Entropy-stable hybridized discontinuous Galerkin methods for the compressible Euler and Navier-Stokes equations

TL;DR

This paper introduces entropy-stable hybridized discontinuous Galerkin methods for compressible fluid equations, improving stability and robustness in complex flow simulations while maintaining high-order accuracy.

Contribution

The paper develops entropy-stable hybridized DG methods in entropy variables for Euler and Navier-Stokes equations, enhancing stability and efficiency over traditional formulations.

Findings

Methods are entropy stable with suitable stabilization.

Schemes demonstrate optimal accuracy and robustness.

Superior performance in shock and under-resolved flows.

Abstract

In the spirit of making high-order discontinuous Galerkin (DG) methods more competitive, researchers have developed the hybridized DG methods, a class of discontinuous Galerkin methods that generalizes the Hybridizable DG (HDG), the Embedded DG (EDG) and the Interior Embedded DG (IEDG) methods. These methods are amenable to hybridization (static condensation) and thus to more computationally efficient implementations. Like other high-order DG methods, however, they may suffer from numerical stability issues in under-resolved fluid flow simulations. In this spirit, we introduce the hybridized DG methods for the compressible Euler and Navier-Stokes equations in entropy variables. Under a suitable choice of the stabilization matrix, the scheme can be shown to be entropy stable and satisfy the Second Law of Thermodynamics in an integral sense. The performance and robustness of the proposed family of schemes are illustrated through a series of steady and unsteady flow problems in subsonic, transonic, and supersonic regimes. The hybridized DG methods in entropy variables show the optimal accuracy order given by the polynomial approximation space, and are significantly superior to their counterparts in conservation variables in terms of stability and robustness, particularly for under-resolved and shock flows.