An Optimization-Based Discontinuous Galerkin Approach for High-Order Accurate Shock Tracking with Guaranteed Mesh Quality

High-order accurate discontinuous Galerkin (DG) methods have recently become popular for the simulation of a wide range of flow problems. However, for shocks and other discontinuities, these methods have to be stabilized by techniques such as limiting or artificial viscosity, leading to a first-order accurate method that requires very aggressive mesh adaptivity and a large number of elements. This work uses an alternative tracking-based technique for accurate solutions of these problems. Inspired by the fact that the DG method allows for discontinuities between all the mesh elements, we formulate a so-called r-adaptive method which moves the mesh in order to align the mesh edges with the jumps in the solution. This has the potential to obtain high-order accuracy with drastically fewer elements and lower computational cost. The proposed tracking method is based on a PDE-constrained optimization formulation, where the representation of the curved high-order mesh is used as optimization variables, the high-order DG discretization is enforced as a constraint, and the objective function is a discontinuity indicator with properties that make it well-suited for a gradient-based optimizer. A full space solver is used to converge the mesh and the solution simultaneously and it does not require solving the discrete PDE on intermediate non-aligned meshes. In previous work, the objective function also included a penalty term to regularize the element qualities and prevent inversion. This introduces a problem-dependent parameter which is difficult to determine, and it also leads to a compromise between the tracking objective and the element qualities. In this work, the mesh qualities are instead controlled by nonlinear constraints, and the shock tracking objective function is minimized among a large class of deformed meshes that satisfy a given quality condition. The mesh quality parameter in this formulation is easier to prescribe and it is essentially problem independent. Furthermore, it allows for a more precise minimization of the tracking objective function and potentially more accurate solutions. The method is demonstrated on Burgers’ equation in 1D and supersonic flow around a cylinder in 2D.