Nonlinear entropy stability and a summation-by-parts framework are used to derive provably stable, polynomial-based spectral collocation element methods of arbitrary order for the compressible Navier-Stokes equations. The new methods are similar to strong form, nodal discontinuous Galerkin spectral elements but conserve entropy for the Euler equations and are entropy stable for the Navier-Stokes equations. Shock capturing follows immediately by combining them with a dissipative companion operator via a comparison approach. Smooth and discontinuous test cases are presented that demonstrate their efficacy. Introduction.Next generation numerical algorithms for use in large eddy simulations (LES) and hybrid Reynolds-averaged Navier-Stokes (RANS)-LES simulations will undoubtedly rely on efficient high-order formulations. Although highorder techniques are well suited for LES, most lack robustness when the solution contains discontinuities or even underresolved physical features. Although a variety of stabilization techniques have been developed for second-order methods (e.g., total variation diminishing (TVD) limiters [35], and entropy stability [40]), extending these techniques to high-order formulations has been problematic. High-order essentially nonoscillatory (ENO) [20,36] and weighted ENO (WENO) [31,25] schemes provide a partial remedy to the problem; they achieve high-order design accuracy away from captured discontinuities and maintain sharp "nearly monotone" captured shocks. Unfortunately, nonoscillatory schemes experience instabilities in less than ideal circumstances (e.g., curvilinear mapped grids or expansion of flows into vacuum). Because nonoscillatory schemes are largely based on stencil biasing heuristics rather than stability analysis, there is little theory to guide further development efforts focused on

Non-linear entropy stability and a summation-by-parts framework are used to derive entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations. A semi-discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator. The data at the boundary are weakly imposed using a penalty flux approach and a simultaneousapproximation-term penalty technique. Although discontinuous spectral collocation operators on unstructured grids are used herein for the purpose of demonstrating their robustness and efficacy, the new boundary conditions are compatible with any diagonal norm summation-by-parts spatial operator, including finite element, finite difference, finite volume, discontinuous Galerkin, and flux reconstruction/correction procedure via reconstruction schemes. The proposed boundary treatment is tested for three-dimensional subsonic and supersonic flows. The numerical computations corroborate the non-linear stability (entropy stability) and accuracy of the boundary conditions.

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