A Family of Second‐Order Regularized Ensemble Algorithms for Incompressible Flow Simulations

In this report, we present a family of regularized ensemble methods for fast simulation of the Navier‐Stokes flow ensembles. The ensemble algorithms are constructed so that all ensemble members share the same coefficient matrix after spatial discretization to facilitate the use of efficient direct or iterative linear solvers for solving the corresponding linear systems. Although this idea has been tested in many papers in recent years and shown to be very effective in reducing the computational cost and storage for computing a large ensemble of flow data, an outstanding issue for computing nonlinear flows is that the fluctuation‐induced timestep condition deteriorates fast as the Reynolds number increases, and the time step needs to be taken small enough for the method to stay stable. To help relax this timestep condition, we propose a family of regularized ensemble methods that are second order in time, featuring a regularization term that is proportional to the discrete curvature of the solution. We prove the long‐time stability and convergence of the methods under a timestep condition. Extensive numerical tests are presented to confirm theoretical results and demonstrate the advantages of using this curvature regularization that mitigates the severity of the timestep condition.