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Enhancing nonlinear solvers for the Navier-Stokes equations with continuous (noisy) data assimilation

Bosco Garcia-Archilla, Xuejian Li, Julia Novo, Leo Rebholz

TL;DR

The paper addresses solving steady incompressible Navier–Stokes equations with partial measurement data by integrating a continuous data assimilation nudging term into the Picard solver (CDA-Picard). It develops an $L^2$-norm convergence analysis and extends the framework to noisy data, proving stability and linear convergence of the iteration residuals while showing the $L^2$-error is ultimately limited by the noise level. Numerical experiments in 2D and 3D lid‑driven cavity problems confirm linear residual decay and error floors at the noise level, and demonstrate that using CDA-Picard iterates to initialize Newton markedly improves convergence for challenging cases. The results offer a practical, data-robust approach to accelerate nonlinear NSE solvers and suggest a hybrid CDA-Picard–Newton strategy for real‑world noisy measurements. Potential future directions include extending to multiphysics settings and reducing required data while maintaining robustness.

Abstract

We consider nonlinear solvers for the incompressible, steady (or at a fixed time step for unsteady) Navier-Stokes equations in the setting where partial measurement data of the solution is available. The measurement data is incorporated/assimilated into the solution through a nudging term addition to the the Picard iteration that penalized the difference between the coarse mesh interpolants of the true solution and solver solution, analogous to how continuous data assimilation (CDA) is implemented for time dependent PDEs. This was considered in the paper [Li et al. {\it CMAME} 2023], and we extend the methodology by improving the analysis to be in the $L^2$ norm instead of a weighted $H^1$ norm where the weight depended on the coarse mesh width, and to the case of noisy measurement data. For noisy measurement data, we prove that the CDA-Picard method is stable and convergent, up to the size of the noise. Numerical tests illustrate the results, and show that a very good strategy when using noisy data is to use CDA-Picard to generate an initial guess for the classical Newton iteration.

Enhancing nonlinear solvers for the Navier-Stokes equations with continuous (noisy) data assimilation

TL;DR

The paper addresses solving steady incompressible Navier–Stokes equations with partial measurement data by integrating a continuous data assimilation nudging term into the Picard solver (CDA-Picard). It develops an -norm convergence analysis and extends the framework to noisy data, proving stability and linear convergence of the iteration residuals while showing the -error is ultimately limited by the noise level. Numerical experiments in 2D and 3D lid‑driven cavity problems confirm linear residual decay and error floors at the noise level, and demonstrate that using CDA-Picard iterates to initialize Newton markedly improves convergence for challenging cases. The results offer a practical, data-robust approach to accelerate nonlinear NSE solvers and suggest a hybrid CDA-Picard–Newton strategy for real‑world noisy measurements. Potential future directions include extending to multiphysics settings and reducing required data while maintaining robustness.

Abstract

We consider nonlinear solvers for the incompressible, steady (or at a fixed time step for unsteady) Navier-Stokes equations in the setting where partial measurement data of the solution is available. The measurement data is incorporated/assimilated into the solution through a nudging term addition to the the Picard iteration that penalized the difference between the coarse mesh interpolants of the true solution and solver solution, analogous to how continuous data assimilation (CDA) is implemented for time dependent PDEs. This was considered in the paper [Li et al. {\it CMAME} 2023], and we extend the methodology by improving the analysis to be in the norm instead of a weighted norm where the weight depended on the coarse mesh width, and to the case of noisy measurement data. For noisy measurement data, we prove that the CDA-Picard method is stable and convergent, up to the size of the noise. Numerical tests illustrate the results, and show that a very good strategy when using noisy data is to use CDA-Picard to generate an initial guess for the classical Newton iteration.
Paper Structure (18 sections, 10 theorems, 84 equations, 8 figures)

This paper contains 18 sections, 10 theorems, 84 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Steady NSE preliminaries
  4. Picard and Newton iterations
  5. CDA and CDA-Picard
  6. An improved analysis for the CDA-Picard iteration
  7. CDA-Picard with noisy data
  8. Stability
  9. Convergence of the CDA-Picard with noisy data iteration residuals
  10. Error analysis of the CDA-Picard with noisy data iteration
  11. Numerical Experiments
  12. 2D driven cavity with varying signal to noise ratio
  13. 3D driven cavity with varying signal to noise ratio
  14. Newton with CDA-Picard generated initial guess
  15. Conclusions and future directions
  16. ...and 3 more sections

Key Result

Lemma 2.1

Let $\alpha=M\nu^{-2}\|f\|_{-1}$. For any $f\in H^{-1}$ and $\nu>0$, solutions to (wd) exist and satisfy Furthermore, if $\alpha<1$, the solution is unique.

Figures (8)

  • Figure 1: The plot above shows streamlines of the solution of the 2D driven cavity problem at $Re=$3,000 (left) and 10,000 (right).
  • Figure 2: Shown above are the error and residual plots for $Re$=3000 driven cavity tests with varying snr.
  • Figure 3: Shown above are the error and residual plots for $Re$=10000 driven cavity tests with varying snr.
  • Figure 4: Shown above are the midsliceplane plots of solutions for the 3D driven cavity simulations at $Re$=1000.
  • Figure 5: Shown above is the method used to split a rectangular box into 6 tetrahedra.
  • ...and 3 more figures

Theorems & Definitions (21)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.1
  • Theorem 3.1
  • Remark 3.1
  • proof
  • Lemma 4.1: Stability result 1
  • proof
  • Lemma 4.2: Stability result 2
  • proof
  • ...and 11 more