An Ensemble-Proper Orthogonal Decomposition Method for the Nonstationary Navier-Stokes Equations


The definition of partial differential equation models usually involves a set of parameters whose values may vary over a wide range. The solution of even a single set of parameter values may be quite expensive. In many cases, e.g., optimization, control, uncertainty quantification, and other settings, solutions are needed for many sets of parameter values. We consider the case of the time-dependent Navier-Stokes equations for which a recently developed ensemble-based method allows for the efficient determination of the multiple solutions corresponding to many parameter sets. The method uses the average of the multiple solutions at any time step to define a linear set of equations that determines the solutions at the next time step. To significantly further reduce the costs of determining multiple solutions of the Navier-Stokes equations, we incorporate a proper orthogonal decomposition (POD) reduced-order model into the ensemble-based method. The stability and convergence results for the ensemble-based method are extended to the ensemble-POD approach.


Mathematics and Statistics

Keywords and Phrases

Principal component analysis; Structural analysis; Viscous flow; Ensemble methods; Partial differential equation models; Proper orthogonal decomposition method; Proper Orthogonal decompositions; Reduced order models; Stability and convergence; Time-dependent Navier-Stokes equations; Uncertainty quantifications; Navier Stokes equations; Navier-stokes equations; Proper orthogonal decomposition; Reduced-order models

International Standard Serial Number (ISSN)

0036-1429; 1095-7170

Document Type

Article - Journal

Document Version


File Type





© 2017 Society for Industrial and Applied Mathematics, All rights reserved.

Publication Date

01 Jan 2017