弱可压缩流动的数据同化数值分析
Numerical analysis of data assimilation for slightly compressible flow
AI总结：
本文对同时同化速度与压力数据的弱可压缩流动数据同化模型及其有限元离散化开展数值分析，建立稳定性与误差估计，验证最优收敛率。
AI中文摘要：
连续数据同化通过持续将模型向可用观测数据靠拢来改进流动预测。对于弱可压缩流动，近期有模型针对仅速度靠拢的局限，同时同化速度与压力数据，并将两者靠拢到不可压缩Navier-Stokes方程[5]；连续时间误差估计和初步实验表明，这种联合靠拢方法有效，相比仅速度靠拢能大幅降低模型误差。受这些结果启发，我们对该模型及其有限元离散化开展数值分析。我们建立了半离散格式和全离散线性化向后欧拉格式的稳定性与误差估计。分析显示存在无限可预测性视界：初始误差的影响随时间指数衰减，模型误差关于观测分辨率H为一阶，关于压力靠拢参数μ₁为μ₁⁻¹/²阶。平衡这两项误差项，选取μ₁=𝒪(H⁻²)，可得到最优收敛率。数值实验验证了预测的收敛率。
英文摘要：
Continuous data assimilation improves flow predictions by continually nudging a model toward available observational data. For slightly compressible flow, a recent model addresses the limitations of velocity-only nudging by assimilating both velocity and pressure data and nudging both quantities into the incompressible Navier-Stokes equations [5]; continuous-in-time error estimates and preliminary experiments show that this joint nudging is effective and substantially reduces the model error relative to velocity-only nudging. Motivated by these results, we carry out the numerical analysis of the model and its finite element discretizations. We establish stability and error estimates for the semi-discrete scheme and for the fully discrete, linearized backward Euler scheme. The analysis shows an infinite predictability horizon: the effect of the initial error decays exponentially in time, and the model error is first order in the observation resolution H and of order $μ_1^{-1/2}$ in the pressure nudging parameter $μ_1$. Balancing these two error terms, we choose $μ_1=\mathcal{O}(H^{-2})$, which yields the optimal convergence rate. Numerical experiments confirm the predicted rates.
