Convergence Analysis of a Fully Discrete Observer For Data Assimilation of the Barotropic Euler Equations
This work addresses data assimilation for quasilinear hyperbolic systems, offering a theoretical foundation for long-time simulations, though it is incremental as it extends existing observer methods to a discrete setting.
The paper tackled the problem of data assimilation for the barotropic Euler equations by analyzing a discrete Luenberger observer, showing an error bound with exponentially decaying initial error and parts proportional to grid sizes and measurement errors, providing the first error estimate for such a system.
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.