Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Data assimilation with model errors

Aytekin Çibik, Rui Fang, William Layton, Farjana Siddiqua

TL;DR

This work analyzes nudging data assimilation for Navier–Stokes equations with an omitted model-error term $\omega R(u)$, showing that the error between the true flow and the nudged solution contains an exponentially decaying component and a model-error contribution that decays as $O(χ^{-1/2})$ as the nudging parameter $χ$ grows. The authors provide a rigorous continuous and discrete analysis, including stability results when $R(u)$ represents a Coriolis force, and derive spatial discretization error bounds under a discrete analog of the continuous assumptions. Numerical experiments validate the theoretical rates, demonstrating second-order convergence in time and the effectiveness of nudging to compensate for rotation-related model errors in a double-pane window natural-convection benchmark. The results support the practical utility of nudging for mitigating model errors in geophysical and engineering flows and inform parameter choices for stable and accurate assimilation.

Abstract

Nudging is a data assimilation method amenable to both analysis and implementation. It also has the (reported) advantage of being insensitive to model errors compared to other assimilation methods. However, nudging behavior in the presence of model errors is little analyzed. This report gives an analysis of nudging to correct model errors. The analysis indicates that the error contribution due to the model error decays as the nudging parameter $χ\to \infty$ like $\mathcal{O}(χ^{-\frac{1}{2}})$, Theorem 3.2. Numerical tests verify the predicted convergence rates and validate the nudging correction to model errors.

Data assimilation with model errors

TL;DR

This work analyzes nudging data assimilation for Navier–Stokes equations with an omitted model-error term , showing that the error between the true flow and the nudged solution contains an exponentially decaying component and a model-error contribution that decays as as the nudging parameter grows. The authors provide a rigorous continuous and discrete analysis, including stability results when represents a Coriolis force, and derive spatial discretization error bounds under a discrete analog of the continuous assumptions. Numerical experiments validate the theoretical rates, demonstrating second-order convergence in time and the effectiveness of nudging to compensate for rotation-related model errors in a double-pane window natural-convection benchmark. The results support the practical utility of nudging for mitigating model errors in geophysical and engineering flows and inform parameter choices for stable and accurate assimilation.

Abstract

Nudging is a data assimilation method amenable to both analysis and implementation. It also has the (reported) advantage of being insensitive to model errors compared to other assimilation methods. However, nudging behavior in the presence of model errors is little analyzed. This report gives an analysis of nudging to correct model errors. The analysis indicates that the error contribution due to the model error decays as the nudging parameter like , Theorem 3.2. Numerical tests verify the predicted convergence rates and validate the nudging correction to model errors.

Paper Structure

This paper contains 13 sections, 8 theorems, 84 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Notation and Preliminaries
  4. Correction of model error
  5. The discrete problems
  6. Stability when $R(u)$ is a Coriolis force
  7. Error estimates for spatial discretization when $R(u)$ is a Coriolis force
  8. Numerical Experiments
  9. Convergence rate study
  10. Double Pane Window Problem
  11. Case I: $Ra=10^4$
  12. Case II: $Ra=10^5$
  13. Conclusion

Key Result

Lemma 2.1

(The Ladyzhenskaya Inequalities, Ladyzhenskaya ladyzhenskaya1969mathematical) For any vector function $u:{{\mathbb{R}}}^{d}\rightarrow {{\mathbb{R}}}^{d}$ with compact support and with the indicated $L^{p}$ norms finite,

Figures (13)

  • Figure 1: $Ra=10^4$ Solution of natural convection without Coriolis.
  • Figure 2: $Ra=10^4$ Solution of natural convection with Coriolis.
  • Figure 3: $Ra=10^4$ Data assimilation solution for $\chi =1$.
  • Figure 4: $Ra=10^4$ Data assimilation solution for $\chi =10^2$.
  • Figure 5: $Ra=10^4$ Data assimilation solution for $\chi =10^4$.
  • ...and 8 more figures

Theorems & Definitions (14)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Remark 3.1
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 4 more