  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.
