A higher-order ensemble/proper orthogonal decomposition method for the nonstationary Navier-Stokes equations
This work provides a more accurate reduced-order modeling technique for ensemble simulations of Navier-Stokes flows, benefiting applications in uncertainty quantification and optimization where multiple parameter values are considered.
The authors extend a proper orthogonal decomposition (POD) reduced-order model to a second-order accurate ensemble algorithm for the nonstationary Navier-Stokes equations, achieving improved accuracy and efficiency over the first-order version. Numerical experiments demonstrate the method's effectiveness.
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.