An Efficient Second-Order-in-Time Penalty-Projection Ensemble Eddy Viscosity Method for Parameterized Navier-Stokes Flows

We propose a novel, robust, and second-order-accurate parameterized penalty-projection ensemble algorithm for incompressible Navier--Stokes flow problems. The resulting linearized algorithm, based on the second-order Backward Differentiation Formula (BDF-2), is computationally efficient because it shares the same coefficient matrix across all realizations for each subproblem at every time step. To enhance robustness in convection-dominated flows, the scheme incorporates Ensemble Eddy Viscosity (EEV) regularization. In addition, it is equipped with grad-div stabilization parameter $γ$, which controls the splitting error; under the assumptions of the analysis, the splitting error decreases and vanishes asymptotically as $γ\to\infty$. We establish the stability of the proposed scheme and rigorously prove its optimal convergence by demonstrating that, as $γ\to\infty$, the scheme converges to an equivalent coupled formulation. We further validate the method through a series of numerical experiments designed to verify the theoretically predicted convergence rates and evaluate its performance on benchmark convection-dominated problems. The numerical results are in excellent agreement with the theoretical analysis and confirm the effectiveness of the proposed scheme.