Error analysis of proper orthogonal decomposition data assimilation schemes for the Navier-Stokes equations

TL;DR

This paper analyzes the error bounds of a stabilized POD data assimilation scheme for Navier-Stokes equations, demonstrating improved accuracy and convergence with large nudging parameters, especially at low viscosities.

Contribution

It introduces a grad-div stabilized POD data assimilation method with error bounds independent of viscosity, and shows its effectiveness through numerical experiments.

Findings

Error bounds independent of inverse viscosity powers.

Rapid convergence for large nudging parameters.

Enhanced accuracy at low viscosities.

Abstract

The error analysis of a proper orthogonal decomposition (POD) data assimilation (DA) scheme for the Navier-Stokes equations is carried out. A grad-div stabilization term is added to the formulation of the POD method. Error bounds with constants independent on inverse powers of the viscosity parameter are derived for the POD algorithm. No upper bounds in the nudging parameter of the data assimilation method are required. Numerical experiments show that, for large values of the nudging parameter, the proposed method rapidly converges to the real solution, and greatly improves the overall accuracy of standard POD schemes up to low viscosities over predictive time intervals.