An r-adaptive, high-order discontinuous Galerkin method for flows with attached shocks

This work extends the method developed in [1, 2] that uses a high-order discontinuous Galerkin discretization and optimization-based r-adaptivity to track discontinuous solutions of conservation laws with the underlying mesh and provide high-order accurate approximations without additional stabilization techniques, e.g., limiting or artificial viscosity, to problems with shocks attached to curved boundaries. Central to the framework is an optimization problem whose solution is a shock-aligned mesh and the correspondingDGapproximation to the flow; in this sense, the framework is an implicit tracking method, which distinguishes it from methods that aim to explicitly mesh the shock surface. The optimization problem is solved using a sequential quadratic programming method that simultaneously converges the mesh and DG solution, which is critical to avoids nonlinear stability issues that would come from computing a DG solution on a unconverged (non-aligned) mesh. In the case of a shock attached to a curved boundary, the mesh coordinates are parametrized in terms of a reduced set of degrees of freedom and a mapping that ensures boundary nodes always conform to the appropriate boundary. We use the proposed method to solve for the inviscid, transonic flow around a NACA0012 airfoil. The framework generates a mesh that successfully tracks the attached shock and provides an accurate flow approximation on a relatively coarse mesh.