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

TL;DR

This paper proves the convergence and error bounds of a fully discrete observer for the 1D barotropic Euler equations, ensuring long-term accuracy in data assimilation with only velocity measurements.

Contribution

It provides the first error estimate for a discrete observer applied to a quasilinear hyperbolic system, demonstrating uniform-in-time accuracy.

Findings

Error bound includes exponential decay, initial difference, grid size, and measurement error contributions.

Constants in error bounds are independent of time and grid sizes.

First such error estimate for a fully discrete observer in hyperbolic PDEs.

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.