Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations

TL;DR

This paper analyzes a continuous data assimilation algorithm for 2D Navier-Stokes equations using a velocity-vorticity formulation, demonstrating stability, accuracy, and effectiveness through theoretical proofs and numerical tests.

Contribution

It provides the first rigorous analysis of CDA applied to a velocity-vorticity formulation, showing stability and optimal accuracy with different nudging strategies.

Findings

CDA preserves long-term stability of the velocity-vorticity method.

Nudging both velocity and vorticity accelerates convergence.

Numerical tests confirm theoretical results and effectiveness in channel flow simulation.

Abstract

We study a continuous data assimilation (CDA) algorithm for a velocity-vorticity formulation of the 2D Navier-Stokes equations in two cases: nudging applied to the velocity and vorticity, and nudging applied to the velocity only. We prove that under a typical finite element spatial discretization and backward Euler temporal discretization, application of CDA preserves the unconditional long-time stability property of the velocity-vorticity method and provides optimal long-time accuracy. These properties hold if nudging is applied only to the velocity, and if nudging is also applied to the vorticity then the optimal long-time accuracy is achieved more rapidly in time. Numerical tests illustrate the theory, and show its effectiveness on an application problem of channel flow past a flat plate.