High order DG methods offer improved accuracy in turbulent and under-resolved flows due to low numerical dissipation and dispersion, but may lose robustness due to the lack of stabilization. A priori stabilization techniques, such as artificial viscosity, slope limiting, and filtering, require heuristic parameter, which does not provide provable guarantees of robustness and may lead to over-dissipation of solutions.

For the compressible Euler and Navier-Stokes equations, physically meaningful solutions must maintain positive physical quantities and satisfy entropy stability. The primary objective of this dissertation is to develop high order DG methods that are provably entropy stable and preserve positivity through a posteriori limiting techniques

We first introduce a more general discretization of the viscous terms in the compressible Navier-Stokes equations, which also enables a simple and explicit imposition of entropy stable wall boundary conditions.

A positivity limiting strategy for entropy-stable discontinuous Galerkin spectral element (ESDGSEM) discretizations is then introduced. The strategy is constructed by blending high order solutions with a low order positivity-preserving and semi-discretely entropy stable discretization through an elementwise limiter. The proposed limiting strategy is both semi-discretely entropy stable and positivity preserving for the compressible Navier-Stokes equations under an appropriate CFL condition.

Another approach we propose is an entropy stable limiting strategy for discontinuous Galerkin spectral element (DGSEM) discretizations. The strategy is an extension to the subcell limiting strategy that satisfies the semi-discrete cell entropy inequality by formulating the limiting factors as solutions to an optimization problem. The optimization problem is efficiently solved using a deterministic greedy algorithm. We discuss the extension of the proposed subcell limiting strategy to preserve positivity and general convex constraints.

Numerical experiments confirm the high order accuracy, entropy stability, and robustness of the proposed strategies.