An Adaptive Shock-Capturing HDG Method for Compressible Flows

Abstract

We introduce a hybridizable discontinuous Galerkin (HDG) method for the numerical solution of the compressible Euler equations with shock waves. By locally condensing the approximate conserved variables the HDG method results in a final system involving only the degrees of freedom of the approximate traces of the conserved variables. The HDG method inherits the geometric flexibility and high-order accuracy of discontinuous Galerkin methods, and offers a significant reduction in the computational cost. In order to treat compressible fluid flows with discontinuities, the HDG method is equipped with an artificial viscosity term based on an extension of existing artificial viscosity methods. Moreover, the artificial viscosity can be used as an indicator for adaptive grid refinement to improve shock profiles. Numerical results for subsonic, transonic, supersonic, and hypersonic flows are presented to demonstrate the performance of the proposed approach.

A class of embedded discontinuous Galerkin methods for computational fluid dynamics

“…Both the HDG and EDG methods have been used to solve RANS flows, shock flows, as well as unsteady flows [44][45][46]55,56]. Future work will focus on extending the IEDG method to simulate these flows and developing effective Newton-Krylov solvers for solving the algebraic systems arising from the IEDG discretization of nonlinear systems of conservation laws in fluid mechanics.…”

“…To put the EDG and IEDG methods in perspective, we compare them to the HDG method [44][45][46]56] and other DG methods such as the LDG method [28], the CDG method [57] or the method of Bassi and Rebay [4]. We consider triangular and tetrahedral discretizations and polynomial approximations of order k = 1, .…”

“…It is then reasonable to expect that in this situation, the EDG methods might prove to be more efficient than the original HDG method. In other words, the EDG method [55] constructed from the HDG method for the compressible Euler and Navier-Stokes equations [44][45][46]56] in the convection-dominated regime, might be more efficient than the HDG method.…”

High-Order Strand Grid Methods for Shock Turbulence Interaction

“…Many approaches have been proposed to handle these flows in a high-order context. The most popular of these approaches include specialized limiting techniques [37] and artificial dissipation [23,25]. Both approaches require substantially different formulations than traditional monotonicity-preserving limiters used in second-order schemes.…”

Assessment of Hybrid High-Order methods on curved meshes and comparison with discontinuous Galerkin methods

“…On the other hand, recent works by Chan et al [13,14] rely on reference frame polynomial spaces introducing weight-adjusted L 2 -inner products in order to recover high-order accuracy. HDG has been employed on meshes with curved boundaries, mainly in the context of compressible flow problems [31,28]; eXtended HDG with level-set description of interfaces has been recently investigated by Gurkan et al [27]. Fidkowski [24] compared DG and HDG methods for unsteady simulations of convection-dominated flows on mapped deforming domains.…”