Numerical analysis of penalty-based ensemble methods

The inherent chaos in fluid flow and the uncertainties in initial conditions restrict the ability to make accurate predictions. Small errors that occur in the initial conditions can grow exponentially until they saturate at O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}$$\end{document}(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, allowing 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 for the Navier-Stokes equations. In addition, we extend the method from deterministic Navier-Stokes equations to Navier-Stokes equations with random variables using Monte Carlo sampling. We validate the method's accuracy and efficiency with three numerical experiments.