Implicit shock tracking using an optimization-based high-order discontinuous Galerkin method

A novel framework for resolving discontinuous solutions of conservation laws, e.g., contact lines, shock waves, and interfaces, using implicit tracking and a high-order discontinuous Galerkin (DG) discretization was introduced in [38]. Central to the framework is an optimization problem whose solution is a discontinuity-aligned mesh and the corresponding high-order approximation to the flow that does not require explicit meshing of the unknown discontinuity surface. The method was shown to deliver highly accurate solutions on coarse, high-order discretizations without nonlinear stabilization and recover optimal convergence rates even for problems with discontinuous solutions. This work extends the implicit tracking framework such that robustness is improved and convergence accelerated. In particular, we introduce an improved formulation of the central optimization problem and an associated sequential quadratic programming (SQP) solver. The new error-based objective function penalizes violation of the DG residual in an enriched test space and is shown to have excellent tracking properties. The SQP solver simultaneously converges the nodal coordinates of the mesh and DG solution to their optimal values and is equipped with a number of features to ensure robust, fast convergence: Levenberg-Marquardt approximation of the Hessian with weighted elliptic regularization, backtracking line search, and rigorous convergence criteria. We use the proposed method to solve a range of inviscid conservation laws of varying difficulty. We show the method is able to deliver accurate solutions on coarse, high-order meshes and the SQP solver is robust and usually able to drive the first-order optimality system to tight tolerances.