AI 中文总结
研究针对Navier-Stokes方程的高阶一致分裂格式在高雷诺数下失效的问题，核心方法是添加谱消失粘性算子进行稳定化，主要贡献是建立了稳定格式的稳定性和误差估计，通过测试证明了其鲁棒性和准确性。
AI 中文摘要
Huang和Shen为不可压缩Navier-Stokes方程开发了一类新型的高阶BDF-IMEX一致分裂格式，首次对时间阶数高于二阶的完全解耦分裂格式进行了严格的稳定性和误差分析。本文将他们的分析从单位粘性扩展到任意粘性，发现误差上界系数包含粘性的逆幂。数值实验表明该格式在高雷诺数下会失效。为解决此问题，通过添加由定向应用的Maday-Kaber-Tadmor核构建的对称半正定谱消失粘性算子来稳定格式，该算子能在不增加渐近成本的情况下选择性地抑制高分辨率不足的模式，且保持误差分析结构不变。建立了稳定格式的稳定性和误差估计。三个二维测试证明了稳定格式的鲁棒性和准确性。对于给定解，稳定格式在k = 2,3,4时保持设计阶数，而非稳定格式发散；对于扰动的Kovasznay流，稳定格式在Re = 10^4时能准确解析边界层并将扰动驱动回稳态，非稳定格式则爆炸；对于Kelvin-Helmholtz不稳定问题，稳定格式在整个可靠区域内再现参考积分诊断，非稳定格式产生虚假解或爆炸。
英文摘要
Huang and Shen developed a novel class of high-order BDF-IMEX consistent-splitting schemes for the incompressible Navier-Stokes equations, giving the first rigorous stability and error analysis for a fully decoupled splitting scheme of temporal order higher than two. Extending their analysis from unit viscosity to arbitrary viscosity, this work reveals that the error upper bound coefficient contains inverse powers of the viscosity. Our numerical experiments show that the scheme can break down at high Reynolds number. To save the scheme from this failure, we stabilize it by adding to the velocity update a symmetric positive-semidefinite spectral vanishing viscosity operator, built from the directionally applied Maday-Kaber-Tadmor kernel, which selectively damps the high, under-resolved modes at no additional asymptotic cost and leaves the structure of the error analysis intact. We establish stability and error estimates for the stabilized scheme in which the spectral vanishing viscosity provides viscosity-independent coercive control of the high modes. Three two-dimensional tests demonstrate the robustness and accuracy of the stabilized scheme. For a manufactured solution, the stabilized scheme retains its design order for k=2,3,4, whereas the unstabilized scheme diverges. For a perturbed Kovasznay flow, it accurately resolves the boundary layer at Re=10^4 and drives the perturbation back to the steady state, while the unstabilized scheme blows up. For the Kelvin-Helmholtz instability problem, it reproduces the reference integral diagnostics throughout the reliable regime, whereas the unstabilized scheme produces spurious solutions or blows up.
Comments32 pages, 11 figures
