AI 中文总结
针对参数化纳维-斯托克斯流，提出时间二阶的惩罚-投影集合涡粘性算法，该算法基于BDF-2，具备grad-div稳定，经数值实验验证其收敛性与有效性。
AI 中文摘要
针对不可压缩纳维-斯托克斯流问题，我们提出了一种新颖、鲁棒且时间二阶精度的参数化惩罚-投影集合算法。该算法基于二阶向后差分公式（BDF-2）构建线性化形式，计算效率高，因为每个子问题在每个时间步的所有实现都共享相同的系数矩阵。为了提高对流主导流的鲁棒性，该方案引入了集合涡粘性（EEV）正则化。此外，该方案配备了控制分裂误差的grad-div稳定参数γ；在分析假设下，当γ→∞时，分裂误差会减小并渐近消失。我们证明了所提方案的稳定性，并通过论证当γ→∞时，该方案收敛到等价耦合形式，严格证明了其最优收敛性。我们还通过一系列数值实验验证了该方法，这些实验旨在验证理论预测的收敛速率并评估其在基准对流主导问题上的性能。数值结果与理论分析吻合良好，证实了所提方案的有效性。
英文摘要
We propose a novel, robust, and second-order-accurate parameterized penalty-projection ensemble algorithm for incompressible Navier--Stokes flow problems. The resulting linearized algorithm, based on the second-order Backward Differentiation Formula (BDF-2), is computationally efficient because it shares the same coefficient matrix across all realizations for each subproblem at every time step. To enhance robustness in convection-dominated flows, the scheme incorporates Ensemble Eddy Viscosity (EEV) regularization. In addition, it is equipped with grad-div stabilization parameter $γ$, which controls the splitting error; under the assumptions of the analysis, the splitting error decreases and vanishes asymptotically as $γ\to\infty$. We establish the stability of the proposed scheme and rigorously prove its optimal convergence by demonstrating that, as $γ\to\infty$, the scheme converges to an equivalent coupled formulation. We further validate the method through a series of numerical experiments designed to verify the theoretically predicted convergence rates and evaluate its performance on benchmark convection-dominated problems. The numerical results are in excellent agreement with the theoretical analysis and confirm the effectiveness of the proposed scheme.
Comments29 pages, 7 figures
