High-order implicit shock tracking (fitting) is a class of high-order numerical methods that use numerical optimization to simultaneously compute a high-order approximation to a conservation law solution and align elements of the computational mesh with non-smooth features. This alignment ensures that 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. In this work, we devise a family of preconditioners for the saddle point linear system that defines the step toward optimality at each iteration of the optimization solver so Krylov solvers can be effectively used. Our preconditioners integrate standard preconditioners from constrained optimization with popular preconditioners for discontinuous Galerkin discretizations such as block Jacobi, block incomplete LU factorizations with minimum discarded fill reordering, and p-multigrid. Thorough studies are performed using two inviscid compressible flow problems to evaluate the effectivity of each preconditioner in this family and their sensitivity to critical shock tracking parameters such as the mesh and Hessian regularization, linearization state, and resolution of the solution space.

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