The chaotic nature of fluid flow and the uncertainties in initial conditions limit predictability. Small errors that occur in the initial condition can grow exponentially until they saturate at $\mathcal{O}$(1). Ensemble forecasting averages multiple runs with slightly different initial conditions and other data to produce more accurate results and extend the predictability horizon. However, they can be computationally expensive. We develop a penalty-based ensemble method with a shared coefficient matrix to reduce required memory and computational cost and thereby allow larger ensemble sizes. Penalty methods relax the incompressibility condition to decouple the pressure and velocity, reducing memory requirements. This report gives stability proof and an error estimate of the penalty-based ensemble method, extends it to the Navier-Stokes equations with random variables using Monte Carlo sampling, and validates the method's accuracy and efficiency with three numerical experiments.

翻译：流体流动的混沌特性及初始条件的不确定性限制了可预测性。初始条件中出现的微小误差会呈指数增长，直至达到$\mathcal{O}$(1)量级的饱和状态。集合预报通过对具有略微不同初始条件及其他数据的多次运行结果进行平均，以产生更精确的结果并延长可预测时限。然而，这类方法通常计算成本高昂。本文提出一种基于惩罚的集合方法，该方法采用共享系数矩阵以降低所需内存和计算成本，从而允许使用更大的集合规模。惩罚方法通过松弛不可压缩条件来解耦压力和速度，减少了内存需求。本报告给出了该惩罚集合方法的稳定性证明与误差估计，将其通过蒙特卡洛采样推广至含随机变量的Navier-Stokes方程，并通过三项数值实验验证了方法的准确性与效率。