Entropy stable Discontinuous Galerkin Scheme for the compressible Navier-Stokes Equations

The class of nonlinear systems of conservation laws contains many important examples. A case in point are the Euler equations and the Navier-Stokes equations, governing inviscid and viscous compressible fluid flow, respectively. In analyzing the stability of numerical schemes for such systems, entropy stability is often the framework of choice. Furthermore, some convergence proofs have been provided for high order space-time discontinuous Galerkin (DG) schemes and more general hyperbolic systems which heavily exploit the property of entropy stability. Compared to entropy stability analysis of hyperbolic conservation laws, much less work has been done in extending entropy stability of DG schemes to nonlinear convection-diffusion systems and most of the literature try to mimic the well-known analysis for the linear Poisson problems e.g., in. To the best knowledge of the authors only very few results are available in this direction (also see). More specifically, in Ref. 35 a difference formulation based on entropy variables has been presented and the entropy stability of the method has been proved in the semi-discrete form. In Ref. 12 the symmetric/nonsymmetric interior penalty DG formulation has been presented for the one-dimensional Navier-Stokes equations realized in terms of entropy variables, and entropy stability has been proved in the semi-discrete settings. In Ref. 27 a different formulation of the interior penalty has been presented employing a different penalty scaling, similar to Ref. 22, for the one-dimensional Navier-Stokes equations. Furthermore, a variant of LDG method is presented for this problem, as well as an entropy stability proof in space-time fully-discrete settings. Another work is the Ref. 15 which presents an entropy-stable space-time formulation for turbulent computation using the Roe flux for the convective part, and the second method of Bassi and Rebay for the viscous part. Though no proof has been provided for entropy stability, the authors have given illustrative examples by comparison between using conservative and entropy variables. The authors highly emphasized the importance of having an entropy stable method as a base-line scheme, despite issues arising from inexact integration. In fact, the authors showed that while de-aliasing process by over-integration is sufficiently helpful for an entropy stable scheme in high Reynolds flow computation, it is unable to provide a stable formulation for under-resolved turbulent simulation if one realize the formulation in terms of conservative variables, which does not enjoy the entropy stability. In this work we consider a more general DG formulation for the Navier-Stokes equations by using a mixed method for discretizing the viscous flow in addition to the entropy stable formulation for the convective flux as we presented in our earlier works (also see e.g.,), which is able to represent a family of different known viscous discretization schemes, extended to the nonlinear diffusion of the Navier-Stokes equations in a canonical way. Most of the schemes in the classic paper by Arnold et al. can be included and proved to be entropy stable. We give a variety of examples in this abstract, augmented with numerical examples.