AI 中文总结
针对谱差分法的二阶通量，开发了一种将模板从五单元减至三单元的紧致中心格式，恢复了偶次多项式阶数的收敛性，在湍流案例中稳定性优于BR1格式。
AI 中文摘要
不连续谱元法（DSEM）中二阶（粘性）项的离散化通常依赖于辅助梯度变量，该变量在单元界面处的处理会影响格式的精度和稳定性。Bassi-Rebay（BR1）格式因简单且无参数而具有吸引力，但在偶次多项式阶数下收敛性欠佳，且需要扩展的五单元模板。受Huynh的通量重构格式启发，我们为谱差分（SD）法内的二阶通量开发了一种紧致、完全中心型的格式。所提方法使用依赖界面的单侧连续通量修改辅助梯度的重构，将模板从五个单元减少到三个单元，同时保留BR1的中心型和无参数特性。该格式在一维中开发并扩展到多维。采用时间特征分析来表征其耗散和色散特性，包括内罚项的影响。数值测试考虑了线性扩散方程、欠分辨的局域狄拉克δ函数、非线性多孔介质方程，以及雷诺数Re=1600和5000下三维泰勒-格林涡的隐式大涡模拟。该紧致格式恢复了所有多项式阶数（包括偶次阶数）的预期收敛阶数，并减少了欠分辨和非线性区域中的伪振荡。由于对高波数数值模式的阻尼得到改善，其在标准格式失效的湍流案例中仍保持稳定。所提方法为SD法中的二阶通量提供了一种有吸引力的、可替代BR1的紧致方案。
英文摘要
The discretization of second-order (viscous) terms in Discontinuous Spectral Element Methods (DSEMs) typically relies on an auxiliary gradient variable, whose treatment at element interfaces affects the accuracy and stability of the scheme. The Bassi-Rebay (BR1) formulation is attractive for its simplicity and parameter-free character, but suffers from sub-optimal convergence at even polynomial orders and requires an extended five-element stencil. Inspired by Huynh's Flux Reconstruction formulation, we develop a compact, fully-centered scheme for second-order fluxes within the Spectral Difference (SD) method. The proposed approach modifies the reconstruction of the auxiliary gradient using interface-dependent, one-sided continuous fluxes, reducing the stencil from five to three elements while preserving the centered and parameter-free nature of BR1. The formulation is developed in one dimension and extended to multiple dimensions. Temporal eigenanalysis is used to characterize its dissipation and dispersion properties, including the effects of interior penalty terms. Numerical tests consider the linear diffusion equation, an under-resolved localized Dirac's delta, the nonlinear porous medium equation, and implicit large-eddy simulations of the three-dimensional Taylor-Green vortex at $\mathrm{Re}=1600$ and $5000$. The compact scheme restores the expected convergence order for all polynomial degrees, including even orders, and reduces spurious oscillations in under-resolved and nonlinear regimes. It also remains stable in turbulent cases where the standard formulation fails, owing to improved damping of high-wavenumber numerical modes. The proposed approach provides an attractive compact alternative to BR1 for second-order fluxes in the SD method.
