A Reynolds-Semi-Robust, Globally Divergence-Free HDG Method for the Smagorinsky Model
Abstract
We develop and analyze a fully discrete, globally divergence-free hybridizable discontinuous Galerkin (HDG) method for a gradient-based Smagorinsky model. The method combines backward Euler time stepping, interior-penalty discretizations of molecular and nonlinear eddy diffusion, and an upwind convective flux. The discrete velocity is H(div)-conforming and pointwise divergence-free, which yields pressure robustness. For sufficiently large penalty parameters, we prove unconditional energy stability and existence of a discrete solution, and establish uniqueness under additional smallness conditions. A velocity error estimate is derived without explicit inverse powers of the molecular viscosity. The nonlinear facet residuals are controlled using local trace-approximation estimates and a viscosity-independent facet penalty. We retain the dependence of the discrete Gronwall factor on the filter scale and the mesh size; a mesh-uniform bound follows under suitable solution regularity, a fixed time-step margin, and the scaling δ=O(h) on quasi-uniform meshes. The reported manufactured-solution results are consistent with the resulting pre-asymptotic error bounds. Further flow examples illustrate the dissipative behavior of the method and are distinguished from the boundary conditions and parameter range covered by the analysis.
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BibTeX
@misc{liu2026reynoldssemirobust,
title = {{A Reynolds-Semi-Robust, Globally Divergence-Free HDG Method for the Smagorinsky Model}},
author = {Shuaijun Liu and Xiaoping Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18652},
eprint = {2609.18652},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18652},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A Reynolds-Semi-Robust, Globally Divergence-Free HDG Method for the Smagorinsky Model
AU - Shuaijun Liu
AU - Xiaoping Xie
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18652
UR - https://arxiv.org/abs/2609.18652
AB - We develop and analyze a fully discrete, globally divergence-free hybridizable discontinuous Galerkin (HDG) method for a gradient-based Smagorinsky model. The method combines backward Euler time stepping, interior-penalty discretizations of molecular and nonlinear eddy diffusion, and an upwind convective flux. The discrete velocity is H(div)-conforming and pointwise divergence-free, which yields pressure robustness. For sufficiently large penalty parameters, we prove unconditional energy stability and existence of a discrete solution, and establish uniqueness under additional smallness conditions. A velocity error estimate is derived without explicit inverse powers of the molecular viscosity. The nonlinear facet residuals are controlled using local trace-approximation estimates and a viscosity-independent facet penalty. We retain the dependence of the discrete Gronwall factor on the filter scale and the mesh size; a mesh-uniform bound follows under suitable solution regularity, a fixed time-step margin, and the scaling δ=O(h) on quasi-uniform meshes. The reported manufactured-solution results are consistent with the resulting pre-asymptotic error bounds. Further flow examples illustrate the dissipative behavior of the method and are distinguished from the boundary conditions and parameter range covered by the analysis.
ER -