  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.
