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Convergence Analysis of a Fully Discrete Observer For Data Assimilation of the Barotropic Euler Equations

Aidan Chaumet, Jan Giesselmann

Abstract

We study the convergence of a discrete Luenberger observer for the barotropic Euler equations in one dimension, for measurements of the velocity only. We use a mixed finite element method in space and implicit Euler integration in time. We use a modified relative energy technique to show an error bound comparing the discrete observer to the original system's solution. The bound is the sum of three parts: an exponentially decaying part, proportional to the difference in initial value, a part proportional to the grid sizes in space and time and a part that is proportional to the size of the measurement errors as well as the nudging parameter. The proportionality constants of the second and third parts are independent of time and grid sizes. To the best of our knowledge, this provides the first error estimate for a discrete observer for a quasilinear hyperbolic system, and implies uniform-in-time accuracy of the discrete observer for long-time simulations.

Convergence Analysis of a Fully Discrete Observer For Data Assimilation of the Barotropic Euler Equations

Abstract

We study the convergence of a discrete Luenberger observer for the barotropic Euler equations in one dimension, for measurements of the velocity only. We use a mixed finite element method in space and implicit Euler integration in time. We use a modified relative energy technique to show an error bound comparing the discrete observer to the original system's solution. The bound is the sum of three parts: an exponentially decaying part, proportional to the difference in initial value, a part proportional to the grid sizes in space and time and a part that is proportional to the size of the measurement errors as well as the nudging parameter. The proportionality constants of the second and third parts are independent of time and grid sizes. To the best of our knowledge, this provides the first error estimate for a discrete observer for a quasilinear hyperbolic system, and implies uniform-in-time accuracy of the discrete observer for long-time simulations.
Paper Structure (14 sections, 24 theorems, 151 equations, 2 figures)

This paper contains 14 sections, 24 theorems, 151 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Outline and Summary of Previous Results
  4. Convergence of the discrete observer.
  5. Additional assumptions
  6. Projection errors.
  7. Time derivative of $\mathop{\mathrm{\symcal{H}}}\nolimits({u}_\mathrm{h},\widehat{u}_\mathrm{h})$.
  8. Modified relative energy.
  9. Uniform Convergence Result
  10. Numerical Experiments
  11. Implementation details
  12. Exponential decay and convergence order
  13. Influence of the nudging parameter
  14. Discussion

Key Result

Theorem 1

(Convergence, c.f. kunkel_obs) Under the assumptions ass:unif -- ass:smallderivs, for any $\mu > 0$, the constants $C_t$ and $\overbar v$ can be chosen small enough, such that there exist constants $C_1,C_2>0$, such that for any $t \in [0,\infty)$, it holds that with ${u} = (\rho,v)$ and $\widehat{u} = (\widehat{\rho},\widehat{v})$.

Figures (2)

  • Figure 1: Absolute $L^2$ error over time for $\mu = 1$. The error is normalized such that $||{u}_\mathrm{h}^0 - \widehat{u}^0||_2 = 1$ at $t=0$.
  • Figure 2: Comparison of the convergence speed for different values of $\mu$. Overly large $\mu$ can lead to slow convergence.

Theorems & Definitions (48)

  • Theorem 1
  • Corollary \ref{col:unif_conv}
  • Lemma \ref{col:unif_conv}: Norm equivalence
  • Lemma \ref{col:unif_conv}: Projection Errors, c.f. egger_ap_22
  • Lemma \ref{col:unif_conv}
  • Remark \ref{col:unif_conv}
  • proof : Proof of Lemma \ref{['lemma:dt_proj_estimates']}
  • Definition \ref{col:unif_conv}: Intermediate values
  • Lemma \ref{col:unif_conv}: Discrete time derivative of the relative energy
  • proof
  • ...and 38 more