumventing cumbersome mesh operations used in conventional IST methods. The approximation space is enriched by XDG basis functions, discontinuous at the interfaces, and the latter are aligned accurately with shocks using IST methodologies. We successfully apply the method to various test scenarios, including steady 2D and unsteady 1D inviscid flow problems. We show that the XDGIST method is more accurate and the only method maintaining high-order convergence properties for 1D shock-acoustic-wave interaction problems, when compared to a traditional DG method employing artificial viscosity.

Second, aiming at extending the use of IST to large-scale problems, we present a family of preconditioners tailored for the saddle point linear systems that define the progression towards optimality in each iteration of the optimization solver. These preconditioners merge traditional constrained optimization techniques with widely-used methods for DG discretizations, including block Jacobi, block incomplete LU factorizations and P-multigrid. Comprehensive evaluations are conducted using two 2D inviscid compressible flow scenarios to assess the effectiveness of each preconditioner within this family, and their responsiveness to key IST parameters, singling out the best preconditioner in terms of performance.