TY - JOUR T1 - A Higher-Order Ensemble/Proper Orthogonal Decomposition Method for the Nonstationary Navier-Stokes Equations AU - Gunzburger , Max AU - Jiang , Nan AU - Schneier , Michael JO - International Journal of Numerical Analysis and Modeling VL - 4-5 SP - 608 EP - 627 PY - 2018 DA - 2018/04 SN - 15 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/12534.html KW - Navier-Stokes equations, ensemble computation, proper orthogonal decomposition, finite element methods. AB -

Partial differential equations (PDE) often involve parameters, such as viscosity or density. An analysis of the PDE may involve considering a large range of parameter values, as occurs in uncertainty quantification, control and optimization, inference, and several statistical techniques. The solution for even a single case may be quite expensive; whereas parallel computing may be applied, this reduces the total elapsed time but not the total computational effort. In the case of flows governed by the Navier-Stokes equations, a method has been devised for computing an ensemble of solutions. Recently, a reduced-order model derived from a proper orthogonal decomposition (POD) approach was incorporated into a first-order accurate in time version of the ensemble algorithm. In this work, we expand on that work by incorporating the POD reduced order model into a second-order accurate ensemble algorithm. Stability and convergence results for this method are updated to account for the POD/ROM approach. Numerical experiments illustrate the accuracy and efficiency of the new approach.