Characteristic boundary conditions for Hybridizable Discontinuous Galerkin methods

TL;DR

The paper embeds characteristic boundary conditions (CBCs) into Hybridizable Discontinuous Galerkin (HDG) discretizations for compressible flows, introducing both Navier–Stokes CBCs (NSCBCs) and a novel generalized characteristic relaxation CBC (GRCBC). By recasting boundary dynamics in terms of incoming/outgoing characteristic waves and incorporating tangential and viscous contributions, the authors achieve non-reflecting or weakly reflecting boundaries in planar and multidimensional weakly compressible regimes. The approach is validated through planar and oblique acoustic pulses, vortical disturbances, and cylinder flow, illustrating when far-field versus target-state boundaries are preferable and how relaxation parameters can tune reflections. Overall, CBCs within HDG provide a flexible, efficient framework for reducing boundary reflections without resorting to mirror elements or extensive sponge layers, improving accuracy in aeroacoustic and low-Mach simulations.

Abstract

In this work we introduce the concept of characteristic boundary conditions (CBCs) within the framework of Hybridizable Discontinuous Galerkin (HDG) methods, including both the Navier-Stokes characteristic boundary conditions (NSCBCs) and a novel approach to generalized characteristic relaxation boundary conditions (GRCBCs). CBCs are based on the characteristic decomposition of the compressible Euler equations and are designed to prevent the reflection of waves at the domain boundaries. We show the effectiveness of the proposed method for weakly compressible flows through a series of numerical experiments by comparing the results with common boundary conditions in the HDG setting and reference solutions available in the literature. In particular, HDG with CBCs show superior performance minimizing the reflection of vortices at artificial boundaries, for both inviscid and viscous flows.

Figures (19)

• Figure 1: Local orthogonal coordinate system $(\xi, \eta)$ with the orthonormal basis vectors ${\bm{n}}$ and ${\bm{t}}$.
• Figure 2: Schematic representation of the incoming, ${{\bm{\mathcal{L}}}}^-_{outflow}$, and outgoing, ${{\bm{\mathcal{L}}}}^+$, characteristic amplitudes at a subsonic outflow boundary $\Gamma$.
• Figure 3: Schematic representation of the incoming, ${{\bm{\mathcal{L}}}}^-_{inflow}$, and outgoing, ${{\bm{\mathcal{L}}}}^+$, characteristic amplitudes at a subsonic inflow boundary $\Gamma$.
• Figure 4: Representation of the domain $\Omega$ with corresponding boundaries $\Gamma$ and the triangulation $\mathcal{T}_h$ with mesh size $h = 0.035$.
• Figure 5: Downstream propagating linear acoustic pulse with strength $\alpha = 0.001$: incoming $\overline{w}^-$ and outgoing $\overline{w}^+$ acoustic characteristics at the boundary $\Gamma_{out}$.

Theorems & Definitions (6)

• Remark 2.1
• Remark 2.2
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 4.1