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The moving discontinuous Galerkin method with interface condition enforcement(MDG-ICE) is a high-order, r-adaptive method that treats the grid as a variableand weakly enforces the conservation law, constitutive law, and correspondinginterface conditions in order to implicitly fit high-gradient flow features. Inthis paper, we develop an optimization solver based on the Levenberg-Marquardtalgorithm that features an anisotropic, locally adaptive penalty method toenhance robustness and prevent cell degeneration in the computation ofhypersonic, viscous flows. Specifically, we incorporate an anisotropic gridregularization based on the mesh-implied metric that inhibits grid motion indirections with small element length scales, an element shape regularizationthat inhibits nonlinear deformations of the high-order elements, and a penaltyregularization that penalizes degenerate elements. Additionally, we introduce aprocedure for locally scaling the regularization operators in an adaptive,elementwise manner in order to maintain grid validity. We apply the proposedMDG-ICE formulation to two- and three-dimensional test cases involving viscousshocks and/or boundary layers, including Mach 17.6 hypersonic viscous flow overa circular cylinder and Mach 5 hypersonic viscous flow over a sphere, which arevery challenging test cases for conventional numerical schemes on simplicialgrids. Even without artificial dissipation, the computed solutions are freefrom spurious oscillations and yield highly symmetric surface heat-fluxprofiles.

The Moving Discontinuous Galerkin Method with Interface Condition Enforcement for the Simulation of Hypersonic, Viscous Flows