Continuous-data-assimilation-enabled fast and robust convergence of an Uzawa-based solver for Navier-Stokes equations with large Reynolds number

TL;DR

This paper introduces a novel data assimilation-enhanced Uzawa solver for Navier-Stokes equations that accelerates convergence and remains robust at high Reynolds numbers, supported by rigorous proofs and numerical experiments.

Contribution

It develops and analyzes a CDA-Uzawa method that provably accelerates convergence and handles large Reynolds numbers in Navier-Stokes simulations, extending the applicability of Uzawa-based solvers.

Findings

CDA-Uzawa accelerates convergence of the Uzawa solver.

More partial data yields greater acceleration.

The method remains effective at high Reynolds numbers, even with noisy data.

Abstract

This paper shows how continuous data assimilation (CDA) can be used to provably enable and accelerate convergence of a (efficient at each iteration due to a physics-splitting, but generally slowly converging and not robust) nonlinear solver for incompressible Navier-Stokes equations (NSE). Herein we develop, analyze and test an Uzawa-based nonlinear solver for incompressible NSE that incorporates partial solution data into the iteration through continuous data assimilation (CDA-Uzawa). We rigorously prove that i) CDA-Uzawa will accelerate a converging Uzawa iteration, and more partial solution data yields more acceleration, and ii) with enough partial solution data CDA-Uzawa will converge for arbitrarily large Reynolds numbers, even if multiple NSE solutions exist. In the case of noisy data, we prove that the convergence results hold down to the size of the noise, and we propose a strategy to pass to Newton once CDA-Uzawa convergence reaches its lower limit. Results of several numerical tests illustrate the theory and show CDA-Uzawa is a very effective and efficient solver. While this paper focuses a particular splitting-based solver for the NSE, the key ideas are quite general and extendable to a wide class of nonlinear solvers.