发表机构
Sichuan University; National Key Laboratory of Fundamental Algorithms and Models for Engineering Simulation, Sichuan University（四川大学; 工程模拟基础算法与模型全国重点实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为 Smagorinsky 模型提出一种全局无散度 HDG 方法，结合后向欧拉与内罚离散，证明能量稳定性并给出无粘性逆幂的误差估计，数值实验验证理论。
AI 中文摘要
我们针对基于梯度的 Smagorinsky 模型开发并分析了一种全离散、全局无散度的混合化间断伽辽金（HDG）方法。该方法结合了后向欧拉时间推进、分子扩散和非线性涡扩散的内罚离散化，以及迎风对流通量。离散速度是 $H(\operatorname{div})$ 相容且逐点无散度的，这保证了压力稳健性。对于足够大的罚参数，我们证明了无条件能量稳定性和离散解的存在性，并在额外的小性条件下建立了唯一性。在不显式涉及分子粘性逆幂的情况下，推导出了速度误差估计。非线性面残差通过局部迹逼近估计和与粘性无关的面罚来控制。我们保留了离散 Gronwall 因子对滤波尺度和网格尺寸的依赖性；在适当的解正则性、固定的时间步长余量以及准均匀网格上的缩放 $\delta=O(h)$ 下，可以得到网格一致界。所报告的制造解结果与由此产生的预渐近误差界一致。进一步的流动实例说明了该方法的耗散行为，并与分析所涵盖的边界条件和参数范围进行了区分。
英文摘要
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 $H(\operatorname{div})$-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.
