A Second-Order Time-Stepping Scheme for Simulating Ensembles of Parameterized Flow Problems


We consider settings for which one needs to perform multiple flow simulations based on the Navier-Stokes equations, each having different initial condition data, boundary condition data, forcing functions, and/or coefficients such as the viscosity. For such settings, we propose a second-order time accurate ensemble-based method that to simulate the whole set of solutions, requires, at each time step, the solution of only a single linear system with multiple right-hand-side vectors. Rigorous analyses are given proving the conditional stability and establishing error estimates for the proposed algorithm. Numerical experiments are provided that illustrate the analyses.


Mathematics and Statistics


This research was partially supported by the U.S. Department of Energy under grants DE-SC0009324 and DE-SC0016540, the U.S. Air Force Office of Scientific Research grant FA9550-15-1-0001, a Defense Advanced Projects Agency contract administered under the Oak Ridge National Laboratory subcontract 4000145366, the U.S. National Science Foundation grants DMS-1522672 and DMS-1720001, and a University of Missouri Research Board grant.

Keywords and Phrases

Linear systems; Viscous flow; Conditional stability; Ensemble methods; Ensemble-based method; Initial conditions; Multiple right-hand sides; Numerical experiments; Parameterized; Time-stepping schemes; Navier Stokes equations; Navier-Stokes Equations; Parameterized Flow

International Standard Serial Number (ISSN)

1609-4840; 1609-9389

Document Type

Article - Journal

Document Version


File Type





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Publication Date

01 Jul 2019