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

TL;DR

This paper introduces an ensemble-POD method for efficiently solving nonstationary Navier-Stokes equations across multiple parameter sets, significantly reducing computational costs while maintaining accuracy.

Contribution

It combines ensemble methods with proper orthogonal decomposition to enhance efficiency and extends stability and convergence analysis for this integrated approach.

Findings

The ensemble-POD method achieves accurate solutions with reduced computational effort.

Numerical experiments demonstrate the method's efficiency and stability.

The approach effectively handles multiple parameter sets in Navier-Stokes simulations.

Abstract

The definition of partial differential equation (PDE) 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. Numerical experiments are provided that illustrate the accuracy and efficiency of computations determined using the new approach.