Penalty Ensembles for Navier-Stokes with Random Initial Conditions and Forcing
Rui Fang
TL;DR
This work addresses uncertainty-driven Navier–Stokes simulations by introducing a penalty-based ensemble method that eliminates pressure to reduce memory consumption and enables larger ensemble sizes via a shared coefficient matrix. The approach uses a penalty NSE with $p^\epsilon = -\frac{1}{\epsilon} \nabla \cdot u^\epsilon$ and a time-stepping scheme that yields a common system across ensemble members, with a CFL-type condition ensuring stability. The authors provide stability and convergence analysis, proving an optimal error bound of $O((\epsilon+\Delta t+h^m)^2)$ under regularity assumptions, and validate the method numerically with convergence tests (Green–Taylor vortex) and stability tests (rotating disk), demonstrating improved predictability horizons from larger ensembles. Overall, the penalty-ensemble framework offers a memory-efficient, scalable tool for reliable NSE simulations under data uncertainty, with open directions including higher Reynolds numbers and adaptive penalty strategies.
Abstract
In many applications, uncertainty in problem data leads to the need for numerous computationally expensive simulations. This report addresses this challenge by developing a penalty-based ensemble algorithm. Building upon Jiang and Layton's work on ensemble algorithms that use a shared coefficient matrix, this report introduces the combination of penalty methods to enhance its capabilities. Penalty methods uncouple velocity and pressure by relaxing the incompressibility condition. Eliminating the pressure results in a system that requires less memory. The reduction in memory allows for larger ensemble sizes, which give more information about the flow and can be used to extend the predictability horizon.