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.
M. Gunzburger et al., "A Higher-Order Ensemble/Proper Orthogonal Decomposition Method for the Nonstationary Navier-Stokes Equations," International Journal of Numerical Analysis and Modeling, vol. 15, no. 2019-04-05, pp. 608 - 627, University of Alberta, Jan 2018.
Mathematics and Statistics
Keywords and Phrases
Ensemble computation; Finite element methods; Navier-Stokes equations; Proper orthogonal decomposition
International Standard Serial Number (ISSN)
Article - Journal
01 Jan 2018