Numerical Analysis of Penalty-based Ensemble Methods

TL;DR

This paper introduces a penalty-based ensemble method for fluid flow prediction that reduces computational costs and memory requirements, enabling larger ensembles and improved forecast accuracy.

Contribution

It develops a novel penalty-based ensemble approach with stability proof, error estimates, and extension to Navier-Stokes equations with stochastic variables.

Findings

Method reduces computational cost and memory usage.

Numerical experiments validate accuracy and efficiency.

Extended to stochastic Navier-Stokes equations.

Abstract

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.