The Moving Discontinuous Galerkin Method with Interface Condition Enforcement for Robust Simulations of High-Speed Viscous Flows

The moving discontinuous Galerkin method with interface condition enforcement (MDG-ICE) is a high-order, r-adaptive method that treats the grid as a variable and weakly enforces the conservation law, constitutive law, and corresponding interface conditions in order to implicitly fit high-gradient flow features. In this paper, we introduce nonlinear solver strategies to more robustly and efficiently compute high-speed viscous flows. Specifically, we incorporate an anisotropic grid regularization based on the mesh-implied metric into the nonlinear least-squares solver that inhibits grid motion in directions with small element length scales. Furthermore, we develop an adaptive elementwise regularization strategy that locally scales the regularization terms as needed to maintain grid validity. We apply the proposed MDG-ICE formulation to test cases involving viscous shocks and/or boundary layers, including Mach 17.6 hypersonic viscous flow over a circular cylinder and Mach 5 hypersonic viscous flow over a sphere, which are very challenging test cases for conventional numerical schemes on simplicial grids. Even without artificial dissipation, the computed solutions are free from spurious oscillations and yield highly symmetric surface heat-flux profiles.