SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2869-A2896, January 2021.
Computing a flow system a number of times with different samples of flow parameters is a common practice in many uncertainty quantification applications, which can be prohibitively expensive for complex nonlinear flow problems. This report presents two second order, stabilized, scalar auxiliary variable (SAV) ensemble algorithms for fast computation of the Navier--Stokes flow ensembles: Stab-SAV-CN and Stab-SAV-BDF2. The proposed ensemble algorithms are based on the ensemble timestepping idea which makes use of a quantity called the ensemble mean to construct a common coefficient matrix for all realizations at the same time step after spatial discretization, in which case efficient block solvers, e.g., block GMRES, can be used to significantly reduce both storage and computational time. The adoption of a recently developed SAV approach that treats the nonlinear term explicitly results in a constant shared coefficient matrix among all realizations at different time steps, which further cuts down the computational cost, yielding an extremely efficient ensemble algorithm for simulating nonlinear flow ensembles with provable long time stability without any time step conditions. The SAV approach for the Navier--Stokes equations for a single realization was proved to be unconditionally stable in [L. Lin, Z. Yang, and S. Dong, J. Comput. Phys., 388 (2019), pp 1--22; X. Li and J. Shen, SIAM J. Numer. Anal., 58 (2020), pp. 2465--2491]. However we found the SAV approach has very low accuracy that compromises its stability in our initial numerical investigations for several commonly tested benchmark flow problems. In this report, we propose to use the stabilization $-\alpha h \Delta (u^{n+1}-u^n)$ in Stab-SAV-CN and $-\alpha h \Delta (3u^{n+1}-4 u^{n}+u^{n-1})$ in Stab-SAV-BDF2 to address this issue. We prove that both of our ensemble algorithms are long time stable under one parameter fluctuation condition, without any time step constraints. For a single realization, both algorithms are unconditionally stable and have better accuracy than the SAV methods studied in [L. Lin, Z. Yang, and S. Dong, J. Comput. Phys., 388 (2019), pp 1--22; X. Li and J. Shen, SIAM J. Numer. Anal., 58 (2020), pp. 2465--2491] for our test problems. Extensive numerical experiments are performed to show the efficiency of the proposed ensemble algorithms and the effectiveness of the stabilization for increasing accuracy and stability.

中文翻译：

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参数化流问题的稳定标量辅助变量集成算法

SIAM 科学计算杂志，第 43 卷，第 4 期，第 A2869-A2896 页，2021 年 1 月。
使用不同的流动参数样本多次计算流动系统是许多不确定性量化应用中的常见做法，这对于复杂的非线性流动问题可能会非常昂贵。本报告介绍了两个二阶稳定标量辅助变量 (SAV) 集成算法，用于快速计算 Navier-Stokes 流集成：Stab-SAV-CN 和 Stab-SAV-BDF2。所提出的集成算法基于集成时间步长思想，该思想利用称为集成均值的数量为空间离散化后的同一时间步长的所有实现构造一个公共系数矩阵，在这种情况下，高效的块求解器，例如块 GMRES , 可用于显着减少存储和计算时间。采用最近开发的 SAV 方法显式地处理非线性项，在不同时间步长的所有实现中产生一个恒定的共享系数矩阵，这进一步降低了计算成本，产生了一种极其有效的集成算法，用于模拟具有可证明性的非线性流集成。没有任何时间步长条件的长期稳定性。单个实现的 Navier-Stokes 方程的 SAV 方法被证明在 [L. Lin, Z. Yang 和 S. Dong, J. Comput。物理学，388 (2019)，第 1--22 页；X. Li 和 J. Shen, SIAM J. Numer。分析，58 (2020)，第 2465--2491 页]。然而，我们发现 SAV 方法的精度非常低，这在我们对几个常用测试基准流问题的初始数值研究中影响了其稳定性。在这份报告中，我们建议在 Stab-SAV-CN 中使用稳定性 $-\alpha h \Delta (u^{n+1}-u^n)$ 和 $-\alpha h \Delta (3u^{n+1}- 4 u^{n}+u^{n-1})$ 在 Stab-SAV-BDF2 中解决这个问题。我们证明了我们的两种集成算法在一个参数波动条件下都是长期稳定的，没有任何时间步长限制。对于单个实现，两种算法都是无条件稳定的，并且比 [L. Lin, Z. Yang 和 S. Dong, J. Comput。物理学，388 (2019)，第 1--22 页；X. Li 和 J. Shen, SIAM J. Numer。Anal., 58 (2020), pp. 2465--2491] 用于我们的测试问题。进行了广泛的数值实验，以显示所提出的集成算法的效率以及稳定性提高准确性和稳定性的有效性。