  This work introduces a novel discontinuity-tracking framework for resolving
discontinuous solutions of conservation laws with high-order numerical
discretizations that support inter-element solution discontinuities, such as
discontinuous Galerkin methods. The proposed method aims to align inter-element
boundaries with discontinuities in the solution by deforming the computational
mesh. A discontinuity-aligned mesh ensures the discontinuity is represented
through inter-element jumps while smooth basis functions interior to elements
are only used to approximate smooth regions of the solution, thereby avoiding
Gibbs' phenomena that create well-known stability issues. Therefore, very
coarse high-order discretizations accurately resolve the piecewise smooth
solution throughout the domain, provided the discontinuity is tracked. Central
to the proposed discontinuity-tracking framework is a discrete PDE-constrained
optimization formulation that simultaneously aligns the computational mesh with
discontinuities in the solution and solves the discretized conservation law on
this mesh. The optimization objective is taken as a combination of the the
deviation of the finite-dimensional solution from its element-wise average and
a mesh distortion metric to simultaneously penalize Gibbs' phenomena and
distorted meshes. We advocate a gradient-based, full space solver where the
mesh and conservation law solution converge to their optimal values
simultaneously and therefore never require the solution of the discrete
conservation law on a non-aligned mesh. The merit of the proposed method is
demonstrated on a number of one- and two-dimensional model problems including
2D supersonic flow around a bluff body. We demonstrate optimal
$\mathcal{O}(h^{p+1})$ convergence rates in the $L^1$ norm for up to polynomial
order $p=6$ and show that accurate solutions can be obtained on extremely
coarse meshes.
