Characteristic boundary conditions for Hybridizable Discontinuous Galerkin methods

2025-03-25

AI Summary

This work addresses a critical issue in hybridizable discontinuous Galerkin (HDG) simulations of weakly compressible flows: spurious wave reflections at artificial boundaries that distort vortical structures. To resolve this, we systematically incorporate characteristic boundary conditions (CBCs)—derived from Euler equation eigenanalysis—into the HDG framework for the first time. We propose two novel CBC formulations: standard Navier–Stokes characteristic boundary conditions (NSCBCs) and their generalized extension, generalized relaxation characteristic boundary conditions (GRCBCs), which employ a dynamic relaxation mechanism to enhance robustness for unsteady, multiscale flows. Numerical experiments demonstrate that the proposed CBCs effectively suppress non-physical boundary reflections. Both inviscid and viscous weakly compressible flow simulations exhibit significantly improved accuracy and stability, with discretization errors reduced by an order of magnitude compared to conventional HDG boundary treatments.

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• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
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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.

Problem

• Prevent wave reflection at domain boundaries using CBCs
• Compare CBC effectiveness with common HDG boundary conditions
• Minimize vortex reflection at artificial boundaries in flows

Innovation

• Characteristic boundary conditions for HDG methods
• Novel generalized characteristic relaxation boundary conditions
• Minimizes vortex reflection at artificial boundaries